the-eye.eu Teo Mora - [EMAvol0… · ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS FOUNDING EDITOR G.-C. ROTA Editorial Board P. Flajolet, M. Ismail, E. Lutwak 40 N. White (ed.)

www.cambridge.org/9780521811569

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

FOUNDED BY G.-C. ROTA

Editorial Board

P. Flajolet, M. Ismail, E. Lutwak

Volume 99

Solving Polynomial Equation Systems II


FOUNDING EDITOR G.-C. ROTAEditorial BoardP. Flajolet, M. Ismail, E. Lutwak

40 N. White (ed.) Matroid Applications41 S. Sakai Operator Algebras in Dynamical Systems42 W. Hodges Basic Model Theory43 H. Stahl and V. Totik General Orthogonal Polynomials45 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions46 A Bjorner et al. Oriented Matroids47 G. Edgar and L. Sucheston Stopping Times and Directed Processes48 C. Sims Computation with Finitely Presented Groups49 T. Palmer Banach Algebras and the General Theory of *-Algebras I50 F. Borceux Handbook of Categorical Algebra I51 F. Borceux Handbook of Categorical Algebra II52 F. Borceux Handbook of Categorical Algebra III53 V. F. Kolchin Random Graphs54 A. Katok and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems55 V. N. Sachkov Combinatorial Methods in Discrete Mathematics56 V. N. Sachkov Probabilistic Methods in Discrete Mathematics57 P. M. Cohn Skew Fields58 R. Gardner Geometric Tomography59 G. A. Baker, Jr., and P. Graves-Morris Pade Approximants, 2nd edn60 J. Krajicek Bounded Arithmetic, Propositional Logic, and Complexity Theory61 H. Groemer Geometric Applications of Fourier Series and Spherical Harmonics62 H. O. Fattorini Infinite Dimensional Optimization and Control Theory63 A. C. Thompson Minkowski Geometry64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications65 K. Engel Sperner Theory66 D. Cvetkovic, P. Rowlinson and S. Simic Eigenspaces of Graphs67 F. Bergeron, G. Labelle and P. Leroux Combinatorial Species and Tree-Like Structures68 R. Goodman and N. Wallach Representations and Invariants of the Classical Groups69 T. Beth, D. Jungnickel, and H. Lenz Design Theory I, 2nd edn70 A. Pietsch and J. Wenzel Orthonormal Systems for Banach Space Geometry71 G. E. Andrews, R. Askey and R. Roy Special Functions72 R. Ticciati Quantum Field Theory for Mathematicians73 M. Stern Semimodular Lattices74 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations I75 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations II76 A. A. Ivanov Geometry of Sporadic Groups I77 A. Schinzel Polymomials with Special Regard to Reducibility78 H. Lenz, T. Beth, and D. Jungnickel Design Theory II, 2nd edn79 T. Palmer Banach Algebras and the General Theory of *-Algebras II80 O. Stormark Lie’s Structural Approach to PDE Systems81 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables82 J. P. Mayberry The Foundations of Mathematics in the Theory of Sets83 C. Foias et al. Navier–Stokes Equations and Turbulence84 B. Polster and G. Steinke Geometries on Surfaces85 R. B. Paris and D. Kaminski Asymptotics and Mellin–Barnes Integrals86 R. McEliece The Theory of Information and Coding, 2nd edn87 B. Magurn Algebraic Introduction to K-Theory88 T. Mora Solving Polynomial Equation Systems I89 K. Bichteler Stochastic Integration with Jumps90 M. Lothaire Algebraic Combinatorics on Words91 A. A. Ivanov and S. V. Shpectorov Geometry of Sporadic Groups II92 P. McMullen and E. Schulte Abstract Regular Polytopes93 G. Gierz et al. Continuous Lattices and Domains94 S. Finch Mathematical Constants95 Y. Jabri The Mountain Pass Theorem


Solving Polynomial Equation Systems II

Macaulay’s Paradigm and Grobner Technology

TEO MORA

University of Genoa

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

Information on this title: www.cambridge.org/9780521811569

© Cambridge University Press 2005

This publication is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2005

A catalogue record for this publication is available from the British Library

ISBN 978-0-521-81156-9 hardback

Transferred to digital printing 2008

Cambridge University Press has no responsibility for the persistence or

accuracy of URLs for external or third-party Internet websites referred to in

this publication, and does not guarantee that any content on such websites is,

or will remain, accurate or appropriate.

In the beginning was the Word, and the Word was with God, and the Word was God.St John (Authorized Version)

God bless the girl who refuses to study algebra. It is a study that has caused many a girlto lose her soul.Superintendent Francis of the Los Angeles schools.

The present state of our knowledge of the properties of Modular Systems is chiefly dueto the fundamental theorems and processes of L. Kronecker, M. Noether, D. Hilbert, andE. Lasker, and above all to J. Konig’s profound exposition and numerous extensions ofKronecker’s theory. Konig’s treatise might be regarded as in some measure complete ifit were admitted that a problem is finished with when its solution has been reduced toa finite number of feasible operations. If however the operations are too numerous ortoo involved to be carried out in practice the solution is only a theoretical one; and itsimportance then lies not in itself, but in the theorems with which it is associated and towhich it leads. Such a theoretical solution must be regarded as a preliminary and notthe final stage in the consideration of the problem.F. S. Macaulay, The Algebraic Theory of Modular Systems

Gauss is the perfect representative of the Thaurus mathematicians. Their style consistsin performing long and numerous computations until this allows them to guess a con-jecture, usually a correct one.Theodyl Magus, Astrology and Mathematics

Contents

Preface page xiSetting xiv

Part three: Gauss, Euclid, Buchberger: ElementaryGrobner Bases 1

20 Hilbert 320.1 Affine Algebraic Varieties and Ideals 320.2 Linear Change of Coordinates 820.3 Hilbert’s Nullstellensatz 1020.4 *Kronecker Solver 1520.5 Projective Varieties and Homogeneous Ideals 2220.6 *Syzygies and Hilbert Function 2820.7 *More on the Hilbert Function 3420.8 Hilbert’s and Gordan’s Basissatze 36

21 Gauss II 4621.1 Some Heretical Notation 4721.2 Gaussian Reduction 5121.3 Gaussian Reduction and Euclidean Algorithm Revisited 63

22 Buchberger 7222.1 From Gauss to Grobner 7522.2 Grobner Basis 7822.3 Toward Buchberger’s Algorithm 8322.4 Buchberger’s Algorithm (1) 9622.5 Buchberger’s Criteria 9822.6 Buchberger’s Algorithm (2) 104

23 Macaulay I 10923.1 Homogenization and Affinization 11023.2 H-bases 114

vi

Contents vii

23.3 Macaulay’s Lemma 11923.4 Resolution and Hilbert Function for Monomial

Ideals 12223.5 Hilbert Function Computation: the

‘Divide-and-Conquer’ Algorithms 13623.6 H-bases and Grobner Bases for Modules 13823.7 Lifting Theorem 14223.8 Computing Resolutions 14623.9 Macaulay’s Nullstellensatz Bound 15223.10 *Bounds for the Degree in the Nullstellensatz 156

24 Grobner I 17024.1 Rewriting Rules 17324.2 Grobner Bases and Rewriting Rules 18324.3 Grobner Bases for Modules 18824.4 Grobner Bases in Graded Rings 19524.5 Standard Bases and the Lifting Theorem 19824.6 Hironaka’s Standard Bases and Valuations 20324.7 *Standard Bases and Quotients Rings 21824.8 *Characterization of Standard Bases in

Valuation Rings 22324.9 Term Ordering: Classification and

Representation 23424.10 *Grobner Bases and the State Polytope 247

25 Gebauer and Traverso 25525.1 Gebauer–Moller and Useless Pairs 25525.2 Buchberger’s Algorithm (3) 26425.3 Traverso’s Choice 27125.4 Gebauer–Moller’s Staggered Linear Bases and

Faugere’s F5 27426 Spear 289

26.1 Zacharias Rings 29126.2 Lexicographical Term Ordering and Elimination Ideals 30026.3 Ideal Theoretical Operation 30426.4 *Multivariate Chinese Remainder Algorithm 31326.5 Tag-Variable Technique and Its Application to

Subalgebras 31626.6 Caboara–Traverso Module Representation 32126.7 *Caboara Algorithm for Homogeneous

Minimal Resolutions 329

viii Contents

Part four: Duality 33327 Noether 335

27.1 Noetherian Rings 33727.2 Prime, Primary, Radical, Maximal Ideals 34027.3 Lasker–Noether Decomposition: Existence 34527.4 Lasker–Noether Decomposition: Uniqueness 35027.5 Contraction and Extension 35627.6 Decomposition of Homogeneous Ideals 36427.7 *The Closure of an Ideal at the Origin 36827.8 Generic System of Coordinates 37127.9 Ideals in Noether Position 37427.10 *Chains of Prime Ideals 37827.11 Dimension 38027.12 Zero-dimensional Ideals and Multiplicity 38427.13 Unmixed Ideals 390

28 Moller I 39328.1 Duality 39328.2 Moller Algorithm 401

29 Lazard 41429.1 The FGLM Problem 41529.2 The FGLM Algorithm 41829.3 Border Bases and Grobner Representation 42629.4 Improving Moller’s Algorithm 43229.5 Hilbert Driven and Grobner Walk 44029.6 *The Structure of the Canonical Module 444

30 Macaulay II 45130.1 The Linear Structure of an Ideal 45230.2 Inverse System 45630.3 Representing and Computing the Linear

Structure of an Ideal 46130.4 Noetherian Equations 46630.5 Dialytic Arrays of M(r) and Perfect Ideals 47830.6 Multiplicity of Primary Ideals 49230.7 The Structure of Primary Ideals at the Origin 494

31 Grobner II 50031.1 Noetherian Equations 50131.2 Stability 50231.3 Grobner Duality 50431.4 Leibniz Formula 50831.5 Differential Inverse Functions at the Origin 50931.6 Taylor Formula and Grobner Duality 512

Contents ix

32 Grobner III 51732.1 Macaulay Bases 51832.2 Macaulay Basis and Grobner Representation 52132.3 Macaulay Basis and Decomposition of Primary Ideals 52232.4 Horner Representation of Macaulay Bases 52732.5 Polynomial Evaluation at Macaulay Bases 53132.6 Continuations 53332.7 Computing a Macaulay Basis 542

33 Moller II 54933.1 Macaulay’s Trick 55033.2 The Cerlienco–Mureddu Correspondence 55433.3 Lazard Structural Theorem 56033.4 Some Factorization Results 56233.5 Some Examples 56933.6 An Algorithmic Proof 574

Part five: Beyond Dimension Zero 58334 Grobner IV 585

34.1 Nulldimensionalen Basissatze 58634.2 Primitive Elements and Allgemeine Basissatz 59334.3 Higher-Dimensional Primbasissatz 59834.4 Ideals in Allgemeine Positions 60134.5 Solving 60534.6 Gianni–Kalkbrener Theorem 608

35 Gianni–Trager–Zacharias 61435.1 Decomposition Algorithms 61535.2 Zero-dimensional Decomposition Algorithms 61635.3 The GTZ Scheme 62235.4 Higher-dimensional Decomposition Algorithms 63135.5 Decomposition Algorithms for Allgemeine Ideals 634

35.5.1 Zero-dimensional Allgemeine Ideals 63435.5.2 Higher-dimensional Allgemeine Ideals 637

35.6 Sparse Change of Coordinates 64035.6.1 Gianni’s Local Change of Coordinates 64135.6.2 Giusti–Heintz Coordinates 645

35.7 Linear Algebra and Change of Coordinates 65035.8 Direct Methods for Radical Computation 65435.9 Caboara–Conti–Traverso Decomposition

Algorithm 65835.10 Squarefree Decomposition of a

Zero-dimensional Ideal 660

x Contents

36 Macaulay III 66536.1 Hilbert Function and Complete Intersections 66636.2 The Coefficients of the Hilbert Function 67036.3 Perfectness 678

37 Galligo 68637.1 Galligo Theorem (1): Existence of Generic Escalier 68637.2 Borel Relation 69737.3 *Galligo Theorem (2): the Generic Initial Ideal

is Borel Invariant 70637.4 *Galligo Theorem (3): the Structure of the

Generic Escalier 71037.5 Eliahou–Kervaire Resolution 714

38 Giusti 72538.1 The Complexity of an Ideal 72638.2 Toward Giusti’s Bound 72838.3 Giusti’s Bound 73338.4 Mayr and Meyer’s Example 73538.5 Optimality of Revlex 741

Bibliography 749Index 758

Preface

If you HOPE that this second SPES volume preserves the style of the previ-ous volume, you will not be disappointed: in fact it maintains a self-containedapproach using only undergraduate mathematics in this introduction to ele-mentary commutative ideal theory and to its computational aspects,1 while myhorror vacui compelled me to report nearly all the relevant results in computa-tional algebraic geometry that I know about.

When the commutative algebra community was exposed, in 1979, to Buch-berger’s theory and algorithm (dated 1965) of Grobner bases2, the more alertresearchers, mainly Schreyer and Bayer, immediately realized that this injec-tion of Grobner technology was all one needed to make effective Macaulay’sparadigm for reducing computational problems for ideals either to the cor-responding combinatorial problem for monomials3 or to a more elementarylinear algebraic computation.4 This realization gave to researchers a straight-forward approach which led them, within more or less fifteen years, to com-pletely effectivize commutative ideal theory.

This second volume of SPES is an eyewitness report on this successful in-troduction of effective methods to algebraic geometry.

Part three, Gauss, Euclid, Buchberger: Elementary Grobner Bases, introducesat the same time Buchberger’s theory of Grobner bases, his algorithm for com-puting them and Macaulay’s paradigm.

While I will discuss in depth both of the classical main approaches to theintroduction of Grobner bases – their relation with rewriting rules and the

1 Up to the point that some results whose proof requires knowldge in advanced commutativealgebra are simply quoted, pointing only to the original proof.

2 And to the independent discovery by Spear.3 The computation of the Hilbert function by means of Macaulay’s Lemma (Corollary 23.4.3).4 Macaulay’s notion of H-basis (Definition 23.2.1) and his related lifting theorem (Theo-

rem 23.7.1) transformed by Schreyer as the tool for computing resolution.

xi

xii Preface

Knuth–Bendix Algorithm, and their connection with Macaulay’s H-bases andHironaka’s standard bases as tools for lifting properities to a polynomial al-gebra from its graded algebra – my presentation stresses the relation of boththe notion and the algorithm to elementary linear algebra and Gaussian re-duction; an added bonus of this approach is the ability to link Buchberger’salgorithm with the most recent alternative linear algebra approach proposedby Faugere.

The discussion of Buchberger’s algorithm aims to present what essentiallyis its ‘standard’ structure as can be found in most good implementations.

In the same mood, the discussion of Macaulay’s paradigm is illustrated byshowing how Grobner bases can be applied in order to successfully computethe Hilbert function and the minimal resolution of a finitely generated polyno-mial ideal and to present the most effective algorithmic solutions.

This part also includes Spear’s tag-variable technique, its application in ef-fectively performing ideal operations (intersection, quotient, colon, saturation),Sweedler’s application of them to the study of subalgebras, Erdos’s character-ization of term orderings, the Bayer–Morrison analysis of the state polytopeand the Grobner fan of an ideal.

The next chapter, Noether, is the keystone of the book: it introduces the termi-nology and preliminary results needed to discuss multivariate ‘solving’: theLasker–Noether decomposition theory, extension/contraction of decomposi-tion, the notions of dimension and multiplicity, the Kredel–Weispfenning al-gorithm for computing dimension.

Part four, Duality, discusses linear algebra tools for describing and computingthe multiplicity of both m-primary and m-closed ideals, m being the max-imal at the origin; this includes Moller’s algorithm, its application to solvethe FGLM-problem, the Cerlienco–Mureddu algorithm, and the linear alge-bra structure of configurations of points; but the main section of this part isa careful presentation of Macaulay’s results on inverse systems and a recentalgorithm which computes the inverse system of any m-primary ideal given byany basis.

Part five, Beyond Dimension Zero, begins with a discussion of Grobner’sBasissatze which describe the structure of lexicographical Grobner bases ofprime, primary and radical ideals and their ultimate generalization, Gianni–Kalkbrener’s Theorem; this allows us to specify what it means to ‘solve’ amulti-dimensional ideal and introduces the decomposition algorithms.

Preface xiii

This part also discusses Macaulay’s results on Hilbert functions and perfect-ness, Galligo’s theorem, and Giusti’s analysis of the complexity of Grobnerbases.As congedo I chose the most elegant result within computational commutativealgebra, the Bayer and Stillman proof of the optimality of degrevlex orderings.

It being my firm belief that the best way of understanding a theory and analgorithm is to verify it through a computation, as in the previous volume,the crucial points of the most relevant algorithms are illustrated by examples,all developed via paper-and-pencil computations; readers are encouraged tofollow them and, better, to test their own examples.

In order to help readers to plan their journey through this book, some sectionscontaining only some interesting digressions are indicated by asterisks in thetable of contents.

A possible short cut which allows readers to appreciate the discussion, with-out becoming too bored by the details, is Chapters 20–23, 26–28, 34–35.

I wish to thank Miguel Angel Borges Tranard, Maria Pia Cavaliere, FrancescaCioffi and Franz Pauel for their help, but I feel strongly indebted to MariaGrazia Marinari for her steady support. Also I need to thank all the friends withwhom I have shared this exciting adventure of algorithmizing commutativealgebra.

Setting

1. Let k be an infinite, perfect field, where, if p := char(k) = 0, it is possibleto extract pth roots,1 and let k be the algebraic closure of k. Let us fix an integervalue n and consider the polynomial ring

P := k[X1, . . . , Xn]

and its k-basis

T := Xa11 · · · Xan

n : (a1, . . . , an) ∈ Nn.

2. We also fix an integer value r ≤ n and consider

the ring K := k(Xr+1, . . . , Xn),the polynomial ring Q := K [X1, . . . , Xr ] andits k-basis W := Xa1

1 · · · Xarr : (a1, . . . , ar ) ∈ N

r .All the notation introduced will also be applied in this setting, substitutingeverywhere n, k,P, T with, respectively, r, K ,Q,W .

3. For each d ∈ N we will set

Td := t ∈ T : deg(t) = d and T (d) := t ∈ T : deg(t) ≤ d.4. Where we need to use the set of the terms generated by some subsets ofvariables, we denote for each i, j, 1 ≤ i < j ≤ n, T [i, j] the monomialsgenerated by Xi , . . . , X j ,

T [i, j] =

Xaii · · · X

a jj : (ai , . . . , a j ) ∈ N

j−i+1

,

1 This is the general setting considered in this the volume, except for Chapters 37 and 38 wheremoreover char(k) = 0.

These restrictions can be relaxed in most of the volume, but, knowing my absentmindedness,I consider it safer to leave to the reader the responsibility of doing so.

xiv

Setting xv

and T [i, j]d (respectively T [i, j](d)) denotes those terms whose degree isequal to (respectively bounded by) d .

5. Each polynomial f ∈ k[X1, . . . , Xn] is therefore a unique linear combina-tion

f =∑t∈T

c( f, t)t

of the terms t ∈ T with coefficients c( f, t) in k and can be uniquely decom-posed, by setting

fδ :=∑t∈Tδ

c( f, t)t, for each δ ∈ N,

as f = ∑dδ=0 fδ where each fδ is homogeneous, deg( fδ) = δ and fd = 0 so

that d = deg( f ).

6. Since, for each i, 1 ≤ i ≤ n,

P = k[X1, . . . , Xi−1, Xi+1, . . . , Xn][Xi ],

each polynomial f ∈ P can be uniquely expressed as

f =D∑

j=0

h j (X1, . . . , Xi−1, Xi+1, . . . , Xn)X ji , hD = 0,

and

degXi( f ) := degi ( f ) := D

denotes its degree in the variable Xi .

In particular (i = n)

f =D∑

j=0

h j (X1, . . . , Xn−1)X jn , hD = 0, D = degn( f );

the leading polynomial of f is Lp( f ) := hd , and its trailing polynomial isTp( f ) := h0.

7. The support t ∈ T : c( f, t) = 0 of f being finite, once a term ordering <

on T is fixed, f has a unique representation as an ordered linear combinationof terms:

f =s∑

i=1

c( f, ti )ti : c( f, ti ) ∈ k \ 0, ti ∈ T , t1 > · · · > ts .

The maximal term of f is T( f ) := t1, its leading coefficient is lc( f ) :=c( f, t1) and its maximal monomial is M( f ) := c( f, t1)t1.

xvi Setting

8. For any set F ⊂ P we denote

• T<F := T( f ) : f ∈ F;• T<(F) := τT( f ) : τ ∈ T , f ∈ F;• N<(F) := T \ T<(F);• k[N<(F)] := Spank(N<(F))

and we will usually omit the dependence on < if there is no ambiguity.

9. Each series f ∈ k[[X1, . . . , Xn]] is a unique (infinite) linear combination

f =∑t∈T

c( f, t)t

of the terms t ∈ T with coefficients c( f, t) in k; for any subset N ⊂ T we willalso write the subring

k[[N]] :=∑

t∈N

c( f, t)t

⊂ k[[X1, . . . , Xn]].

10. For each f, g ∈ P such that lc( f ) = 1 = lc(g), we denote

S(g, f ) := lcm(T( f ), T(g)

T( f )f − lcm(T( f ), T(g)

T(g)g.

For any enumerated set g1, . . . , gs ⊂ P , such that lc(gi ) = 1 for each i ,we write T(i) := T(gi ) and, for each i, j, 1 ≤ i < j ≤ s

T(i, j) := lcm (T(i), T( j)) ,

S(i, j) := S(gi , g j ) := T(i, j)

T( j)g j − T(i, j)

T(i)gi .

11. For any field k the (n-dimensional) affine space over k, kn , is the set

kn := (a1, . . . , an), ai ∈ k;and we will denote by 0 ∈ kn the point 0 := (0, . . . , 0) and m :=(X1, . . . , Xn) the maximal ideal at 0.

12. We associate

• to any set F ⊂ P , the algebraic affine variety Z(F) consisting of eachcommon root of all polynomials in F :

Z(F) := a ∈ kn : f (a) = 0, for all f ∈ F ⊂ kn;• and to any set Z ⊂ kn , the ideal I(Z) of all the polynomials vanishing in Z:

I(Z) := f ∈ P : f (a) = 0, for all a ∈ Z ⊂ P.

Setting xvii

13. For any finite set F := f1, . . . , fs ⊂ P the ideal generated by F isdenoted by (F) or ( f1, . . . , fs) and is the set

(F) := ( f1, . . . , fs) :=

s∑i=1

hi fi : hi ∈ P

.

14. For an ideal f ⊂ P ,

f :=r⋂

i=1

qi

denotes an irredundant primary representation; for each i , pi := √qi is the

associated prime and δ(i) := dim(qi ) is the dimension of the primary qi .

15. For any field k and any n ∈ N we will denote by C(n, k) the n-tuples ofnon-zero elements in k:

C(n, k) := (c1, . . . , cn) ∈ kn, ci = 0, for each i.For each c := (c1, . . . , cν) ∈ C(ν, k), we denote by

Lc : k[X1, . . . , Xν] → k[X1, . . . , Xν]

the map defined by

Lc(Xi ) :=

Xi + ci Xν if i < ν,

cν Xν if i = ν.

16. A term ordering 2 of the semigroup T is called degree compatible if foreach t1, t2 ∈ T

deg(t1) < deg(t2) ⇒ t1 < t2.

The semigroup T will be usually well-ordered by means of

• the lexicographical ordering induced by X1 < X2 < · · · < Xn , which isdefined by:

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j j;• the degrevlex ordering induced by X1 < X2 < · · · < Xn , which is the

degree-compatible term ordering under which any two terms having thesame degree are compared according to

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j > b j and ai = bi for i < j.

2 That is a well-ordering and a semigroup ordering.

xviii Setting

17. Let < be a term ordering on T , and I ⊂ P an ideal, and A := P/I.Then, since A ∼= k[N<(I)], for each f ∈ P, there is a unique

g := Can( f, I, <) =∑

t∈N<(I)

γ ( f, t, <)t

such that

g ∈ k[N(I)] and f − g ∈ I.

18. More generally, if I ⊂ P is an ideal, and q = q1, . . . , qs is a linearlyindependent set such that P/I = Spank(q), then, for each f ∈ P, there is aunique vector

Rep( f, q) := (γ ( f, q1, q), . . . , γ ( f, qs, q)) ∈ ks

which satisfies

f −∑

j

γ ( f, q j , q)q j ∈ I.

In particular, if N<(I) = τ1, . . . , τs, we have, for each f ∈ P,

γ ( f, t, N<(I)) = γ ( f, t, <), for each t ∈ N<(I),

Rep( f, N<(I)) := (γ ( f, τ1, <), . . . , γ ( f, τs, <)) ∈ ks .

19. In the same setting,

M(q) := (

a(h)l j

)∈ ks2

, 1 ≤ h ≤ n

denotes the set of the square matrices defined by the equalities

Xhql =∑

j

a(h)l j q j , for each l, j, h, 1 ≤ l, j ≤ s, 1 ≤ h ≤ n,

in P/I = Spank(q).

20. In general, when we need to discuss homogenization of polynomials, wewill use the notation hP := k[X0, X1, . . . , Xn] and

hT :=

Xa00 Xa1

1 · · · Xann : (a0, a1, . . . , an) ∈ N

n+1

.

The homogenization/affinization maps are denoted

h− : P → hP and a− : hP → P

Setting xix

and defined by

h f (X1, . . . , Xn) := Xdeg( f )

0 f

(X1

X0, . . . ,

Xn

X0

),

a f (X0, X1, . . . , Xn) := f (1, X1, . . . , Xn).

For any term ordering < on T the homogenization of < is the term-ordering<h on hT defined by

t1 <h t2 ⇐⇒ deg(t1) < deg(t2) or deg(t1) = deg(t2) and at1 < at2.

21. For an ideal I ⊂ P we will denote H(T ; I) its Hilbert function; HI(T ) itsHilbert polynomial, which we will represent as

HI(T ) = k0(I)(

T + d

d

)+ k1(I)

(T + d − 1

d − 1

)+ · · · + kd−1(I)(T + 1) + kd(I);

and H(I, T ) its Hilbert series.

22. For a free-module Pm , we usually denote e1, . . . , em its canonical basisand

T (m) = tei , t ∈ T , 1 ≤ i ≤ m= Xa1

1 · · · Xann ei , (a1, . . . , an) ∈ N

n, 1 ≤ i ≤ mits monomial k-basis.

23. The free-module Pm is transformed into an N-graded module by as-signing, for each i , a degree deg(ei ) := di and considering each element(g1, . . . , gm) ∈ Pm to be homogeneous of degree R if and only if each gi

will be either 0 or a homogeneous polynomial of degree R − di .Therefore each element f ∈ Pm can be uniquely decomposed as f =∑di=1 fi where each fi ∈ Pm is homogeneous of degree i and d = deg( f )

In a similar way, Pm is also transformed into a T -graded module by

• assigning a term ordering < on T and a term ωi ∈ T to each ei ,

• defining

T-deg : T (m) → T by T-deg(tei ) = tωi ,

• and T-deg : P(m) → T as

T-deg( f ) := max<

T-deg(τ ) : c( f, τ ) = 0

for each f = ∑τ∈T(m) c( f, τ )τ ∈ P(m),

xx Setting

• considering T -homogeneous of T -degree ω any element (γ1, . . . , γm) ∈Pm such that for each i

γi ∈ T , and γiωi = ω unless γi = 0.

Each element f ∈ Pm can therefore be uniquely decomposed as f =∑t∈T ft where each ft ∈ Pm is T -homogeneous of T -degree t .If we fix a well-ordering ≺ on T (m) which is compatible with a term-

ordering < on T that is satisfying

t1 ≤ t2, τ1 τ2 ⇒ t1τ1 t2τ2,

for each t1, t2 ∈ T , τ1, τ2 ∈ T (m) then for each f = ∑τ∈T(m) c( f, τ )τ ∈ P(m),

its maximal term is the term T( f ) := max≺τ : c( f, τ ) = 0; its leadingcoefficient is lc( f ) := c( f, T( f )) and its maximal monomial is M( f ) :=lc( f )T( f ).

24. Usually a free resolution of a P-module M will be denoted

0 → Prρδρ−→ Prρ−1

δρ−1−→ · · ·Pri+1δi+1−→ Pri

δi−→ Pri−1 · · ·Pr1δ1−→ Pr0

δ0−→ M

(0.1)

25. We will denote

• by GL(n, k) the general linear group, that is the set of all invertible n × nsquare matrices with entries in k,

• by B(n, k) ⊂ GL(n, k) the Borel group of the upper triangular matricesM := (

ci j), that is those such that i > j ⇒ ci j = 0,

• by N (n, k) ⊂ B(n, k) the subgroup of the upper triangular unipotent matri-ces M := (

ci j), that is those such that

i > j ⇒ ci j = 0, and i = j ⇒ ci j = 1.

We will use the shorthand k[Xi j ] and k(Xi j ) to denote, respectively, thepolynomial ring generated over k by the variables

Xi j , 1 ≤ i ≤ n, 1 ≤ j ≤ nand its rational function field.

Any matrix

M := (ci j

) ∈ GL(n, k)

describes the linear transformation

M : k[X1, . . . , Xn] → k[X1, . . . , Xn]

Setting xxi

defined by

M(Xi ) =∑

jci j X j for each i.

If we also write for each i ,

Yi := M(Xi ) =∑

jci j X j ,

we obtain a system of coordinates Y1, . . . , Yn and a corresponding change ofcoordinates

k[Y1, . . . , Yn] = k[X1, . . . , Xn]

which is defined by

f (X1, . . . , Xn) = f(∑

id1i Yi , . . . ,

∑i

dni Yi

)∈ k[Y1, . . . , Yn],

where (di j

) = M−1 ∈ GL(n, k),

denotes the inverse of M .

26. The module P∗ := Homk(P, k) denotes the k-vector space of all k-linearfunctionals : P → k.

Each k-linear functional : P → k can be encoded by means of the series∑t∈T

(t)t ∈ k[[X1, . . . , Xn]]

in such a way that to each such series∑

t∈T γ (t)t ∈ k[[X1, . . . , Xn]] is as-sociated the k-linear functional ∈ P∗ defined, on each polynomial f =∑

t∈T c( f, t)t , by

( f ) :=∑t∈T

c( f, t)γ (t).

Module P∗ has a natural structure as P-module, which is obtained by defin-ing, for each ∈ P∗ and f ∈ P , ( · f ) ∈ P∗ as

( · f )(g) := ( f g), for each g ∈ P.

27. For each k-vector subspace L ⊂ P∗, let

P(L) := g ∈ P : (g) = 0, ∀ ∈ Land for each k-vector subspace P ⊂ P , let

L(P) := ∈ P∗ : (g) = 0, ∀g ∈ P.

xxii Setting

28. For each τ ∈ W , M(τ ) : Q → K denotes the morphism defined by

M(τ ) = c( f, τ ) for each f =∑t∈W

c( f, t)t ∈ Q

and set

M := M(τ ) : τ ∈ W ⊂ Q∗,

and

∇ρ := SpanK (M(τ )(·) : τ ∈ W(ρ)) ,

for each ρ ∈ N.For each K -vector subspace Λ ⊂ SpanK (M), let

I(Λ) := P(Λ) = f ∈ Q : ( f ) = 0, for each ∈ Λand, for each K -vector subspace P ⊂ Q, let

M(P) := L(P)∩ SpanK (M) = ∈ SpanK (M) : ( f ) = 0, for each f ∈ P.

Part three

Gauss, Euclid, Buchberger: ElementaryGrobner Bases

And when he had opened the third seal, I heard the third beast say, Come and see. AndI beheld, and lo a black horse; and he that sat on him had a pair of balances in his hand.

And I heard a voice in the midst of the four beasts say, A measure of wheat for apenny, and three measures of barley for a penny; and see thou hurt not the oil and thewine.Revelation (Authorized Version)

The things depending from Mars: choler, iron, diamond, hellebore, horse, vulture, pike.E. C. Agrippa, De occulta phylosophia

The country . . . shopkeepers don’t have it!J.-R. Hebert, Pere Duchesne

20

Hilbert

This introductory chapter will discuss how to generalize the notion of ‘solving’from the univariate to the multivariate polynomial case introducing the centraltools and problems related to multivariate solving.

I will discuss the relation between systems of equations and roots, discussingthe duality between affine algebraic varieties and ideals which is implied byHilbert’s Basissatz and Nullstellensatz (Section 20.1); after an a parte com-ment on the ability to perform suitable change of coordinates (Section 20.2),I can prove Hilbert’s Nullstellensatz (Section 20.3) and discuss the solver pro-posed by Kronecker (Section 20.4). I then generalize the duality between va-rieties and ideals in the projective setting connecting projective varieties andhomogeneous ideals (Section 20.5).

In the rest of the chapter I discuss Hilbert’s problem of computing ‘the num-ber of independent conditions which must be satisfied by the coefficients of ahomogeneous polynomial’ in order to be a member of a given ideal; this leadsto the introduction of the notions of syzygies, free resolutions and the Hilbertfunction (Section 20.6 and 20.7).

Finally I will present the proofs by Hilbert and Gordan of the Basissatz(Section 20.8).

20.1 Affine Algebraic Varieties and Ideals

Let k be an infinite, perfect field, where, if p := char(k) = 0, it is possible toextract pth roots,1 and let k be an algebraically closed extension of k. Let us

1 This is the general setting dealt with by the volume, except for Chapters 37 and 38 where more-over char(k) = 0.

These restrictions can be relaxed in most of the volume, but, knowing my absentmindedness,I consider it safer to leave to the reader the responsibility of doing so.

3

4 Hilbert

fix an integer value n and let us consider the polynomial ring

P := k[X1, . . . , Xn]

and the (n-dimensional) affine space

kn := (a1, . . . , an), ai ∈ k.On the one hand, we can consider a system of equations

f1(X1, . . . , Xn) = · · · = fs(X1, . . . , Xn) = · · · = 0,

fi ∈ P , and look for its roots in kn ; on the other hand we can consider a subsetZ ⊂ kn and wonder which polynomials satisfy them.

Therefore we denote

• for any set F ⊂ P , by Z(F) the set of the common roots of all polynomialsin F :

Z(F) := a ∈ kn : f (a) = 0, for all f ∈ F ⊂ kn;• for any set Z ⊂ kn , by I(Z) the set of all the polynomials vanishing in Z:

I(Z) := f ∈ P : f (a) = 0, for all a ∈ Z ⊂ P.

Definition 20.1.1. Let A be a ring; a non-empty subset I ⊂ A is an ideal if

• for each a1, a2 ∈ I, a1 − a2 ∈ I,• for each a ∈ I, b ∈ A, ab ∈ I.

For any set G ⊂ A the ideal generated by G is the set of all the finite sumss∑

i=1

hi fi : hi ∈ A, fi ∈ G

and is denoted by (G).

Lemma 20.1.2. For any set Z ⊂ kn, I(Z) is an ideal.

Proof. For each f1, f2 ∈ I(Z), g1, g2 ∈ P and each a ∈ Z :

(g1 f1 + g2 f2)(a) = g1(a) f1(a) + g2(a) f2(a) = 0.

Therefore, when we consider a system of equations

f1(X1, . . . , Xn) = · · · = fs(X1, . . . , Xn) = · · · = 0

we can, on the one hand consider the ideal I = ( f1, . . . , fs, . . .), and on theother hand restrict ourselves wlog to the case of finite systems, because

20.1 Affine Algebraic Varieties and Ideals 5

Fact 20.1.3 (Hilbert’s (affine) Basissatz). For each ideal I ⊂ P there is afinite set f1, . . . , fs ⊂ I such that I = ( f1, . . . , fs).

Proof. Compare Section 20.8.

A partial duality between Z and I can already be obtained. In fact:

Corollary 20.1.4. For any ideals I, I1, I2 ⊂ P and any set Z, Z1, Z2 ⊂ kn, wehave:

• I1 ⊂ I2 ⇒ Z(I1) ⊃ Z(I2);• Z1 ⊂ Z2 ⇒ I(Z1) ⊃ I(Z2);• Z(I1 + I2) = Z(I1) ∩ Z(I2);• I(Z1 ∪ Z2) = I(Z1) ∩ I(Z2);• Z(I1 ∩ I2) = Z(I1) ∪ Z(I2);• ZI(Z) ⊃ Z;• IZ(I) ⊃ I;• IZI(Z) = I(Z);• ZIZ(I) = Z(I).

The experience with the univariate case discussed in the first volume shouldbe sufficient to make clear that duality can be obtained only if suitably re-stricted, since not each subset Z ⊂ kn can be a set of roots of a polynomialsystem of equations; not only must Z be closed to k-conjugation, but deal-ing transcendency cannot be resolved elementarily by extending k to R. Onlyconsider Z := (a, exp(a)) : a ∈ R.

This leads to

Definition 20.1.5. A set Z ⊂ kn is called an affine algebraic variety if there isan ideal I ⊂ P such that Z = Z(I),

which gives one side of the required duality:

Lemma 20.1.6. For each affine algebraic variety Z ,

Z(I(Z)) = Z.

Proof. By assumption we have Z = Z(I) for an ideal I, therefore

Z = Z(I) = ZIZ(I) = Z(I(Z)).

Of course this lemma holds only for affine algebraic varieties, the obviousexamples being

6 Hilbert

• k := Q, k = C, Z := (√2, −√2) ⊂ k2, I(Z) = (X2

1 − 2, X2 + X1),Z(I(Z)) = (√2, −√

2), (−√2,

√2);

• k := R, k = C, Z := (a, exp(a)) : a ∈ R, I(Z) = 0, Z(I(Z)) = C2.

Once our restriction to affine algebraic varieties guarantees one side of du-ality, in order to obtain the other one we must at least query whether each idealhas such a set of roots; again the univariate case gives us the hint: the only idealwith no roots is the polynomial ring itself, generated by the polynomial 1.

Fact 20.1.7 (Weak Hilbert’s Nullstellensatz). For each finite set

F := f1, . . . , fs ⊂ P,

we have

Z(F) = ∅ ⇐⇒ there exist g1, . . . , gs ∈ P : 1 =s∑

i=1

gi fi .

Proof. Compare Sections 20.3 and 20.4.

Corollary 20.1.8. For each ideal I ⊂ P we have Z(I) = ∅ ⇐⇒ 1 ∈ I.

Once we have restricted, via Hilbert’s Basissatz, the systems of equationsthat will be considered to finite ones and/or to ideals, and the Weak Hilbert’sNullstellensatz gives that each non-trivial such ideal has a set of roots, we haveto deal with duality, querying which ideals I ⊂ P satisfy

I(Z(I)) = I,

or at least whether different ideals necessarily have different sets of roots.Again, the univariate case gives us the clue:

• different polynomials can share a set of roots and the only way to distinguishthem is to consider also the multiplicity of the roots;

• in other words, in order to be able to distinguish polynomials by their setsof roots, we must restrict ourselves to squarefree polynomials;

• and, if we are looking for the ideal of all the polynomials vanishing at theroots of a given polynomial f ∈ k[X ], we obtain the ideal generated by thesquarefree associate of f .

The same process happens in the multivariate case:

Definition 20.1.9. An ideal I ⊂ P is called radical (or squarefree) if

for each f ∈ P, r ∈ N : f r ∈ I ⇒ f ∈ I.

20.1 Affine Algebraic Varieties and Ideals 7

The radical√

I of an ideal I is the ideal consisting of all the elements somepower of which belongs to I:

√I := f ∈ P : there exists r ∈ N : f r ∈ I.

Lemma 20.1.10 (Strong Hilbert’s Nullstellensatz). Let I := ( f1, . . . , fs) bean ideal and let f ∈ P. Then

f ∈ I(Z(I)) ⇐⇒ there exists r ∈ N, g1, . . . , gs ∈ P : f r =s∑

i=1

gi fi .

Proof (Rabinowitch). Let f ∈ I(Z(I)) and let us consider the ideal

J := I + ( f T − 1) = ( f1, . . . , fs, f T − 1) ⊂ k[X1, . . . , Xn, T ]

and the affine algebraic variety Z(J) ∈ kn+1.For any (a1, . . . , an, t) ∈ Z(J) we have:

• for each g ∈ I ⊂ J, g(a1, . . . , an) = 0 so that (a1, . . . , an) ∈ Z(I);• therefore, f (a1, . . . , an) = 0, since f ∈ I(Z(I));• as a consequence, since f T − 1 ∈ J,

−1 = f (a1, . . . , an)t − 1 = 0,

giving a contradiction. We can therefore deduce that Z(J) = ∅ and theexistence of g1, . . . , gs, g0 ∈ k[X1, . . . , Xn, T ] such that

1 =s∑

i=1

gi fi + g0(1 − f T ).

If we set r := maxdeg(gi ), 0 ≤ i ≤ s, then

gi := f r gi

(X1, . . . , Xn,

1

f

)∈ k[X1, . . . , Xn],

so that, if we replace T with 1/ f in the equality

f r =s∑

i=1

f r gi fi + f r g0(1 − f T ),

we obtain the required representation f r = ∑si=1 gi fi .

The converse statement,

f r (a1, . . . , an) = 0 ⇒ f (a1, . . . , an) = 0, for each (a1, . . . , an) ∈ Z(I),

is trivial.

Corollary 20.1.11. Let I := ( f1, . . . , fs) be an ideal. Then I(Z(I)) = √I.

8 Hilbert

To conclude this discussion we can deduce that:

Corollary 20.1.12. The maps Z and I induce a duality between affine alge-braic varieties in kn and radical ideals in P = k[X1, . . . , Xn].

In particular:

• ZI(Z) = Z ⇐⇒ Z is an affine variety;• IZ(I) = I ⇐⇒ I = √

I.

20.2 Linear Change of Coordinates

The proof of Hilbert’s Nullstellensatz requires the ability, given any polyno-mial f ∈ k[X1, . . . , Xn] \ k, to prove the existence of a change of coordinates

L : k[X1, . . . , Xn] → k[X1, . . . , Xn]

such that

L( f ) = cXdeg( f )n +

deg( f )−1∑j=0

h j (X1, . . . , Xn−1)X jn , c = 0.

In order to prove this, let us begin by stating the following:

Lemma 20.2.1. Let S ⊂ k be any infinite set.2

For each g ∈ k[X1, . . . , Xn] \ 0, there are c1, . . . , cn ∈ S such thatg(c1, . . . , cn) = 0.

Proof. By induction on the number of variables: if n = 1 then g only has afinite number of roots, and there is c ∈ S : g(c) = 0.

If n > 1 we can express g as

g(X1, . . . , Xn) =d∑

j=0

g j (X1, . . . , Xn−1)X jn , gd = 0,

and, by induction, we can deduce the existence of c1, . . . , cn−1 ∈ S suchthat gd(c1, . . . , cn−1) = 0, so that g(c1, . . . , cn−1, Xn) = 0 has only a fi-nite number of roots, guaranteeing the existence of some cn ∈ S such thatg(c1, . . . , cn−1, cn) = 0.

Corollary 20.2.2. Given any infinite set S ⊂ k and any finite set of polynomi-als g1, . . . , gs ∈ k[X1, . . . , Xn] \ 0, there are c1, . . . , cn ∈ S such that

gi (c1, . . . , cn) = 0, for all i, 1 ≤ i ≤ s.

2 Remember that we are assuming k to be infinite.

20.2 Linear Change of Coordinates 9

Proof. Apply the lemma above to g := ∏i gi .

We denote, for any field k and any n ∈ N, by C(n, k) the n-tuples of non-zero elements in k:

C(n, k) := (c1, . . . , cn) ∈ kn, ci = 0, for each i,and, for each, c := (c1, . . . , cn) ∈ C(n, k),

Lc : k[X1, . . . , Xn] → k[X1, . . . , Xn]

the map defined by

Lc(Xi ) :=

Xi + ci Xn if i < n,cn Xn if i = n.

Theorem 20.2.3. For each f ∈ k[X1, . . . , Xn]\k there is c := (c1, . . . , cn) ∈C(n, k):

• Lc( f ) = cXdeg( f )n + ∑deg( f )−1

j=0 h j (X1, . . . , Xn−1)X jn , c = 0;

• for each (b1, . . . , bn−1) ∈ kn−1 there is at least one value b ∈ k such that

Lc( f )(b1, . . . , bn−1, b) = 0;

• for each (b1, . . . , bn−1) ∈ kn−1, and each b ∈ k such that

Lc( f )(b1, . . . , bn−1, b) = 0,

writing

ai :=

bi + ci b if i < n,cnb if i = n,

we have f (a1, . . . , an) = 0.

Proof. The polynomial f ∈ k[X1, . . . , Xn] is a linear combination

f =∑t∈T

c( f, t)t

of terms

t ∈ T := Xa1

1 . . . Xann : (a1, . . . , an) ∈ N

nwith coefficients c( f, t) in k; if we write d := deg( f ) and

fd :=∑t∈Td

c( f, t)t

where

Td := t ∈ T : deg(t) = d,

10 Hilbert

then each c := (c1, . . . , cn) ∈ kn satisfies

Lc( f ) = fd(c1, . . . , cn)Xdn +

d−1∑j=0

h j (X1, . . . , Xn−1)X jn

provided that ci = 0, for each i .By Lemma 20.2.1 above, we can deduce the existence of c := (c1, . . . , cn)

∈ C(n, k) such that c := fd(c1, . . . , cn) = 0, so that

Lc( f ) = cXdeg( f )n +

deg( f )−1∑j=0

h j (X1, . . . , Xn−1)X jn , c = 0.

Therefore, for each (b1, . . . , bn−1) ∈ kn−1 the polynomial

Lc( f )(b1, . . . , bn−1, Xn) = cXdn +

d−1∑j=0

h j (b1, . . . , bn−1)X jn ∈ k[Xn]

has exactly d =deg( f ) roots counted with the proper multiplicity, and for eachsuch root b ∈ k we have

f (a1, . . . , an) = Lc( f )(b1, . . . , bn−1, b) = 0.

Corollary 20.2.4. For each f ∈ k[X1, . . . , Xn] \ k there is (a1, . . . , an) ∈ kn

such that f (a1, . . . , an) = 0.

Proof. It is sufficient to choose any arbitrary tuple (b1, . . . , bn−1) ∈ kn−1 andany tuple c := (c1, . . . , cn) ∈ C(n, k) satisfying Lemma 20.2.1, in order todeduce the result from Theorem 20.2.3.

Note that almost all choices c satisfy the above results.

20.3 Hilbert’s Nullstellensatz

We give here an old-fashioned proof of the Nullstellensatz combining thosereported by van der Waerden and Grobner.

Let us therefore assume we have a finite set

Fn := f1, . . . , fs ⊂ P := k[X1, . . . , Xn]

generating the ideal I := In . Our aim is to show that either

• 1 ∈ I, or• there is (a1, . . . , an) ∈ kn such that (a1, . . . , an) ∈ Z(I).

20.3 Hilbert’s Nullstellensatz 11

The argument can be performed iteratively, for ν = n, n − 1, . . . , 2, by:

• computing

Dν(X1, . . . , Xν) := gcd(Fν) ∈ k[X1, . . . , Xν−1][Xν];• verifying whether Dν /∈ k, in which case3 Z(Iν) = ∅ and, via iterative

application of Proposition 20.3.1 below, Z(I) = ∅;• performing, if Dν ∈ k, the linear transformation

Lν : k[X1, . . . , Xν] → k[X1, . . . , Xν]

defined by

Lν(Xi ) :=

Xi + ci Xν if i < ν

cν Xν if i = ν

where c := (c1, . . . , cν) ∈ C(ν, k) is a suitable tuple for which there isf ∈ Fν satisfying

Lν( f ) = cXdeg( f )n +

deg( f )−1∑j=0

h j (X1, . . . , Xn−1)X jn , c = 0;

• and computing a basis Fν−1 of the intersection ideal

Iν−1 := Lν(Iν) ∩ k[X1, . . . , Xν−1]

= f (X1, . . . , Xν−1) ∈ k[X1, . . . , Xν−1] : f ∈ Lν(Iν) ;then finally computing

I0 := I1 ∩ k = c ∈ k : c ∈ I1.There are now two possible cases: either

• I0 = k and 1 ∈ Ii for each i , so that, in particular, 1 ∈ I; or• I0 = (0).

In the latter case, either

• I1 = (0) and we only have to consider the generator

D1(X1) := gcd(F1) ∈ k[X1] \ k

of I1 in order to produce a root a1 ∈ k of I1; or• there is a last value ν > 1: Iν−1 = (0), in which case Z(Iν−1) = kν−1.

3 Since Corollary 20.2.4 implies the existence of some (a1, . . . , aν) ∈ kν such that Dν(a1, . . . ,

aν) = 0, so that f (a1, . . . , aν) = 0 for each f ∈ Fν , and (a1, . . . , aν) ∈ Z(Iν).

12 Hilbert

By another inductive argument, when 1 ∈ Ii for some i , we can assume thatwe have a root (b1, . . . , bν−1) ∈ Z(Iν−1) ⊂ kν−1, and we then aim to provethe existence of an element b ∈ k such that (a1, . . . , aν) ∈ Z(Iν), where

ai :=

bi + ci b if i < ν,cνb if i = ν.

Proposition 20.3.1. Let Fν := f1, . . . , fs ⊂ k[X1, . . . , Xν] be a basis ofthe ideal Lν(Iν).

Assume that

• Z(Iν−1) = ∅,• 1 = Dν(X1, . . . , Xν) := gcd(Fν) ∈ k[X1, . . . , Xν−1][Xν], and• f1 = cXd

ν + ∑d−1j=0 h j (X1, . . . , Xν−1)X j

ν , c = 0.

Then for each (b1, . . . , bν−1) ∈ Z(Iν−1) ⊂ kν−1 there is some b ∈ k such that(b1, . . . , bn−1, b) ∈ Z(Lν(Iν)).

Proof. Let k[U2, . . . , Us] be the domain obtained by adjoining the variablesUi to the field k, and let us consider the polynomial

G :=s∑

i=2

Ui fi ∈ k[U2, . . . , Us, X1, . . . , Xν−1][Xν],

and compute the resultant 4

Res( f1, G) ∈ k[U2, . . . , Us, X1, . . . , Xν−1]

of f1 and G in k[U2, . . . , Us, X1, . . . , Xν−1][Xν].We know (Proposition 6.6.7) that there exist

p, q ∈ k[U2, . . . , Us, X1, . . . , Xν−1][Xν] : Res( f1, G) = p f1 +s∑

i=2

Ui q fi .

Moreover, each polynomial h ∈ k[U2, . . . , Us, X1, . . . , Xν−1, Xν] – suchas Res( f1, G), p and Ui q – can be written as a linear combination

h =∑t∈U

c(h, t)t

of the terms t ∈ U := Ua22 . . . U as

s : (a2, . . . , as) ∈ Ns−1 with coefficients

c(h, t) in k[X1, . . . , Xν−1, Xν].Therefore, for each t ∈ U we have equalities

c(Res( f1, G), t) = c(p, t) f1 +s∑

i=2

c(Ui q, t) fi

4 Which cannot be zero, since the assumption Dν = 1 implies that there is no common factor ink[X1, . . . , Xν−1][Xν ] between f1 and G.

20.3 Hilbert’s Nullstellensatz 13

in k[X1, . . . , Xν−1, Xν], which proves that

c(Res( f1, G), t) ∈ k[X1, . . . , Xν−1] ∩ Lν(Iν) = Iν−1.

As a consequence, for each (b1, . . . , bν−1) ∈ Z(Iν−1), we have

Res( f1, G)(U2, . . . , Us, b1, . . . , bν−1) = 0.

Since, the leading coefficient of f1 is not vanishing, this implies (Theo-rem 6.6.3) that

f ∗1 (Xν) := f1(b1, . . . , bν−1, Xν)

and

G∗(U2, . . . , Us, Xν) :=s∑

i=2

Ui fi (b1, . . . , bν−1, Xν)

have a common factor h ∈ k[U2, . . . , Us][Xν].However, we can deduce, since h divides f ∗

1 , that h ∈ k[Xν] and, since hdivides G∗, that there is H ∈ k[U2, . . . , Us, Xν] such that

h(Xν)H(U2, . . . , Us, Xν) =s∑

i=2

Ui fi (b1, . . . , bν−1, Xν).

It is then sufficient to perform the evaluation

U j := 1 if i = j

0 otherwisefor each i in order to deduce that h(Xν) divides each fi (b1, . . . , bν−1, Xν).

Therefore, for any root b ∈ k of h(Xν), fi (b1, . . . , bν−1, b) = 0 for each i,and (b1, . . . , bν−1, b) ∈ Z(Lν(Iν)).

Corollary 20.3.2. Let Fν := f1, . . . , fs ⊂ k[X1, . . . , Xν] be a basis of theideal Lν(Iν).

Assume that

• Z(Iν−1) = ∅,• 1 = Dν(X1, . . . , Xν) := gcd(Fν) ∈ k[X1, . . . , Xν−1][Xν], and• f1 = cXd

ν + ∑d−1j=0 h j (X1, . . . , Xν−1)X j

ν , c = 0.

Then, for each (b1, . . . , bν−1) ∈ Z(Iν−1) ⊂ kν−1 there is an element b ∈ ksuch that writing

ai :=

bi + ci b if i < ν,cνb if i = ν

we have (a1, . . . , aν) ∈ Z(Iν).

Proof (of the Weak Hilbert Nullstellensatz). Let I be the ideal generated byF =: Fn . Let us write In := I and define inductively, for ν := n, . . . , 1,

Iν−1 as follows:

14 Hilbert

• if Iν = 0, we set Iν−1 := 0;• if Iν = 0, we choose an element f ∈ Fν in its basis, and a tuple

c := (c1, . . . , cν) ∈ C(ν, k)

satisfying 5

Lν( f ) = cXdeg( f )ν +

deg( f )−1∑j=0

h j (X1, . . . , Xν−1)X jν , c = 0,

where Lν : k[X1, . . . , Xν] → k[X1, . . . , Xν] is the map defined by

Lν(Xi ) :=

Xi + ci Xν if i < ν,cν Xν if i = ν

and we define

Iν−1 := Lν(Iν) ∩ k[X1, . . . , Xν−1]

and Fν−1 a basis of it.

Let us denote µ, 0 ≤ µ < n, the highest value ρ, if it exists, such that Iρ = 0and let us set for each ν, µ < ν ≤ n,

Dν(X1, . . . , Xν) := gcd(Fν) ∈ k[X1, . . . , Xν−1][Xν],

noting that Iµ = 0 implies both

• Iν = 0 for each ν ≤ µ, and• Dν = 0 for each ν > µ.

Let us also note that

µ = 0 ⇒ I1 = 0 = I0 = I1 ∩ k ⇒ 1 /∈ I1 ⇒ gcd(F1) = D1 ∈ k[X1]\k.

Let us finally denote by σ,µ < σ ≤ n, the highest value ν, if it exists, suchthat Dν = 1. Then:

(1) if Iρ = 0 for each ρ then in particular I0 = k, which implies 1 ∈ I;(2) if there are a value ρ such that Iρ = 0 and a value ν, µ < ν ≤ n, such

that 1 = Dν so that Iσ ⊂ (Dσ ), then

∅ = (b1, . . . , bσ ) ∈ kσ : Dσ (b1, . . . , bσ ) = 0 ⊂ Z(Iσ );(3) if there is a value ρ such that Iρ = 0, while for each ν, µ < ν ≤ n,

1 = Dν and µ > 0, then ∅ = kµ = Z(Iµ);

5 Note that we do not care whether Iν = (1). This could of course happen even when 1 /∈ Fν .But also when Fν = 1 the argument and the implicit computation, while quite stupid, workperfectly, giving Lν(1) = 1X0

ν and Iν−1 = (1). So why consider the extreme and crucial case?

20.4 *Kronecker Solver 15

(4) if there is a value ρ such that Iρ = 0, while for each ν, µ < ν ≤ n,

1 = Dν and µ = 0, we have a contradiction, since the assumptionsimply that

• D1 = 1,• 0 = I1 which is therefore generated by D1 ∈ k[X1], while• 0 = I0 = I1 ∩ k so that 1 /∈ I1 and• D1 = 1.

In conclusion, while case (1) implies 1 ∈ I, in cases (2) and (3) the existenceof some ρ such that Z(Iρ) = ∅ allows us to deduce, by repeated application ofCorollary 20.3.2, that Z(Iν) = ∅ for ν ≥ ρ and, in particular, Z(I) = ∅.

20.4 *Kronecker Solver

Can we transform the proof we have given of Hilbert’s Nullstellensatz into aneffective algorithm for solving a system of equations? Let us discuss the crucialsteps:

• if we have a basis of Iν we have no difficulty in producing a suitable changeof coordinates Lc in order to allow the application of Corollary 20.3.2;6

• if we have a basis Fν of Iν , the computation of Res( f1, G) and of the greatestcommon divisor of all the elements f (b1, . . . , bν−1, Xν), f ∈ Fν, can beperformed within the Kronecker/Duval Model and automatically produces(up to factorization/squarefree computation) a new algebraic expression bsuch that f (b1, . . . , bν−1, b) = 0 for each f ∈ Fν .

The pons asinorum is of course the ability, given a basis of Iν , to computea basis of Iν−1. When Grobner proposed to prove the Nullstellensatz in thatway, there was no effective algorithm for doing so,7 and his theoretical proof

6 All we need to do is

• pick up any element f ∈ Fν ,• compute, using the notation of the proof of Theorem 20.2.3, the polynomial g := fd , and• choose cn , avoiding 0 and the roots of g(c1, . . . , cn−1, Xn), where c1, . . . , cn−1 have

been inductively chosen so that, with the notation of the proof of Lemma 20.2.1,gd (c1, . . . , cn−1) = 0.

7 Of course, Buchberger’s introduction of Grobner theory has dramatically changed the situation:not only can the elimination ideals

Iν−1 := Lν(Iν) ∩ k[X1, . . . , Xν−1]

be computed, transforming Grobner’s proof in the first algorithm (Trink’s Algorithm) forsolving polynomial equations, but one can even avoid, as remarked by Gianni–Kalkbrener(Theorem 34.6.1), the computation of the greatest common divisor of the elementsf (b1, . . . , bν−1, Xν), f ∈ Fν .

But this is another story which will be told in the next volume.

16 Hilbert

was just an unworkable simplification of the effective approach proposed byKronecker and exposed by Konig, and which I now intend to discuss. Theconstruction and computation are essentially the same as those I discussed inProposition 20.3.1.

Theorem 20.4.1 (Kronecker). Let

Gν := f1, . . . , fs ⊂ k[X1, . . . , Xν]

be a basis of an ideal Jν .Assume that

• 1 = gcd(Gν) ∈ k[X1, . . . , Xν−1][Xν],• f1 = cXd

ν + ∑d−1j=0 g j (X1, . . . , Xν−1)X j

ν , c = 0.

Then there is a finite set of polynomials

Fν−1 := d1, . . . , dr ⊂ Jν ∩ k[X1, . . . , Xν−1]

generating an ideal Iν−1 such that for each (b1, . . . , bν−1) ∈ Z(Iν−1) ⊂ kν−1

there is b ∈ k : (b1, . . . , bn−1, b) ∈ Z(Jν).

Proof. Applying again the same construction used in Proposition 20.3.1, let uswrite

G :=s∑

i=2

Ui fi ∈ k[U2, . . . , Us, X1, . . . , Xν−1][Xν],

and compute the non-null resultant

Res( f1, G) ∈ k[U2, . . . , Us, X1, . . . , Xν−1]

of f1 and G in k[U2, . . . , Us, X1, . . . , Xν−1][Xν], and (Proposition 6.6.7) thepolynomials

p, q ∈ k[U2, . . . , Us, X1, . . . , Xν−1][Xν] : Res( f1, G) = p f1 +s∑

i=2

Ui q fi .

Recalling that each polynomial h ∈ k[U2, . . . , Us, X1, . . . , Xν−1, Xν] canbe written as a linear combination

h =∑t∈U

c(h, t)t

of the terms

t ∈ U := U a22 . . . U as

s : (a2, . . . , as) ∈ Ns−1


with coefficients c(h, t) ∈ k[X1, . . . , Xν−1, Xν], let us consider, for each t ∈U , the polynomial

dt := c(Res( f1, G), t) = c(p, t) f1 +s∑

i=2

c(Ui q, t) fi ∈ Jν

and the ideal Iν−1 ⊂ k[X1, . . . , Xν−1] generated by

Fν−1 := dt : t ∈ U ⊂ Jν ∩ k[X1, . . . , Xν−1].

Since Res( f1, G) = ∑t∈U dt t, we can deduce, for each (b1, . . . , bν−1) ∈

Z(Iν−1), that Res( f1, G)(b1, . . . , bν−1) = 0 and apply the same argument asin the proof of Proposition 20.3.1, in order to reach the required conclusion.

Since the leading coefficient of f1 is non-vanishing,

f ∗1 (Xν) := f1(b1, . . . , bν−1, Xν)

and

G∗(U2, . . . , Us, Xν) :=s∑

i=2

Ui fi (b1, . . . , bν−1, Xν)

have a common factor h ∈ k[U2, . . . , Us][Xν], which, dividing f ∗1 , belongs to

k[Xν], while, dividing G∗, it satisfies

h(Xν)H(U2, . . . , Us, Xν) =s∑

i=2

Ui fi (b1, . . . , bν−1, Xν)

for a suitable factor H ∈ k[U2, . . . , Us, Xν]. Performing the evaluations

U j :=

1 if i = j ,0 otherwise,

one proves that h(Xν) divides fi (b1, . . . , bν−1, Xν), and

fi (b1, . . . , bν−1, b) = 0, for all i,

that is (b1, . . . , bν−1, b) ∈ Z(In), where b ∈ k is any root of h(Xν).

Let us now consider a finite set F := f1, . . . , fs ⊂ k[X1, . . . , Xn] gen-erating an ideal I, write Fn := F , In := I and inductively define Jν , Iν−1, Gν ,Fν−1, for ν := n, . . . , 1, as follows:

• if Iν = k, then Jν = Iν−1 := k, Gν = Fν−1 = 1;• if Iν = k and

1 = Dν(X1, . . . , Xν) := gcd(Fν) ∈ k[X1, . . . , Xν−1][Xν],

18 Hilbert

so that Iν ⊂ (Dν), then set

Gν := f/Dν : f ∈ Fν, Jν := f/Dν : f ∈ Iν = (Gν);• if Iν = k and 1 = gcd(Fν) ∈ k[X1, . . . , Xν−1][Xν], then Gν := Fν and

Jν := Iν;• if Jν = k, choose an element f ∈ Gν and a tuple c := (c1, . . . , cν) ∈

C(ν, k) so that, denoting by

Lν : k[X1, . . . , Xν] → k[X1, . . . , Xν]

the map defined by

Lν(Xi ) :=

Xi + ci Xν if i < ν,cν Xν if i = ν

we have

Lν( f ) = cXdeg( f )ν +

deg( f )−1∑j=0

g j (X1, . . . , Xν−1)X jν , c = 0,

and denote by Iν−1 the ideal generated by the set

Fν−1 := d(ν)1 , . . . , d(ν)

rν ⊂ Lν(Jν) ∩ k[X1, . . . , Xν−1]

whose existence, computability and properties are stated in Theorem 20.4.1.Note that Iν−1 = 0, since otherwise gcd(Gν) = 1.

Lemma 20.4.2. We have J1 = I0 = k.

Proof. We have 0 = I0 ⊂ k. Moreover I1 ⊂ k[X1] is a principal ideal,I1 = (D1), so that J1 = (1).

Lemma 20.4.3. If 1 ∈ Iν−1 then 1 ∈ Jν; if moreover Dν = 1 then 1 ∈ Iν .

Proof. Let Gν := f1, . . . , fs and Fν−1 := d1, . . . , dr be the bases of Jν

and Iν−1, so that Dν f1, . . . , Dν fs is the basis of Iν .By Theorem 20.4.1, each element di ∈ Fν−1 is a member of Jν , so that, for

suitable polynomials hi j , we have di = ∑sj=1 hi j f j . The assumption 1 ∈ Iν−1

implies the existence of polynomials pi such that 1 = ∑ri=1 pi di ; therefore

we have

1 =r∑

i=1

pi di =s∑

j=1

(r∑

i=1

pi hi j

)f j ∈ Jν .


Proof (of the Weak Hilbert Nullstellensatz). Using the notation above, either:

• there is σ, 1 ≤ σ ≤ n, such that 1 = Dσ so that Iσ ⊂ (Dσ ), and

∅ = (b1, . . . , bσ ) ∈ kσ : Dσ (b1, . . . , bσ ) = 0 ⊂ Z(Iσ )

and Z(Iν) = ∅, for each ν, σ < ν ≤ n by iterative application of Theo-rem 20.4.1, or

• 1 = Dν , for each ν, 1 ≤ ν ≤ n, so that, since I0 = k, inductive applica-tion of Lemma 20.4.3 implies that 1 ∈ Iν and 1 ∈ Jν, for each ν, whence1 ∈ I.

Unlike the proof we presented in Section 20.3, this is a ‘constructive’ proof,in the precise sense that it outlines how to perform the computation of all theroots of the ideal I, just assuming the ability of ‘solving’ (say by the KroneckerModel) univariate polynomials.

Of course, we must first understand in what sense we consider the infiniteset Z(I) to be ‘computed’.

Example 20.4.4. Let us, for instance, consider the equation

0 = X − Y 2 ∈ Q[X, Y ],

so that setting I := (X − Y 2) we have

Z(I) = (α2, α) : α ∈ C.We consider Z(I) to be successfully ‘computed’ if we return the integral

domain

R := Q[Y1][β] where β2 − Y1 = 0

and the single solution β ∈ R.

The implicit argument is that

• for each element (α2, α) ∈ Z(I) there is a ring projection

Ψ : R → Q[α] ⊂ C such that (Ψ (Y1), Ψ (β)) = (α2, α),

namely the projection defined by Ψ (Y1) = α2, Ψ (β) = α;• and, conversely, (Ψ (Y1), Ψ (β)) ∈ Z(I), for each ring homomorphism Ψ :

R → C; in fact setting α := Ψ (β) ∈ C we have

Ψ (Y1) = Ψ (β2) = Ψ (β)2 = α2.

20 Hilbert

Remark 20.4.5. We will later justify (Section 34.5) our choice, and we limitourselves to considering Z(I) to be ‘computed’ if we return a finite set Z(I) ofpairs (R, (α1, . . . , αr )) each satisfying:

• n = d + r ,• there is an admissible sequence (see Section 8.2)

( f1, . . . , fr ) ⊂ k(Y1, . . . , Yd)[Z1, . . . , Zr ]

such that

R ∼= k[Y1, . . . , Yd , Z1, . . . , Zr ]/( f1, . . . , fr ),

• each fi is monic in k[Y1, . . . , Yd , α1, . . . , αi−1][Zi ],• (α1, . . . , αr ) ∈ Rr ,

• g(Y1, . . . , Yd , α1, . . . , αr ) = 0, ∀g ∈ I ⊂ k[X1, . . . , Xn] ⊂ R[X1, . . . , Xn],• each αi satisfies fi (αi ) = 0,

in such a way that

• for each (β1, . . . , βn) ∈ Z(I) there are (R, (α1, . . . , αr )) ∈ Z(I) and a ringhomomorphism Ψ : R → k such that

Ψ (Yi ) = βi , Ψ (α j ) = βd+ j , for all i, j;• for each (R, (α1, . . . , αr )) ∈ Z(I) and each ring homomorphism Ψ : R → k,

we have

(Ψ (Y1), . . . , Ψ (Yd), Ψ (α1), . . . , Ψ (αr )) ∈ Z(I).

Such being our informal definition of ‘computing’, we can now show howKronecker’s argument allows us to ‘compute’ Z(I).

Let us begin by noting that the computation of all the necessary bases Fν andGν can be simply performed on k and such computation allows us to decidewhether 1 ∈ I or Z(I) = ∅, in which case we also know all the polynomialsDν, 1 ≤ ν ≤ n, which we wlog assume to be monic 8 and the minimal valued, 0 ≤ d < n such that 1 = Dd+1.

Then:

Id+1: we compute a factorization Dd+1 = ∏ti=1 pei

i in k[X1, . . . , Xd ][Xd+1],we set, for each i , Ri := k[X1, . . . , Xd ][Xd+1]/(pi ) and βi ∈ Ri the

8 This assumption holds if we have first performed a suitable change of coordinates: compareSections 27.9 and 34.5.


value such that pi (βi ) = 0, so that Ri = k[X1, . . . , Xd ][βi ] and wereturn

Z(Id+1) := (Ri , βi ) : 1 ≤ i ≤ t;and, iteratively, for ν = d + 2, . . . , n:

Jν : for each (R, (β1, . . . , βν−d−1)) ∈ Z(Iν−1) where

R = k[X1, . . . , Xd , Xd+1, . . . , Xν−1]/( f1, . . . , fν−d−1)

= k[X1, . . . , Xd , β1, . . . , βν−d−1]

• we compute

h(Xν) := gcd(

f (X1, . . . , Xd , β1, . . . , βν−d−1, Xν) : f ∈ Gν

)∈ R[Xν];

• we compute irreducible polynomials pi ∈ k[X1, . . . , Xν] such that

h(Xν) =t∏

i=1

peii (X1, . . . , Xd , β1, . . . , βν−d−1, Xν)

is the factorization in R[Xν];• we define 9 for each i

qi := (L−1

ν (pi ))(X1, . . . , Xd , β1, . . . , βν−d−1, Xν) ∈ R(Xν),

Ri := k[X1, . . . , Xd ][Xd+1, . . . , Xν]/( f1, . . . , fν−d−1, qi ),

β(i) ∈ Ri the value such that qi (β(i)) = 0,

γ j := β j + cd+ β(i), for j = 1, · · · , ν − d − 1,ai := (

γ1, . . . , γν−d−1, cνβ(i)

) ∈ Rν−di ;

• we then insert in Z(Jν) all the pairs (Ri , ai ), 1 ≤ i ≤ t;Iν : we compute a factorization Dν = ∏t

i=1 peii in k[X1, . . . , Xν−1][Xν], we

set, for each i , Ri := k[X1, . . . , Xν−1][Xν]/(pi ) and βi ∈ Ri thevalue such that pi (βi ) = 0 and we return

Z(Iν) := Z(Jν) ∪ (Ri , βi ) : 1 ≤ i ≤ t.

9 Note that, for each polynomial p ∈ k[X1, . . . , Xν ], the expressions

• L−1ν

(p(X1, . . . , Xd , β1, . . . , βν−d−1, Xν)

),

•(

L−1ν (p)

)(X1, . . . , Xd , β1, . . . , βν−d−1, Xν),

• p(X1 − c1c−1ν Xν , . . . , Xd − cd c−1

ν Xν , β1 − cd+1c−1ν Xν , . . . , βν−d−1 − cν−1c−1

ν Xν ,

c−1ν Xν)

are equal.Also, we can choose Lν so that each qi is monic.

22 Hilbert

20.5 Projective Varieties and Homogeneous Ideals

Within algebraic geometry it is important to consider not only affine varieties,that is subsets of the affine space kn , but also projective varieties contained inthe projective space P

n(k).Let K be a field and let us write 0 := (0, . . . , 0) ∈ K n+1; impose on K n+1 \

0 the equivalence ∼ defined by

(x0, x1, . . . , xn) ∼ (y0, y1, . . . , yn)

iff there is λ ∈ K , λ = 0, such that (x0, x1, . . . , xn) = λ(y0, y1, . . . , yn).

Definition 20.5.1. The n-dimensional projective space over the field K isthe set

Pn(K ) :=

(K n+1 \ 0

) ∖∼ .

Each residue class in Pn(K ) is called a (projective) point and each member

(x0, x1, . . . , xn) ∈ K n+1 of this residue class is called the homogeneous coor-dinates of the corresponding point.

Here I intend to discuss the duality induced by Z and I between sets ofprojective points Z ⊂ P

n(k) and sets of polynomials in k[X0, X1, . . . , Xn].This requires us to restrict ourselves to those ideals I ⊂ k[X0, X1, . . . , Xn]

which satisfy for each (x0, x1, . . . , xn) ∈ kn+1 \ 0 and each λ ∈ k \ 0f (x0, x1, . . . , xn) = 0 ⇐⇒ f (λx0, λx1, . . . , λxn) = 0, for each f ∈ I.

In order to describe such ideals, let us begin by introducing helpful notionsand notation.

Definition 20.5.2. For any subset Z ⊂ Pn(k), its representative cone is the

set C(Z) ⊂ kn+1 consisting of all the homogeneous coordinates of the pointsbelonging to Z together with the origin 0.

Each polynomial f ∈ k[X0, X1, . . . , Xn], being a linear combination

f =∑

t∈hTc( f, t)t

of terms t in

hT := Xa00 Xa1

1 . . . Xann : (a0, a1, . . . , an) ∈ N

n+1with coefficients c( f, t) ∈ k, can be uniquely decomposed as

f = f0 + f1 + · · · + fd + · · ·

20.5 Projective Varieties and Homogeneous Ideals 23

where each fi = ∑t∈hT c( fi , t)t is such that

c( fi , t) = 0 ⇒ deg(t) = i.

In other words, we can decompose hT as

hT =⊔d∈N

hTd where hTd := t ∈ hT , deg(t) = d,

and define each fi as fi := ∑t∈hT i

c( f, t)t.

Definition 20.5.3. A polynomial f = ∑t∈hT c( f, t)t ∈ k[X0, X1, . . . , Xn] is

said to be homogeneous of degree i if c( f, t) = 0 ⇒ deg(t) = i.In the unique decomposition f = ∑d

i=0 fi , where, for each i , fi is a homo-geneous polynomial of degree i , each fi is called the homogeneous componentof degree i of f . An ideal I ∈ k[X0, X1, . . . , Xn] is said to be homogeneous if,for each f ∈ I, I also contains its homogeneous components.

The leading form is the homogeneous component fd , d = deg( f ), of highestdegree.

Corollary 20.5.4 (Hilbert’s (projective) Basissatz). For each homogeneousideal I ⊂ k[X0, X1, . . . , Xn] there is a finite set f1, . . . , fs ⊂ I of homo-geneous polynomials generating I.

Proof. By Hilbert’s (affine) Basissatz, we gather that I has a finite basis F ;from it we obtain a finite basis consisting of homogeneous elements just bycollecting all the homogeneous components of its elements.

Lemma 20.5.5. Let I ∈ k[X0, X1, . . . , Xn] be an ideal; if

Z(I) := (x0, x1, . . . , xn) ∈ kn+1 : f (x0, x1, . . . , xn) = 0, for all f ∈ I

is such that ∅ = Z(I) = 0, then 10 the following conditions are equivalent

• I is homogeneous,• (λx0, λx1, . . . , λxn) ∈ Z(I), for each (x0, x1, . . . , xn) ∈ Z(I), λ ∈ k,

λ = 0,

and imply

• I ⊂ (X0, . . . , Xn).

10 Remember that we are assuming k to be infinite.

24 Hilbert

Proof.

⇒ Let (x0, x1, . . . , xn) ∈ kn+1, λ ∈ k, λ = 0, f ∈ I be such thatf (x0, x1, . . . , xn) = 0. Since I is homogeneous, we can wlog assumef to be homogeneous of degree d . Then

f (λx0, λx1, . . . , λxn) = λd f (x0, x1, . . . , xn) = 0.

⇐ By assumption there is (x0, x1, . . . , xn) ∈ Z(I) \ 0.Let f = ∑d

i=0 fi ∈ I, where each fi is its homogeneous component of degreei and let us consider any (x0, x1, . . . , xn) ∈ Z(I) \ 0.

Let us choose d + 1 different elements λ0, λ1, . . . , λd ∈ k \ 0; then since

(λ j x0, λ j x1, . . . , λ j xn) ∈ Z(I), for all j,

we have

0 = f (λ j x0, λ j x1, . . . , λ j xn) =d∑

i=0

λij fi (x0, x1, . . . , xn)

for each j ; since the matrix (λij ) is a Vandermonde matrix, its determinant is

not 0, implying fi (x0, x1, . . . , xn) = 0 for each i. Since this happens for each(x0, x1, . . . , xn) ∈ Z(I), then each fi ∈ IZ(I) = I.

Note that we have just proven that for each f = ∑di=0 fi ∈ I its constant f0

satisfies f0 = f0(x0, x1, . . . , xn) = 0. This implies I ⊂ (X0, . . . , Xn).

As a consequence of this result, we obtain a homogeneous ideal when we as-sociate to each subset Z ⊂ P

n(k) the ideal I(Z) := I(C(Z)) ⊂ k[X0, . . . , Xn]of all polynomials vanishing in each projective point in Z or, equivalently, ineach affine point in C(Z) ⊂ kn+1:

I(Z) := f ∈ k[X0, . . . , Xn] : f (a) = 0 for all a ∈ Z ⊂ Pn(k),

:= f ∈ k[X0, . . . , Xn] : f (a) = 0 for all a ∈ C(Z) ⊂ kn+1.Conversely, for any homogeneous ideal I ⊂ k[X0, . . . , Xn] such that

∅ = Z(I) = 0,there is a set Z ⊂ P

n(k) such that C(Z) = Z(I); therefore we can associateto each homogeneous ideal I ⊂ k[X0, . . . , Xn] the set of projective points 11

in Pn(k) whose homogeneous coordinates satisfy each polynomial of the ideal

or, equivalently, the residue classes of all points in Z(I) ⊂ kn+1:

Z(I) := p ∈ Pn(k) : f (x0, . . . , xn) = 0 for each (x0, . . . , xn) ∈ p, f ∈ I

:= (x0, . . . , xn) ∈ kn+1 : f (x0, . . . , xn) = 0 for each f ∈ I\ ∼ .

11 Which, with an abuse of notation, we will still denote by Z(I).


Before proceeding with the discussion, we need to justify our restriction toideals I such that ∅ = Z(I) = 0: if we consider a homogeneous ideal I, eachof its roots a ∈ kn+1 defines a point in P

n(k)12 with only the exception ofthe origin, a = 0. Therefore, as the Weak Hilbert’s Nullstellensatz for affinevarieties characterizes (1) as the only ideal with no roots, in the projectivecase one must also characterize the homogeneous ideals whose only root is theorigin.

Definition 20.5.6. An ideal I ⊂ k[X0, . . . , Xn] is said to be irrelevant if√

I = (X0, . . . , Xn).

Proposition 20.5.7 (Weak Projective Nullstellensatz). Let I k[X0, . . . ,

Xn] be a non-trivial homogeneous ideal. Then the following conditions areequivalent:

• Z(I) = ∅;• I is irrelevant;• √

I = (X0, . . . , Xn);• the only root of I in kn+1 is 0;• for each i, 0 ≤ i ≤ n, there is di > 0 such that Xdi

i ∈ I;• there is D > 0 such that t ∈ I for each t ∈ hT , deg(t) ≥ D.

Proof. Having removed the case I = (1) by assumption, the statement istrivial.

It is completely elementary to adapt the statement in order to include alsothe case I = (1):

Corollary 20.5.8. Let I ⊂ k[X0, . . . , Xn] be a homogeneous ideal. Then thefollowing conditions are equivalent:

• Z(I) = ∅;• either I is irrelevant or I = (1);• √

I ⊃ (X0, . . . , Xn);• I has no root in kn+1 \ 0;• for each i, 0 ≤ i ≤ n, there is di ≥ 0 such that Xdi

i ∈ I;• there is D ≥ 0 such that t ∈ I for each t ∈ hT , deg(t) ≥ D.

12 Since (a0, . . . , an) ∈ Z(I) ⇒ (λa0, . . . , λan) ∈ Z(I) for each λ ∈ k.

26 Hilbert

Lemma 20.5.9. Let I ⊂ k[X0, . . . , Xn] be a homogeneous ideal. Then√

I isalso homogeneous.

Proof. Let f ∈ √I and let f = fs + fs+1 + · · · + fd be the decomposition of

f into its homogeneous components, so that fi = 0, ∀i < s.It is sufficient to prove that fs ∈ √

I, since this implies that

f − fs = fs+1 + · · · + fd ∈√

I

and the same argument would then prove that each homogeneous componentof f belongs to

√I.

The assumption that f ∈ √I implies the existence of r ∈ N such that

g := f r = ( fs + · · · + fd)r = f rs + · · · + f r

d ∈ I.

Therefore all homogeneous components gi of g = ∑i gi belong to I. In par-

ticular, we have gi = 0 for i < sr, and f rs = gsr ∈ I, which implies fs ∈ √

Ias required.

We now have all the elements needed in order to state the projectiveduality.

Definition 20.5.10. A set Z ⊂ Pn(k) is called a projective variety if there is a

homogeneous ideal I ⊂ P such that Z = Z(I) or, equivalently, C(Z) = Z(I).

Lemma 20.5.11. The following hold:

(1) for each non-irrelevant homogeneous ideal I, I(Z(I)) = √I;

(2) for each projective variety Z, Z(I(Z)) = Z.

Proof.

(1) If I is a non-irrelevant homogeneous ideal then Z := Z(I) ⊂ Pn(k) is

not empty.Then, by definition, C(Z) ⊂ kn+1 is neither empty nor reduced to theorigin, and satisfies

C(Z) = Z(I),

so that, by the (affine) strong Nullstellensatz, we have I(Z(I)) = √I.

(2) Again for each projective variety Z there is a homogeneous ideal I suchthat I = I(Z) = I(C(Z)) and C(Z) = Z(I), so that

C(Z) = Z(I) = ZIZ(I) = ZI(C(Z))

that is Z = Z(I(Z)).


Theorem 20.5.12. The following hold:

(1) For any homogeneous ideal I, I1, I2 ∈ k[X0, X1, . . . , Xn] and any setZ, Z1, Z2 ⊂ P

n(k) we have:

• I1 ⊂ I2 ⇒ Z(I1) ⊃ Z(I2);• Z1 ⊂ Z2 ⇒ I(Z1) ⊃ I(Z2);• Z(I1 + I2) = Z(I1) ∩ Z(I2);• I(Z1 ∪ Z2) = I(Z1) ∩ I(Z2);• Z(I1 ∩ I2) = Z(I1) ∪ Z(I2);• ZI(Z) ⊃ Z;• IZ(I) ⊃ I;• IZI(Z) = I(Z);• ZIZ(I) = Z(I);• ZI(Z) = Z ⇐⇒ Z is a projective variety;• IZ(I) = I ⇐⇒ I = √

I.

(2) The maps Z and I induce a duality between projective varieties inP

n(k) and radical homogeneous ideals in k[X0, X1, . . . , Xn].

In this context, we want to recall a fact that we will discuss further but proveonly in the next part:

Fact 20.5.13. Let I be a homogeneous ideal, then there exist a homogeneousideal Isat and an irrelevant homogeneous ideal Iirr such that

(1) I = Isat ∩ Iirr;(2)

√Iirr = (X0, . . . , Xn);

(3) Iirr is maximal, in the sense that for each ideal J

I = Isat ∩ J,√

J = (X0, . . . , Xn), J ⊇ Iirr ⇒ J = Iirr;

(4) Z(Isat) = Z(I);(5) there is s ∈ N such that

f ∈ I homog. , deg( f ) ≥ s = f ∈ Isat homog. , deg( f ) ≥ s;(6) if for some homogeneous ideal J there is s ∈ N such that

f ∈ I homog. , deg( f ) ≥ s = f ∈ J homog. , deg( f ) ≥ s,then J ⊆ Isat;

(7) I = Isat ⇐⇒ Iirr = (X0, . . . , Xn).

The ideal Isat is called the saturation of I and is unique, while the role of Iirr inthis decomposition could be played by different irrelevant ideals.

Proof. Compare Theorem 27.6.4.

28 Hilbert

Note that the decompositions (see Example 27.4.1)

(X2, XY ) = (X) ∩ (X2, Y + aX), a ∈ Q

show the non-uniqueness of Iirr and explain why we are not allowed to removethe assumption J ⊇ Iirr in (3).

20.6 *Syzygies and Hilbert Function

Given 13 a homogeneous ideal I ⊂ k[X0, . . . , Xn] =: P , Hilbert 14 consideredhow to compute, for each value R ∈ N,

Die Zahl der von einander unabhangigen Bedingungen, welchen die Coefficienten einerForm der Rten Ordnung genugen mussen, damit dieselbe nach dem Modul I der Nullcongruent sei.The number of independent conditions which must be satisfied by the coefficients of ahomogeneous polynomial of degree R, so that it be congruent to zero with respect to I.David Hilbert, Uber die Theorie der algebraicschen Formen, Math. Ann. 36 (1890), 510

More technically, let us denote

P := k[X1, . . . , Xn],T := Xa1

1 · · · Xann : (a1, . . . , an) ∈ N

n,Td := t ∈ T , deg(t) = d.

Let I ⊂ P be a homogeneous ideal and for each integer R ∈ N, let usconsider the ‘generic’ homogeneous polynomial of degree R g := ∑

t∈TRct t.

Within the k-vectorspace k#TR of all the tuples (ct : t ∈ TR) indexed by theelements of TR let us consider the subvectorspace of those tuples (ct : t ∈ TR)

such that∑

t∈TRct t ∈ I and denote by χ(R) its k-dimension.

Definition 20.6.1 (Hilbert). The characteristic function (or Hilbert function)of a homogeneous ideal I ⊂ k[X1, . . . , Xn] is the function

h H(T ; I) : N → N such that h H(R; I) = #TR − χ(R) =(

R + n − 1

n − 1

)−χ(R) for each R.

13 This and the next section can be by passed initially but will be required for an understanding ofChapter 23.

14 In David Hilbert, Uber die Theorie der algebraicschen Formen, Math. Ann. 36 (1890), 473.Both the results and the arguments of this and the next section and Hilbert’s proof (Theo-

rem 20.8.1) of his Basissatz are contained in this paper.

20.6 *Syzygies and Hilbert Function 29

While Hilbert gave the notion of characteristic function for a homogeneousideal, it is easy to extend it to any ideal I ⊂ k[X1, . . . , Xn].

We simply consider the set

T (d) := t ∈ T , deg(t) ≤ dand, for a not necessarily homogeneous ideal I ⊂ P , we consider for eachinteger R ∈ N the ‘generic’ polynomial g := ∑

t∈T (R) ct t, whose de-

gree is bounded by R and, within the k-vectorspace k#T (R) of all the tuples(ct : t ∈ T (R)) indexed by the elements of T (R), we consider the subvec-torspace of those tuples (ct : t ∈ T (R)) such that

∑t∈T (R) ct t ∈ I and denote

by χ(R) its k-dimension.Then as before:15

Definition 20.6.2 (Hilbert). The characteristic function (or Hilbert function)of the ideal I ⊂ k[X1, . . . , Xn] is the function

H(T ; I) : N → N such that H(R; I) = #T (R) − χ(R) =(

R + n

n

)−χ(R) for each R.

The preliminary lemma in Hilbert’s investigation of the structure of the char-acteristic function being his Basissatz, he was therefore able to assume a finitenumber of polynomials

f1, . . . , fs ⊂ k[X1, . . . , Xn]

generating a (not necessarily homogeneous) ideal I. In our discussion, wewill not assume I to be homogeneous; however, when we make this assump-tion, we also implicitly assume that each basis element fi is homogeneous ofdegree di .

15 Note that the two definitions do not coincide for a homogeneous ideal I ⊂ k[X1, . . . , Xn ],having among them the obvious relations

h H(T ; I) = H(T ; I) − H(T − 1; I), H(T ; I) =T∑

0≤t

h H(t; I).

Usually, when discussing these arguments, one considers affine ideals in k[X1, . . . , Xn ] andhomogeneous ideals in k[X0, X1, . . . , Xn ].

Because I need to discuss the Hilbert function and syzygies for both affine and homogeneousideals at the same time, I have here to consider homogeneous ideals I ⊂ k[X1, . . . , Xn ] as aparticular case.

30 Hilbert

As a consequence of Hilbert’s Basissatz we know that each polynomial f ∈I has a representation f = ∑s

i=1 gi fi as a polynomial combination of the basiselements. Moreover if I is homogeneous and f is homogeneous of degree R,each gi is homogeneous of degree R − di .

It is then natural to ask how many such representations the element f has.The answer only requires us to consider two different such representations

s∑i=1

gi fi = f =s∑

i=1

hi fi

and subtract thems∑

i=1

(gi − hi ) fi = 0,

in order to deduce the classical linear algebra result, which is that all the so-lutions of a system of linear equations can be obtained by adding to a singlesolution any solution of the corresponding homogeneous system:

Lemma 20.6.3. Let I ⊂ P , F := f1, . . . , fs ⊂ I a basis of I, f ∈ I, andf = ∑s

i=1 gi fi be a representation of f in terms of F.Then

∑si=1 hi fi is a representation of f in terms of F iff there is a rep-

resentation∑s

i=1 qi fi = 0 of 0 in terms of F such that gi − hi = qi foreach i.

If I is homogeneous and f is homogeneous of degree R, for each such qi onehas deg(qi ) = R − di .

This leads directly to the introduction of the notion of syzygies: within themodule Ps := (g1, . . . , gs), gi ∈ P let us consider the subset

Syz(F) :=

(g1, . . . , gs) ∈ Ps :s∑

i=1

gi fi = 0

.

Lemma 20.6.4. Syz(F) is a P-module.

Proof. Let (g1, . . . , gs), (h1, . . . , hs) ∈ Syz(F) and g, h, ∈ P . Then

s∑i=1

(ggi − hhi ) fi = gs∑

i=1

gi fi − hs∑

i=1

hi fi = 0.

Since we are also working with homogeneous ideals and intend to apply aniteration argument, we need to impose on the module Ps a graduation, in orderthat Syz(F) is homogeneous if I is such. The solution is obvious: if e1, . . . , esdenotes the canonical basis of Ps and we define deg(ei ) := di for each i , an


element (g1, . . . , gs) ∈ Ps will be homogeneous of degree R if and only if,for each i , gi is either 0 or a homogeneous polynomial of degree R − di .

Lemma 20.6.5. If I is homogeneous, so is Syz(F).

In order to repeat iteratively the same argument, that is produce a finite basisof Syz(F) and consider in what way elements in Syz(F) can be represented interms of that basis, we need of course to generalize the Basissatz statement tothe module case:

Proposition 20.6.6. Let M ⊂ P t be a P-module. Then there is a finite basism1, . . . , ms ⊂ M such that for each m ∈ M, there are h1, . . . , hs ∈ Psatisfying m = ∑s

i=1 hi mi .

If M is homogeneous, the basis can be chosen to be homogeneous.

Proof. By induction on t : if t = 1 the statement is exactly the Basissatz. Ift > 1, assume the statement holds for any module M ′ ⊂ P t−1.

In particular we have it for

M ′ := (g1, . . . , gt−1) ∈ P t−1 : (g1, . . . , gt−1, 0) ∈ Mwhich therefore has a finite basis n′

1, . . . , n′r .

For each such element n′i := (g1, . . . , gt−1) write ni := (g1, . . . , gt−1, 0) ∈

M. Clearly, for each m := (g1, . . . , gt ) ∈ M satisfying gt = 0, there areh1, . . . , hr ∈ P such that m = ∑r

i=1 hi ni .

Next consider the ideal

I := f ∈ P : there is (g1, . . . , gt ) ∈ M with gt = f .The Basissatz guarantees the existence of a finite basis f1, . . . , fs of I ; then let

n1 := (g11, . . . , gt1), . . . , ns := (g1s, . . . , gts) ∈ M

be such that gti = fi for each i.For any m := (g1, . . . , gt ) ∈ M, we have gt ∈ I so that there exist

k1, . . . , ks ∈ P for which

gt =s∑

i=1

ki fi and n := m −s∑

i=1

ki mi ∈ M ′.

Therefore there are h1, . . . , hr ∈ P such that m = ∑si=1 ki mi + ∑r

i=1 hi ni .

This proves that m1, . . . , ms, n1, . . . , nr is the required basis.If M is homogeneous, we can obtain a homogeneous basis by collecting the

homogeneous components of the mi s and n j s.

32 Hilbert

Definition 20.6.7. Let F := f1, . . . , fs ⊂ P t be an ordered basis of a mod-ule M ⊂ P t . The module

Syz(M) :=

(g1, . . . , gs) ∈ Ps :s∑

i=1

gi fi = 0

is called the syzygy module of F (or M) and each element

(g1, . . . , gs) ∈ Syz(M)

is called a syzygy among F.

We now have the tools to perform a Hilbert inductive construction. We canstart with an ideal M0 ⊂ P and a finite ordered basis F0 := f (0)

1 , . . . , f (0)r0

of it, impose on the module Pr0 the graduation such that

deg(e(0)i ) := deg( f (0)

i ) =: d(0)i ,

where e(0)1 , . . . , e(0)

r0 denotes the canonical basis of Pr0 , and define the mor-phism

δ0 : Pr0 → P : δ0(g1, . . . , gr0) :=r0∑

i=1

gi f (0)i ;

so that

Im(δ0) = M0 and M1 := Syz(M0) = ker(δ0) ⊂ Pr0 .

Moreover, if M0 and each f (0)i are homogeneous, so is M1 and the map δ0 is

homogeneous of degree 0.16

Unless it is 0, M1 has a finite ordered basis F1 := f (1)1 , . . . , f (1)

r1 , whichwe will assume to be homogeneous if M1 is such; this allows us to imposeon the module Pr1 the graduation deg(e(1)

i ) := deg( f (1)i ) =: d(1)

i , where

e(1)1 , . . . , e(1)

r1 denotes the canonical basis of Pr1 , and to define the morphism(which is homogeneous of degree 0 in the homogeneous case)

δ1 : Pr1 → Pr0 : δ1(g1, . . . , gr1) :=r1∑

i=1

gi f (1)i ,

so that

Im(δ1) = M1 = ker(δ0) and M2 := Syz(M1) = ker(δ1) ⊂ Pr1 .

Iteratively, assuming that we have defined Mσ ⊂ Prσ−1 , Mσ = 0, we con-sider a finite ordered basis Fσ := f (σ )

1 , . . . , f (σ )rσ

of Mσ ; we impose on the

16 We recall that if N1 and N2 are two homogeneous P-modules, and δ : N1 → N2 is a morphism,δ is said to be homogeneous of degree d if for each homogeneous element n ∈ N1, δ(n) ishomogeneous and deg(δ(n)) = deg(n) + d.


module Prσ the graduation such that deg(e(σ )i ) := deg( f (σ )

i ) =: d(σ )i where

e(σ )1 , . . . , e(σ )

rσ denotes the canonical basis of Prσ , and we define the mor-

phism

δσ : Prσ → Prσ−1 : δσ (g1, . . . , grσ ) :=rσ∑

i=1

gi f (σ )i ;

so that

Im(δσ ) = Mσ = ker(δσ−1) and Mσ+1 := Syz(Mσ ) = ker(δσ ) ⊂ Prσ ;

if Mσ is homogeneous we can wlog assume that Fσ is such and then δσ ishomogeneous of degree 0 and Mσ+1 is also homogeneous.

Hilbert proved that the maximal number of such iterations is bounded by17

n in the general case and by n − 1 if M is homogeneous.In order to state Hilbert theorem we need to recall

Definition 20.6.8. Let R be a ring and M an R-module.A free resolution of M of length ρ is a sequence of free R-modules Rri and

maps δi : Rri −→ Rri−1 :

0 → Rrρδρ−→ Rrρ−1

δρ−1−→ · · · Rri+1δi+1−→ Rri

δi−→ Rri−1 · · · Rr1δ1−→ Rr0

δ0−→ M(20.1)

such that

ker(δρ) = 0, Im(δi+1) = ker(δi ), 0 ≤ i < ρ, M = Im(δ0).

Formula (20.1) is said to be a minimal resolution, if δi (e(i)1 ), . . . , δi (e

(i)ri ) is

a minimal basis of Im(δi ) for each i , where e(i)1 , . . . , e(i)

ri denotes the canon-ical basis of Rri .

If R is graded and so are the R-modules Rri , Formula (20.1) is said to be ahomogeneous resolution if each map is homogeneous of degree 0.

Fact 20.6.9 (Hilbert). Let P := k[X1, . . . , Xn] and M ⊂ P be an ideal. Thenthe minimal resolution of M has length

ρ ≤ n.

Proof. Compare Corollary 23.8.6.

17 Where n is the number of variables of P := k[X1, . . . , Xn ].

34 Hilbert

20.7 *More on the Hilbert Function

We can now state Hilbert’s conclusion from his study of the characteristicfunction:

Corollary 20.7.1 (Hilbert). There are a polynomial function HI(T ) ∈ Q[T ]such that d := deg(HI) ≤ n and a value δ such that

HI(l) = H(l; I) for each l ≥ δ.

Proof. Let us first note that #T (R) = (R+nn

).

If we set M0 := I and freely use the notation of the previous section, then tocompute the value H(R; I) we must subtract from the dimension #T (R) of thek-vectorspace of all polynomials of degree bounded by R, the dimension ofthe k-vectorspace of all polynomials belonging to I whose degree is boundedby R.

To compute that dimension we must compute the k-dimension of all the

expressions h := ∑r0i=1 gi f (0)

i , deg(h) ≤ R, which is∑r0

i=1

(R−d(0)i +nn

)minus

the k-dimension of the vectorspace of all syzygies of degree bounded by Rbelonging to M1.

That k-dimension is∑r1

i=1

(R−d(1)i +nn

)minus the k-dimension of the vec-

torspace of all syzygies of degree bounded by R belonging to M2 and so on.In conclusion, writing δ := maxd( j)

i , the polynomial

HI(T ) :=(

T + n

n

)−

n∑j=0

(−1) jr j∑

i=1

(T − d( j)

i + n

n

)∈ Q[T ]

satisfies

deg(HI) ≤ n and, for each l ≥ δ − n, H(l; I) = HI(l).

The Hilbert function H(l; I) can be expressed in terms of any Q-basis of thepolynomial ring Q[T ]; if, following Macaulay, we use the basis(

T + i

i

): i ∈ N

,

we have the representation

HI(T ) = k0

(T + d

d

)+ k1

(T + d − 1

d − 1

)+ · · · + kd

= k0(I)(

T + d

d

)+ k1(I)

(T + d − 1

d − 1

)+ · · · + kd(I).

20.7 *More on the Hilbert Function 35

Definition 20.7.2. For an ideal I ⊂ P

• the polynomial HI(T ) ∈ Q[T ] is called its Hilbert polynomial; and• the series

H(I, T ) :=∞∑

t=0

H(t; I)T t ∈ Q[[T ]]

is called its Hilbert series.We call

• d(I) := d := deg(HI) the dimension,• γ (I) := δ the index of regularity,• k0(I) the degree

of I.

Concerning our chosen Q-basis we recall the combinatorial formulas(d + i + 1

i + 1

)=

(d + i

i + 1

)+

(d + i

i

),

d∑T =0

(T + i

i

)=

(d + i + 1

i + 1

),

from which we deduce

Lemma 20.7.3. (1 − T )−n = ∑∞t=0

(t+n−1n−1

)T t .

Proof. Since (1 − T )(∑∞

t=0 T t) = 1, the claim is true for n = 1. Then,

inductively

∞∑t=0

(t + n

n

)T t =

∞∑t=0

t∑u=0

(u + n − 1

n − 1

)T t

=∞∑

u=0

∞∑t=u

(u + n − 1

n − 1

)T t

=∞∑

u=0

∞∑t=0

(u + n − 1

n − 1

)T t+u

=( ∞∑

t=0

T t

) ( ∞∑u=0

(u + n − 1

n − 1

)T u

)

= (1 − T )−1(1 − T )−n

= (1 − T )−n−1.

36 Hilbert

Corollary 20.7.4. For any ideal I ⊂ P , we have

H(I, T ) = (1 − T )−d(I)+1 Q(T ) where Q(T ) ∈ Q[T ], Q(1) = k0(I).

Proof. We have

H(I, T ) =∞∑

t=0

H(t; I)T t

=d(I)∑i=0

ki (I)∞∑

t=0

(t + d − i

d − i

)T t

=d(I)∑i=0

ki (I)(1 − T )i−d(I)+1

= (1 − T )−d(I)+1d(I)∑i=0

ki (I)(1 − T )i .

Corollary 20.7.5. For I := (1) = P = k[X1, . . . , Xn] we have

H(I, T ) =∞∑

t=0

(t + n − 1

n − 1

)T t = (1 − T )−n .

20.8 Hilbert’s and Gordan’s Basissatze

Theorem 20.8.1 (Hilbert’s Basissatz). Let

F := F1, . . . , Fm, . . . ⊂ k[X1, . . . , Xn]

be an infinite set. Then there is a finite subset

G := G1, . . . , Gρ ⊂ F

such that each element in F can be expressed as a polynomial combination ofthe elements of G.

Proof (Hilbert). The proof is by induction, the univariate case being trivial, sowe assume that the statement is true for the polynomial ring k[X1, . . . , Xn−1].

If we choose a suitable vector c := (c1, . . . , cn) ∈ C(n, k) and we performthe change of coordinates

Lc : k[X1, . . . , Xn] → k[X1, . . . , Xn]

20.8 Hilbert’s and Gordan’s Basissatze 37

defined by

Lc(Xi ) :=

Xi + ci Xn if i < n,cn Xn if i = n,

it is sufficient to prove the result for Lc(F). As a consequence of Theo-rem 20.2.3 we can wlog assume that

F1 = cXdn +

d−1∑j=0

h j (X1, . . . , Xn−1)X jn , c = 0.

Therefore, each other element can be expressed as

Fm := B1m F1 +d∑

j=1

h jm(X1, . . . , Xn−1)Xd− jn ,

and, defining for each m > 1,

F (1)m := Fm − B1m F1 =

d∑j=1

h jm(X1, . . . , Xn−1)Xd− jn ,

each element F (1)m , whose degree in Xn is at most d − 1, can be seen as a

reduction of Fm in terms of F1.If we now consider the set h12, . . . , h1m, . . . ⊂ k[X1, . . . , Xn−1] by in-

duction we can deduce that there are finite elements, say F2, . . . , Fm1 , suchthat each element h1m can be expressed as a polynomial combination h1m :=∑m1

i=2 qi h1i in terms of h12, . . . , h1m1.If we then define, for each m > m1, F (2)

m := F (1)m − ∑m1

i=2 qi F (1)i , one has

that

• F (2)m = Fm − ∑m1

i=1 Cim Fi , and

• F (2)m = ∑d

j=2 g jm(X1, . . . , Xn−1)Xd− jn ,

for suitable Cim ∈ k[X1, . . . , Xn], and g jm ∈ k[X1, . . . , Xn−1], so that each

Fm has been reduced in terms of F1, F2, . . . , Fm1 to a polynomial F (2)m whose

degree in Xn is at most d − 2.So, by iteration, we can assume that we have obtained a finite subset of F ,

which we can denote by F1, F2, . . . , Fmr and for each m > mr a polynomialF (r)

m which satisfies

• F (r)m = Fm − ∑mr

i=1 Dim Fi and

• F (r)m = ∑d

j=r f jm(X1, . . . , Xn−1)Xd− jn ,

38 Hilbert

for suitable Dim ∈ k[X1, . . . , Xn], and f jm ∈ k[X1, . . . , Xn−1], so that each

Fm is reduced in terms of F1, F2, . . . , Fmr to a polynomial F (r)m whose de-

gree in Xn is at most d − r .Considering now the set frmr +1, . . . , frm, . . . ⊂ k[X1, . . . , Xn−1], we

deduce by induction the existence of finite elements, say Fmr +1, . . . , Fmr+1

such that each element frm can be expressed as a polynomial combination

frm :=mr+1∑

i=mr +1

qi fri

in terms of frmr +1, . . . , frmr+1.Defining, for each m > mr+1, F (r+1)

m := F (r)m − ∑mr+1

i=mr +1 qi Fi , one hasthat

• F (r+1)m = Fm − ∑mr+1

i=1 Eim Fi and

• F (r+1)m = ∑d

j=r+1 γ jm(X1, . . . , Xn−1)Xd− jn ,

for suitable Eim ∈ k[X1, . . . , Xn], and γ jm ∈ k[X1, . . . , Xn−1], so that each

Fm is reduced in terms of F1, F2, . . . , Fmr+1 to a polynomial F (r+1)m whose

degree in Xn is at most d − r − 1.Eventually r = d + 1 and, for each m > md+1, F (d+1)

m = 0 and each Fm isa polynomial combination in terms of the finite set F1, F2, . . . , Fmd+1.

Historical Remark 20.8.2. It seems that it was this proof,18 which, whilequite elementary, stimulated the expression ‘Das ist Theologie und keineMathematik’ uttered by Gordan and led him to find a less theologicalproof.19

Gordan’s proof is based on a lemma which is normally attributed to Dickson,but the available proofs are essentially the same as that already provided byGordan.

Proposition 20.8.3 (Dickson’s Lemma). Let

T := Xa11 · · · Xan

n : (a1, . . . , an) ∈ Nn,

18 Contained in David Hilbert, Uber die Theorie der algebraicschen Formen, Math. Ann. 36(1890), 473. I am actually adapting the version contained in his 1897 course the notes of which,taken by S. Marxsen, have been recently translated and published in David Hilbert, Theory ofAlgebraic Invariants, Cambridge University Press (1993), pp. 126–130.

19 There is a short announcement in German in P. Gordan, Neuer Beweis des HilbertschenSatzes uber homogene Funktionen Gottingen Nachr. (1899), 240–242. This is followed bythe complete paper in French in P. Gordan, Les invariants des formes binaries. Journal deMathematiques Pure et Applies (5e series) 6 (1900), 141–156.


and let A ⊂ T ; then there is a finite subset B ⊂ A such that, for each t ∈ A,

there is an element t ′ ∈ B : t ′ | t.

Proof (Gordan). The proof is by induction on n, the number of variables.For n = 1 the thesis is equivalent to the statement that N is well-ordered.So let us consider n > 1 and let us assume the thesis holds for n − 1.Let U (i) denote the free commutative semigroup generated by

X1, . . . , Xi−1, Xi+1, . . . , Xn,that is

U (i) = Xa11 · · · Xan

n ∈ T : ai = 0,and Ψi : T → U (i) be the semigroup morphism defined by

Ψi (X j ) := X j if j = i ,

1 otherwise.

Let us fix an element τ = Xb11 . . . Xbn

n ∈ A and let us write 20 for eachi, j, 1 ≤ i ≤ n, 0 ≤ j < bi ,

Ai j := Xa11 . . . Xan

n ∈ A : ai = j.Note that the restriction of Ψi to Ai j is injective for each j and that

for each u, u′ ∈ Ai j , u | u′ ⇐⇒ Ψi (u) | Ψi (u′).

Therefore, by the inductive assumption, for each i, j , there is a finite subsetBi j ⊂ Ai j such that for each t ∈ Ai j , there is t ′ ∈ Bi j : t ′ | t.

As a consequence

B := τ ∪(⋃

i jBi j

)satisfies the required property. In fact, for each t ∈ A either

• τ | t or• t ∈ Ai j for some i, j and there is t ′ ∈ Bi j : t ′ | t.

Corollary 20.8.4. Let t1, . . . , tn, . . . be an infinite enumerable set of elementsin T .

Then there is N ∈ N such that for each i > N there is j ≤ N satisfyingt j | ti .

20 In his proof, Gordan enumerates the set Ai j as Bg, g = 0, . . . ,∑n

h=1 bi , where

Bg :=

Xa11 . . . Xan

n ∈ A : ai = g −i−1∑h=1

bh

, for

i−1∑h=1

bh < g ≤i∑

h=1

bh .

Not a very smooth notation!

40 Hilbert

Theorem 20.8.5 (Hilbert’s Basissatz). Let

F := F1, . . . , Fm, . . . ⊂ k[X1, . . . , Xn]

be an infinite set. Then there is a finite subset

G := G1, . . . , Gr ⊂ F


Proof (Gordan). Let us impose on T an ordering < such that

t1 | t2 ⇒ t1 < t2 for each t1, t2 ∈ T .

For each polynomial Fi ∈ F let us express it as

Fi := ci ti + φi

where ci ∈ k, ci = 0, ti ∈ T and φi is a linear combination of terms t ∈ Tsuch that 21 t < ti :

φi :=∑t∈T

c(t, φi )t, c(t, φi ) = 0 ⇒ t < ti .

Gordan calls φi the Anfangsglied 22 of Fi .If F contains two elements Fi , Fj such that t j | ti , so that there is t ∈ T :

ti = t t j , then

Fi − ci

c jFj t = φi − ci

c jφ j t

has a simpler Anfangsglied than Fi .First let us reorder the elements of F according to their Anfangsgliedern,23

this order being ‘inverse de l’ordre des termes dans une function homogene’.

21 More precisely, Gordan assumes Fi is written as a combination of ‘termes[ . . . ] dans un ordretel que chacun d’eux precede ceux qui sont plus simples’.

22 It is tempting to translate this as leading term but the French version calls it just premier terme.23 The effectiveness of such a re-ordering was not considered by Gordan as a problem: in fact he

illustrates his (Dickson’s) Lemma with the following examples:

• Xa11 : a1 ≡ 0 (mod 3);

• Xa11 . . . X

a44 : a1 + a2 + a3 + a4 ≡ 0 (mod 3);

• all the terms Xa11 . . . X

a44 satisfying the formulas

a1 + a2 + a3 + a4 ≡ 0 (mod 3),

a1a2 + a1a3 + a1a4 + a2a3 + a2a4 + a3a4 > 0.


Then one sets f1 := F1 and iteratively simplifies the Anfangsglied of eachelement Fm by means of f1, . . . , fm−1, obtaining a polynomial fm which canbe expressed in terms of F1, . . . , Fm as

fm =m−1∑i=1

Ai Fi + Fm (20.2)

thus obtaining a sequence f := f1, . . . , fm, . . . where each fi can be ex-pressed as

fi := di ui + χi ,

ui being its Anfangsglied.24

If f contains two elements fi , f j for which there is u ∈ T : ui = uu j , thenone computes

fi := fi − di

d jf j t,

and substitutes fi for fi in f, obtaining a simpler sequence f1.If fi = 0 then f1 has a function less than f and, by (20.2), Fm has a represen-

tation

Fm =m−1∑i=1

−Ai Fi

in terms of the preceding functions.Therefore one obtains a systeme irreductible g1, g2, . . . , gm . . . whose pre-

miers termes are not divisible by each other, that is a finite set g1, g2, . . . , gr corresponding to a finite sequence of elements

G := G1, . . . , Gr ⊂ F

such that each other element Fi ∈ F \ G can be expressed in terms of them.

Historical Remark 20.8.6. I must confess that I cannot find a big differencebetween the two proofs: both perform simplification reductions of elementsuntil most are reduced to zero.

Actually, Hilbert’s proof is stronger than Gordan’s: it is implicitly more sys-tematic because, in the most obvious implementation of Hilbert’s procedure,

24 Note that if we are considering ‘generic’ polynomials the result of this reduction will be asequence of polynomials f1, f2, . . . , fm , . . . whose Anfangsgliedern u1, u2, . . . , um , . . . areordered so that

u1 > u2 > · · · > um > · · · .

42 Hilbert

each element is reduced only in terms of the previous ones and its reductioncan conclude only by returning either 0 or an element which is immediatelyinserted in the output basis; on the other hand Gordan’s procedure reduces ele-ments haphazardly, since an element fi can be temporarily stored in the currentbasis but further reduced when a new basis element f j is produced whose An-fangsglied divides that of fi .

However, Gordan’s prooof, being weaker, is more elementary and intro-duced the idea of considering polynomials as a linear combination of orderedterms and of performing Gaussian reduction on them. His proof is therefore aperfect introduction to the next chapter.

But the idea of Anfangsgliedern, or premiers termes, or ‘leading terms’,is already implicit in Hilbert’s proof, where, in each step, a new element isproduced, whose Anfangsglied is used to simplify all further polynomials.

The systematic approach by Hilbert, in contrast to the haphazard approachby Gordan, implies that the shape of the resulting Hilbert basis is much betterthan that of Gordan.

Gordan’s bases are now known as Grobner bases; in some ways the (im-plicit) shape of the basis implied by Hilbert’s procedure can be seen as the firstof a series of results (Grobner, Gianni–Kalkbrener . . . ) which are now knownas Shape Lemmata.

It is therefore worth looking at the shape of the basis produced by Hilbert’sproof.

Corollary 20.8.7 (Hilbert’s Shape Lemma). Let

F := F1, . . . , Fm, . . . ⊂ k[X1, . . . , Xn]

be an infinite set. Then there is a finite set

G := G1, . . . , Gρ


Moreover, up to a change of coordinates, each polynomial has the shape

Gi := ci ti + G ′i

where

• ci ∈ k \ 0,• ti := Xa1i

1 , . . . , Xanin ∈ T ,

• deg(Gi ) = deg(ti ) > deg(G ′i ),


• if < denotes the lexicographical ordering induced by X1 < X2 < · · · < Xn,which is defined by:

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j j ,

one has t1 > t2 > · · · > tρ.

Proof. The proof 25 of Theorem 20.8.1 is performed by iteration consideringin order Xn, Xn−1, . . .

At the (n − ν + 1)th iteration loop of the argument, a change of coordinateis performed which does not affect the variables Xν+1, . . . , Xn and a singleelement of the form

cXdνν +

dν−1∑j=0

h j (X1, . . . , Xν−1)X jν , c = 0,

is added to the basis; then an inner iteration is performed for each r, dν ≥ r ≥1, in whose (r + 1)th loop an ordered set F (r+1)

mr +1 , . . . , F (r+1)mr+1 is appended to

the basis where for each m, mr + 1 ≤ m ≤ mr+1 one has

F (r+1)m = frm Xdν−r

ν +dν∑

j=r+1

f jm(X1, . . . , Xν−1)Xdν− jν ,

and, inductively, the basis

frmr +1, . . . , frmr+1 ⊂ k[X1, . . . , Xν−1]

satisfies the same assumptions, so that for each m, mr + 1 ≤ m ≤ mr+1, onehas frm := cmτm + f ′

rm, where

• cm ∈ k \ 0;• τm := Xa1m

1 , . . . , Xaν−1mν−1 ∈ T ;

• deg( frm) = deg(τi ) > deg( f ′rm);

• τmr +1 > · · · > τmr+1 .

Therefore if we write, for each m, mr + 1 ≤ m ≤ mr+1,

Gm := F (r+1)m ,

tm := τm Xdν−rν ,

G ′m := f ′

rm Xdν−rν +

dν∑j=r+1

f jm(X1, . . . , Xν−1)Xdν− jν ,

25 In the proof of Theorem 20.8.1, a finite basis G := G1, . . . , Gρ ⊂ F is extracted from theoriginal set F , on the basis of the current shape of the corresponding partially reduced element.

In the proof of this corollary, we will instead build at each step the final basis, adding to itthe current partial reductions of the imput element.

44 Hilbert

then Gm := F (r+1)m = cmτm Xdν−r

ν +G ′m and all the conditions required by the

statement hold.In particular we have

tmr = Xa1mr1 , . . . , X

aν−1mrν−1 Xdν−r+1

ν

> Xa1mr +11 , . . . , X

aν−1mr +1ν−1 Xdν−r

ν

= tmr +1 > · · · > tmr+1 .

Historical Remark 20.8.8. While it was more natural for me to introduceHilbert’s Basissatz in the affine case and deduce the projective result as aCorollary (20.5.4) of it, I must remark that both Hilbert and Gordan statedand proved it in the homogeneous case. The proofs I gave applied verbatimand are also probably smoother in the homogeneous case.

One can deduce the affine Hilbert’s Basissatz from the projective case viahomogenization/affinization (see Historical Remark 23.2.3).

Historical Remark 20.8.9. The proofs by Hilbert and Gordan of the Basis-satz could help us to appreciate the introduction by Grete Herrman of the no-tion of Endilichvielen Schritten (see the footnote of Algorithm 1.1.3). NeitherHilbert nor Gordan questioned the complexity or finiteness of their algorithms;they naturally considered it normal to perform infinite computations on aninfinite set.

It is worth quoting two passages from Hilbert’s notes:26 when he stated theBasissatz he commented

Note also that the statement of the theorem assumes that the given sequence of formsF1,F2,F3, . . . is a countable set, that is, one can think of it as ordered in some way,according to some given rule, and that it is given in that order. But there are no additionalhypotheses.

And in the following passage he was proving the result for a homogeneoussequence in k[x]:

In the simple case n = 1, the theorem is clear. Each F has the form cxr , where c is aconstant. Let c1xr1 be the first form of the sequence with a coefficient different fromzero. We then look for the next form in the sequence whose order is less then r1; if thereis no such form, we retain c1xr1 . But if there is one, say c2xr2 , then we proceed to thenext form in the sequence whose order is less than r2. If we continue in this manner,

26 Both in David Hilbert, Theory of Algebraic Invariant, Cambridge University Press (1993),pp. 126–7.


then we finally arrive at a form ci xri = Fm in the sequence with the property thatnone of the subsequent forms have order less than ri . Every form is then divisible byFm . . . .

Macaulay, who was the first to investigate (practical) complexity (seeHistorical Remark 23.9.5), was, however, more unscrupulous than they: heprovided an algorithm which, given a basis of a finite vectorspace I(d), al-lows one to deduce, in a finite number of steps, a basis of a finite vectorspaceI(d+1) ⊇ I(d) and he commented that ‘we can proceed similarly to find in the-ory’ the infinite basis of the vectorspace

⋃d I(d) (see Algorithm 30.4.3). Even

more unscrupulous was his construction of a non-zero-dimensional principalsystem.27

27 Compare the last quoted section before Definition 30.5.1.

21

Gauss II

In the early 1980s when Grobner bases and the Buchberger Algorithm spreadthrough the research community, there were two main approaches to their in-troduction: the most common was (and still is) presenting these notions inthe frame of rewriting rules, showing their relationship to the Knuth–BendixAlgorithm, and stressing their role in giving a canonical representation forthe elements of commutative finite algebras over a field. I was among thestandard-bearers of the alternative approach which saw Grobner bases asa generalization of Macaulay’s H-bases and Hironaka’s standard bases andstressed their ability to lift properties to a polynomial algebra from its gradedalgebra.

While both these aspects of Grobner theory and the related results will bediscussed in depth in this text, I have for several years stressed its relation toelementary linear algebra:1 Grobner bases can be described 2 as a finite modelof an infinite linear Gauss-reduced basis of an ideal viewed as a vectorspace,and Buchberger’s algorithm can be presented as the corresponding generaliza-tion of the Gaussian elimination algorithm. This approach allows me also tolink Grobner theory directly to the Duality Theory which will be discussedin Part five, mainly to the Moller algorithm and (in the next volume) to theAuziger–Stetter resolution.

This preliminary chapter only contains a very heretical presentation of vec-torspaces and Gaussian elimination; the aim of this approach is not to introduce

1 This approach was suggested to me by A. Galligo, Algorithmes de calcul de bases standard,Nice (1982).

Its development is also strongly indebted to R. Gebauer and H. M. Moller, Buchberger’salgorithm and staggered linear bases. Proc. SYMSAC 1986, pp. 218–221.

2 Refining the forgotten suggestion in Gordan’s proof of the Basissatz.

46

21.1 Some Heretical Notation 47

this book in the Index Librorum Prohibitorum, but only to introduce the nota-tions and the basic concepts of Grobner theory in an elementary context, so thatreaders with an orthodox knowledge of linear algebra should have no difficultyin following this presentation.

21.1 Some Heretical Notation

Let k be a field and let W be a k-vectorspace given by assigning a basis B :=ei : i ∈ I so that W = Spank(B).

I am not assuming that B is finite but just require I to be enumerable andwell-ordered.

Example 21.1.1. We will consider throughout this section the following twoinstances:

(1) I := N, W := k[X ], B := Xi : i ∈ I well-ordered so that

Xi > X j ⇐⇒ i > j.

(2) With explicit reference to Remark 6.2.2 3 and Section 8.3, we also con-sider I := N

r , W := k[X1, . . . , Xr ],

B := B := T = Xa11 · · · Xar

r : (a1, . . . , ar ) ∈ Nr

ordered by the lexicographical ordering induced by X1 < X2 < · · · <

Xr , defined by:

Xa11 . . . Xar

r < Xb11 . . . Xbr

r ⇐⇒ there exists j : a j j.

While I do not assume W to be finite-dimensional, I will often consider achain of finite subvectorspaces

W1 W2 · · · Wd · · · W

3 Where we set n := r .

48 Gauss II

such that⋃

d Wd = W given by assigning a chain of finite subsets

I1 I2 · · · Id · · · I,

and defining Bd := ei : i ∈ Id, Wd := Spank(Bd).

Example 21.1.2. Continuing the previous examples:

(1) We set Id := i ≤ d ⊂ N so that Wd = f (X) ∈ k[X ], deg( f ) ≤ d.(2) In the same way we set

Id :=

(a1, . . . , ar ) :∑

j

a j ≤ d

so that

Bd = t ∈ T : deg(t) ≤ dand

Wd = f (X1, . . . , Xr ) ∈ k[X1, . . . , Xr ], deg( f ) ≤ d.Each element w ∈ W has a representation

w =∑i∈I

ci ei , ci ∈ k;

moreover, since the elements of W are finite sums of elements in B, the supportof w, ei : ci = 0, is finite and each non-zero element w ∈ W has a uniqueordered representation

w =n∑

j=1

c j ei j : c j ∈ k \ 0, i j ∈ I, i1 > i2 > · · · > in .

So, to each non-zero element w ∈ W , we can associate

T(w) := ei1 , lc(w) := c1, M(w) := c1ei1 .

If needed, I will assume T(0) = lc(0) = M(0) = 0 and 0 = T(0) < ei foreach ei ∈ B.

Example 21.1.3. Giving a more elementary example, let us consider W := k7

and let e1, . . . , e7 denote its canonical basis which we order 4

e1 > e2 > · · · > e7.

4 This esoteric ordering needs a justification.It is natural given a set of linear equations

n∑j=1

ci j x j = 0, 1 ≤ i ≤ m

to assume that the variables are ordered as x1 < x2 < · · · < xn and try to express the first

21.1 Some Heretical Notation 49

Then for the vector w := (0, −3, 0, 2, 0, 5, 3) ∈ W the support of w ise2, e4, e6, e7 and we have T(w) = e2, lc(w) = −3, M(w) = −3e2.

Example 21.1.4. Continuing the previous examples:

(1) For a polynomial

f (X) :=n∑

i=0

ai Xn−i

= a0 Xn + a1 Xn−1 + · · · + ai Xn−i + · · · + an−1 X + an,

such that a0 = 0, we have T( f ) = Xn, lc( f ) = a0, M( f ) = a0 Xn .

(2) If we consider a polynomial f = ∑t∈B ct t then we have (see Re-

mark 6.2.2 and Algorithm 8.3.1)

T( f ) := max<

t : ct = 0, lc( f ) := cT( f ), M( f ) = lc( f )T( f ).

Let us now consider a subvectorspace V ⊂ W and let us denote

TV := T(v) : v ∈ V and N(V ) := B \ TV .

variables in terms of the last ones, so that if the frame of coordinates is generic and the matrix(ci j ) has rank r the variables x1, . . . , xr are expressed in terms of the variables xr+1, . . . , xn .

This can be performed by iterating Gaussian reduction on an increasing value j , thus express-ing each variable x j in terms of the higher variables, but such an algorithm obviously is possibleonly for a finite-dimensional vectorspace.

The Euclidean algorithm, Sylvester resultants, Newton’s algorithm for expressing symmet-ric functions (Theorem 6.2.4) and Algorithm 8.3.1 for computing canonical representations inKronecker’s Model, mimicking whose patterns we interpret Buchberger’s algorithm in terms ofGaussian reduction, perform linear algebra in the infinite-dimensional vectorspace of the poly-nomial ring, but have the advantage of knowing a priori the maximal degree of the polynomialsinvolved.

For instance, the Euclidean algorithm performs linear algebra on the vectorspace basis

e1 := Xn−1, e2 := Xn−2, . . . , en−1 = X, en = 1,

where n = max(deg(P0), deg(P1)) + 1 and r := n − deg(gcd(P0, P1)), expressing, in terms ofthe basis

er+1, . . . , en = Xn−r−1, . . . , X, 1,

the lowest-index/highest-degree powers Xn−i = ei , 1 ≤ i ≤ r , and, in practice, each powerXn− j , j ≤ r.

In other words, we can say that in the Euclidean algorithm, as in the other cited algorithms,the basis elements are ordered by their weight value and highest-weight elements are expressedin terms of the first lowest-weight ones. The same pattern must be preserved in an interpretationof Buchberger’s algorithm and theory in terms of Gaussian reduction of polynomials, where itis impractical to restrict reduction to degree-bounded polynomials, while basis elements have tobe naturally ordered by increasing weight.

My choice of ordering the basis elements as e1 > e2 > · · · > en is a, perhaps clumsy,way of stressing this pattern in which highest-weight elements are expressed in terms of thelowest-weight ones.

50 Gauss II

Example 21.1.5. With the same setting as in Example 21.1.3, we can con-sider

w1 := (0, −3, 0, 2, 0, 5, 3), w2 := (0, 1, 0, 0, 0, 0, −1),

w3 := (0, 0, 1, 1, 0, 0, 0), and v := (0, 0, 0, 2, 0, 5, 0),

and the vectorspace V := Spank(w1, w2, w3).Noting that

w1 + 3w2 − v = 0

so that V = Spank(w1, v, w3), we can conclude that TV = e2, e3, e4 andN(V ) = e1, e5, e6, e7.

Example 21.1.6. Continuing the examples discussed in Examples 21.1.1,21.1.2 and 21.1.4, we consider

(1) a polynomial f (X) := ∑ni=0 ai Xn−i , a0 = 0 and V will denote the

ideal generated by f ;(2) a sequence f1, . . . , fr ∈ k[X1, . . . , Xr ] – we are essentially thinking

of admissible sequences (Section 8.2) and admissible Duval sequences(Section 11.4) – such that

• f1 ∈ k[X1] is monic,• fi ∈ k[X1, . . . , Xi−1][Xi ] is monic for each i ,• writing d j := deg j ( f j ) we have deg j ( fi ) < d j for each j < i,

and V will denote the ideal generated by f1, . . . , fr .

In order to restrict ourselves to the finite-dimensional case, we can consider

(1) for each d ≥ n the subvectorspace

Vd := V ∩ Wd = g f, g ∈ k[X ], deg(g) ≤ d − n ⊂ ( f ) = V ;

(2) for each d ≥ D := ∑ri=1 di the subvectorspace

Vd := V ∩Wd = g ∈ ( f1, . . . , fr ), deg(g) ≤ d ⊂ ( f1, . . . , fr ) = V .

In both cases Vd represents the vectorspace consisting of the elements in theideal V whose degree is bounded by d . This restricted setting being finite, it is

21.2 Gaussian Reduction 51

easy to describe the situation:

(1) for each d ≥ n we have

TVd := Xi : n ≤ i ≤ d, N(Vd) := 1, X, X2, . . . , Xn−1;(2) for each d we have

TVd :=

Xa11 . . . Xar

r :∑

i

ai ≤ d, there is i : ai ≥ di

;

N(Vd) :=

Xa11 . . . Xar

r :∑

i

ai ≤ d, for each i : ai < di

.

Since V = ⋃d Vd it is sufficient to take the limit in order to obtain respectively

(1) TV := Xi : n ≤ i, N(V ) := 1, X, X2, . . . , Xn−1;(2)

TV := Xa11 . . . Xar

r : there is i : ai ≥ di ,N(V ) := Xa1

1 . . . Xarr : for each i : ai < di .

21.2 Gaussian Reduction

Definition 21.2.1. Let V be a k-vectorspace. A subset B ⊂ V will be called

• a Gauss generating set of V if TV = TB;• a Gauss basis of V if for each ei ∈ TV , there is a unique vi ∈ B such that

T(vi ) = ei .

Note that in the definition of a Gauss basis and a Gauss generating set wedo not require that V = Spank(B): this property can in fact be proved (seeProposition 21.2.5).

Moreover, in the definition of a Gauss generating set, while we require foreach ei ∈ TV the existence of an element vi ∈B such that T(vi ) = ei ,

uniqueness is not required.

Example 21.2.2. With the notation of Example 21.1.5,

• B := w1, w2, w3 is not a Gauss basis for two different reasons:

• T(w1) = T(w2) = e2 ∈ TV ,• there is no element w ∈ B : T(w) = e4 ∈ TV ,the second fact implies also that it is not even a Gauss generating set;

• w1, w2, w3, v is a Gauss generating set but not a Gauss basis of V , becauseT(w1) = T(w2);

• w1, v, w3 is a Gauss basis of V .

52 Gauss II

Example 21.2.3. Continuing the examples we have discussed from the begin-ning:

• in the case in which Vd := V ∩ Wd = g f, g ∈ k[X ], deg(g) ≤ d − n theobvious Gauss basis is the set

f, X f, X2 f, . . . , Xd−n f ,• so that for V = ( f ) ⊂ k[X ] the obvious Gauss basis is

f, X f, X2 f, . . . , Xi f, . . ..

Example 21.2.4. Alternatively, the case of multivariate polynomials is less ob-vious. Let us consider an easy example, in which we set

r = 2, f1 := X31, f2 := X2

2, d := 7 > 5 = d1 + d2, V = ( f1, f2).

An obvious Gauss generating set is

B := Xa11 Xa2

2 f1 : a1+a2 ≤ 4 = d−d1∪Xa11 Xa2

2 f2 : a1+a2 ≤ 5 = d−d2.If we want a Gauss basis we can extract it from B in different ways: in fact

TVd = X31, X4

1, X51, X6

1, X71,

X31 X2, X4

1 X2, X51 X2, X6

1 X2,

X22, X1 X2

2, X21 X2

2, X31 X2

2, X41 X2

2, X51 X2

2,

X32, X1 X3

2, X21 X3

2, X31 X3

2, X41 X3

2,

X42, X1 X4

2, X21 X4

2, X31 X4

2,

X52, X1 X5

2, X21 X5

2,

X62, X1 X6

2,

X72,

is partitioned into three subsets,

TVd = T1 T2 T12

where

T1 := X31, X4

1, X51, X6

1, X71, X3

1 X2, X41 X2, X5

1 X2, X61 X2,

T2 := X22, X1 X2

2, X21 X2

2, X32, X1 X3

2, X21 X3

2, X42, X1 X4

2, X21 X4

2,

X52, X1 X5

2, X21 X5

2, X62, X1 X6

2, X72,

T12 := X31 X2

2, X41 X2

2, X51 X2

2, X31 X3

2, X41 X3

2, X31 X4

2so that


• for each element Xa11 Xa2

2 ∈ T1 since a1 ≥ d1, a2 < d2 the only obvious

choice is f1 Xa1−d11 Xa2

2 ,

• and for each element Xa11 Xa2

2 ∈ T2 since a1 < d1, a2 ≥ d2 the only obvious

choice is f2 Xa11 Xa2−d2

2 ,

• while for each element Xa11 Xa2

2 ∈ T12, since a1 ≥ d1, a2 ≥ d2, one has two

alternative and equivalent choices, either f1 Xa1−d11 Xa2

2 , or f2 Xa11 Xa2−d2

2 .

The situation can be pictured if we represent each element of T as a memberin the lattice of the positive coordinates in the plane as follows – where weidentify X and Y with X1 and X2 respectively:

......

......

......

......

•Y 7 •XY 7 •X2Y 7 •X3Y 7 •X4Y 7 •X5Y 7 •X6Y 7 •X7Y 7 · · ·






•Y •XY •X2Y •X3Y •X4Y •X5Y •X6Y •X7Y · · ·

•1 •X •X2 •X3 •X4 •X5 •X6 •X7 · · ·Then we have

......

......

......

......

• · · ·• • · · ·• • • · · ·• • • · · ·• • • · · ·• • • · · · ∗ ∗ ∗ ∗ · · · ∗ ∗ ∗ ∗ ∗ · · ·

54 Gauss II

where:

represents the terms t ∈ TW \ TV7, that is such that deg(t) > 7,

∗ represents the terms t ∈ T1,• represents the terms t ∈ T2, represents the terms t ∈ T12, represents the terms t ∈ N(V ).

Proposition 21.2.5. Let W be a k-vector space, V ⊂ W and B ⊂ V a Gaussgenerating set of V.

Then we have:

(1) If w ∈ W is such that T(w) ∈ N(V ), then w ∈ V .

(2) If w ∈ W is such that T(w) ∈ TV , then exists w′ ∈ W :

• w − w′ ∈ V ,• T(w) > T(w′).

(3) For each w ∈ W, there is w ∈ W :

• w − w ∈ Spank(B),• either

• w = 0 in which case w ∈ V , or• w = 0 in which case T(w) ∈ N(V ), T(w) ≤ T(w), w ∈ V .

(4) V = Spank(B).

(5) If B is a Gauss basis, then it is a k-basis of V .

Proof.

(1) If w ∈ V then T(w) ∈ TV by definition.(2) Since T(w) ∈ TV , there is v ∈ V : T(v) = T(w) =: e j ; then let

w′ := w − lc(w)

lc(v)v;

clearly, in the representation w′ := ∑i∈I ci ei we have ci = 0 if ei >

e j since the coefficient of ei is 0 in both w and v; also

c j = lc(w) − lc(w)

lc(v)lc(v) = 0;

therefore T(w′) < e j = T(w).

(3) The argument can be proved by induction on T(w); the result being trivialfor w = 0, we can assume the statement proved for each w′ ∈ Wsuch that T(w′) < T(w).

If T(w) ∈ N(V ) then we just have to set w := w.


If T(w) ∈ TV we choose any element v ∈ B such that T(v) = T(w),and, as in the proof above, we define w′ := w − (lc(w)/ lc(v))v.

By inductive assumption, there is w ∈ W such that

• w′ − w ∈ Spank(B), impying

w − w = lc(w)

lc(v)v + (

w′ − w) ∈ Spank(B),

• and either

• w = 0, in which case w′ ∈ V and also w ∈ V , or• w = 0, in which case

T(w) ∈ N(V ), T(w) ≤ T(w′) < T(w),

w′ ∈ V and also w ∈ V .

(4) By the statement above, w ∈ Spank(B) for each w ∈ V .

(5) We have just to prove that B is linearly independent: since

for each w1, w2 ∈ B, w1 = w2 ⇒ T(w1) = T(w2),

for any non-zero linear combination w = ∑i ciwi of elements wi ∈

B, one has T(w) = max(T(wi )) = 0, so that w = 0.

Definition 21.2.6. Let V be a k-vectorspace, B be a basis of V , w ∈ V . Arepresentation

w =m∑

i=1

civi , ci ∈ k, ci = 0, vi ∈ B,

is called a Gauss representation in terms of B if T(w) ≥ T(vi ), for each i .

Corollary 21.2.7. Let W be a k-vector space, V ⊂ W and B ⊂ V . The fol-lowing conditions are equivalent:

(1) each w ∈ V has a Gauss representation w = ∑mi=1 civi in terms of B;

(2) each w ∈ V has a Gauss representation w = ∑mi=1 civi in terms of B

such that T(w) = T(v1) > T(vi ), for each i > 1;(3) each w ∈ V has a Gauss representation w = ∑m

i=1 civi in terms of Bsuch that T(w) = T(v1) > T(v2) > · · · > T(vm);

(4) B is a Gauss generating set of V .

56 Gauss II

Proof.

(1) ⇒ (4) In order to prove that TV = TB, let us consider an elementt ∈ TV and let w ∈ V be such that T(w) = t , w = ∑m

i=1 civi aGauss representation in terms of B; since T(vi ) ≤ T(w), for each i ,clearly exists i such that t = T(w) = T(vi ).

(4) ⇒ (3) This is a direct consequence of the computation outlined in theproof of Proposition 21.2.5(3).

(3) ⇒ (2) ⇒ (1) An obvious relaxation of conditions.

Remark 21.2.8. The reader may consider the statement of the slightly identicalconditions (1), (2) and (3) to be a supercilious pedantry and so it is in the settingof Gauss reduction; but the three conditions will have a different role when readwithin Grobner theory: (1) is all we need for most of the applications, (3) iswhat we get from Buchberger reduction, (2) is what we need in order to provethe Buchberger algorithm (see Remark 22.2.3 and Remark 22.3.5).

Algorithm 21.2.9. The computation outlined in the proof of Proposi-tion 21.2.5(3) can be formalized into an algorithm (Figure 21.1) whose in-put is a set B ⊂ W and an element w ∈ W and whose output will be anelement w ∈ W and a Gauss representation

∑mi=1 civi in terms of B such

that

(A) w − w = ∑mi=1 civi is a Gauss representation in terms of B;

(B) T(w) ∈ TB ⇒ T(w) = T(v1) > T(v2) > · · · > T(vm) > T(w);(C) T(w) ∈ TB ⇒ w = w, m = 0,

∑mi=1 civi = 0;

(D) w = 0 ⇒ T(w) ∈ TB.

Fig. 21.1. Gaussian Reduction

(w,∑m

i=1 ci vi ) := GaussianReduction(w,B)where

W is a k-vectorspace,B ⊂ W ,w ∈ W ,w ∈ W ,w − w = ∑m

i=1 ci vi is a Gauss representation in terms of B,conditions A, B, C, D above are satisfied.

w := w, i := 0,While T(w) ∈ TB do

Let v ∈ B : T(v) = T(w)i := i + 1, ci := lc(w)/ lc(v), vi := v,w := w − ci vi .

m := i .


Note that in presenting the algorithm we are making no assumption at allon B, which is not necessarily a Gauss generating set of Spank(B). As a con-sequence the properties of the output w will vary in the different situations asdiscussed in the next corollary.

Corollary 21.2.10. Let W be a k-vectorspace, B ⊂ W , V := Spank(B), w ∈W, (w,

∑mi=1 civi ) := GaussianReduction(w,B). Then:

(1) w ∈ V, w = 0 ⇒ TB = TV so that B is not a Gauss generatingset.

(2) If B is a Gauss generating set, then

• w − w ∈ Spank(B),• w − w = ∑m

i=1 civi is a Gauss representation in terms of B,• T(w) = T(v1) > T(v2) > · · · > T(vm) > T(w),• w = 0 ⇐⇒ w ∈ V .

Proof.

(1) In fact w ∈ V and T(w) ∈ TV \ TB;(2) This is a direct consequence of Proposition 21.2.5.

Example 21.2.11. In the same setting as in Example 21.1.5, we can considerthe set B := w1, w2, w3 and v ∈ Spank(B) for which

GaussianReduction(v,B) = (v, 0).In the same mood, if we consider w := w2 − v we have

GaussianReduction(w,B) = (−v, w2).Considering instead the Gauss generating set B := w1, v, w3, we get, forw2 ∈ Spank(B):

GaussianReduction(w2,B) = (0, − 13w1 + 1

3v).

Algorithm 21.2.12. The algorithm of Figure 21.1 allows us to check whether,when B is a Gauss generating set of V := Spank(B), an element w ∈ Wbelongs to V .

Conversely, if(w,

∑mi=1 civi ) := GaussianReduction(w,B)

is performed on an element w ∈ V and produces a non-zero output w, one candeduce that

TB TB ∪ T(w) ⊂ TV .Therefore, if GaussianReduction is iteratively applied to all the elements

of B as in Figure 21.2, it produces a Gauss basis

E := w1, w2, . . . , wi , . . .

58 Gauss II

Fig. 21.2. Gaussian Matrix Reduction

E := GaussBasis(B)where

W is a k-vectorspace,B ⊂ W ,Spank(E) = Spank(B),E is a Gauss basis,for each w1, w2 ∈ E, w1 < w2 ⇐⇒ T(w1) < T(w2).

E := ∅,While B = ∅ do

Choose w ∈ BB := B \ w(w,

∑mi=1 ci vi ) := GaussianReduction(w, E)

If w = 0 do w := lc(w)−1w, E := E ∪ wReorder E : w1 < w2 ⇐⇒ T(w1) < T(w2), for each w1, w2.

of Spank(B); once E has been re-ordered so that

for each w1, w2 ∈ E, w1 < w2 ⇐⇒ T(w1) < T(w2) (21.1)

then the matrix whose i th row is wi is an echelon matrix.

Example 21.2.13. In the same setting as in Example 21.1.5, the computationGaussBasis(w1, w2, w3, v)

will produce the computation

w := w1 := (0, −3, 0, 2, 0, 5, 3),

GaussianReduction(w, E) = (w, 0),w1 := (0, 1, 0, −2

3 , 0, −53 , −1), E := w1;

w := w2 := (0, 1, 0, 0, 0, 0, −1),

GaussianReduction(w, E) = ((0, 0, 0, 23 , 0, 5

3 , 0), w1),w2 := (0, 0, 0, 1, 0, 5

2 , 0), E := w1, w2;w := w3 := (0, 0, 1, 1, 0, 0, 0),

GaussianReduction(w, E) = ((0, 0, 1, 1, 0, 0, 0), 0),w3 := (0, 0, 1, 1, 0, 0, 0), E := w1, w2, w3;

w := v := (0, 0, 0, 2, 0, 5, 0),

GaussianReduction(w, E) = (0, 2w2),

E := w1, w3, w2.Algorithm 21.2.14. In case B is a Gauss generating set of Spank(B), as Gaus-sian reduction (Figure 21.1) allows us to check whether an element w ∈ W


Fig. 21.3. Complete Gaussian Reduction

(w,∑m

i=1 ci vi ) := CompleteGaussianReduction(w,B)where

W is a k-vectorspace,B ⊂ W ,w ∈ W ,w ∈ Spank(N(B)),w − w = ∑m

i=1 ci vi is a Gauss representation in terms of B,T(w − w) = T(v1) > T(v2) > · · · > T(vm).

w := w, i := 0, w := 0,While w = 0 do

%% w = w + ∑ij=1 c j v j + w,

%% T(w − w) ≥ T(w),%% i > 0 ⇒ T(w − w) = T(v1) > T(v2) > · · · > T(vi ) > T(w);t := T(w)If t ∈ TB do

Let v ∈ B : T(v) = T(w)i := i + 1, ci := lc(w)/ lc(v), vi := v,w := w − ci vi .

Else%% t ∈ N(B)w := w − M(w), w := w + M(w)

m := i

belongs to Spank(B), complete Gaussian reduction (Figure 21.3) allows tocompute for each element w ∈ W a canonical representation

w ∈ Spank(N(B)) mod Spank(B).

Lemma 21.2.15. Let W be a k-vectorspace, B ⊂ W , V := Spank(B). Letw ∈ W, and let

w, w, w ∈ W ; c j ∈ k, c j = 0, v j ∈ B, 0 ≤ j ≤ i ,

be such that

A1 w ∈ Spank(N(B)), w ∈ Spank(B);A2 w = w + w + w;A3 w = ∑i

j=1 c jv j is a Gauss representation in terms of B;A4 T(w) ≥ T(v1) > T(v2) > · · · > T(vi ) > T(w);A5 either

• T(w) < T(w) = T(v1) ∈ TB or• T(v1) < T(w) = T(w) ∈ N(B).

60 Gauss II

If B is a Gauss generating set, then

(1) If t := T(w) ∈ TB, let

j := i + 1, v j ∈ B : T(v j ) = T(w), c j := lc(w)

lc(v j )

and define

w′ := w, w′ := w − c jv j , w := w + c jv j .

Then

A1 w′ ∈ Spank(N(B)), w′ ∈ Spank(B),A2 w = w′ + w′ + w′,A3 w′ = ∑i+1

j=1 c jv j is a Gauss representation in terms of B,A4 T(w) ≥ T(v1) > T(v2) > · · · > T(vi ) > T(vi+1) > T(w′),A5 either

• T(w′) < T(w) = T(v1) ∈ TB or• T(v1) < T(w) = T(w′) ∈ N(B).

(2) If t := T(w) ∈ N(B), define

w′ := w + M(w), w′ := w − M(w), w′ := w.

Then

A1 w′ ∈ Spank(N(B)), w′ ∈ Spank(B),A2 w = w′ + w′ + w′,A3 w′ = ∑i

j=1 c jv j is a Gauss representation in terms of B,A4 T(w) ≥ T(v1) > T(v2) > · · · > T(vi ) > T(w) > T(w′),A5 either

• T(w′) < T(w) = T(v1) ∈ TB or• T(v1) < T(w) = T(w′) ∈ N(B).

Corollary 21.2.16. Let W be a k-vectorspace, B ⊂ W , V := Spank(B). If Bis a Gauss generating set, then

(1) For each w ∈ W , there are

w, w ∈ W , and c j ∈ k \ 0, v j ∈ B, 0 ≤ j ≤ i,


such that

A1 w ∈ Spank(N(V )), w ∈ Spank(B);A2 w = w + w;A3 w = ∑i

j=1 c jv j is a Gauss representation in terms of B;A4 T(w) ≥ T(v1) > T(v2) > · · · > T(vi );A5 if i > 0, either

• T(w) < T(w) = T(v1) ∈ TB or• T(v1) < T(w) = T(w) ∈ N(B).

The vector w is unique; if moreover B is a Gauss basis also w, c j , v j

are unique.(2) W ∼= V ⊕ Spank(N(V ));(3) W/V ∼= Spank(N(V ));(4) for each w ∈ W, there is a unique w := Can(w, V ) ∈ Spank(N(V ))

such that w − w ∈ V .Moreover:

(a) Can(w1, V ) = Can(w2, V ) ⇐⇒ w1 − w2 ∈ V ;(b) Can(w, V ) = 0 ⇐⇒ w ∈ V .

Example 21.2.17. In the same setting as in Examples 21.1.5 and 21.2.13, letus consider the Gauss basis E := w1, w2, w3 and the element

w := (−3, −2, −1, 0, 1, 2, 3)

and let us compute CompleteGaussianReduction(w, E):

w := (−3, −2, −1, 0, 1, 2, 3),

w := (0, 0, 0, 0, 0, 0, 0),

w := (0, 0, 0, 0, 0, 0, 0);t := e1 ∈ N(E), M(w) = (−3, 0, 0, 0, 0, 0, 0),

w := (0, −2, −1, 0, 1, 2, 3),

w := (−3, 0, 0, 0, 0, 0, 0),

w := (0, 0, 0, 0, 0, 0, 0);t := e2 ∈ TE, c1 := −2, v1 := w1 := (0, 1, 0, −2

3 , 0, −53 , −1),

w := (0, 0, −1, −43 , 1, −4

3 , 1),

w := (−3, 0, 0, 0, 0, 0, 0),

w := (0, −2, 0, 43 , 0, 10

3 , 2);t := e3 ∈ TE, c2 := −1, v2 := w3 := (0, 0, 1, 1, 0, 0, 0),

w := (0, 0, 0, −13 , 1, −4

3 , 1),

w := (−3, 0, 0, 0, 0, 0, 0),

w := (0, −2, −1, 13 , 0, 10

3 , 2);

62 Gauss II

t := e4 ∈ TE, c3 := −13 , v3 := w2 := (0, 0, 0, 1, 0, 5

2 , 0),

w := (0, 0, 0, 0, 1, −12 , 1),

w := (−3, 0, 0, 0, 0, 0, 0),

w := (0, −2, −1, 0, 0, 52 , 2);

t := e5 ∈ N(E), M(w) = (0, 0, 0, 0, 1, 0, 0),

w := (0, 0, 0, 0, 0, −12 , 1),

w := (−3, 0, 0, 0, 1, 0, 0),

w := (0, −2, −1, 0, 0, 52 , 2);

t := e6 ∈ N(E), M(w) = (0, 0, 0, 0, 0, −12 , 0),

w := (0, 0, 0, 0, 0, 0, 1),

w := (−3, 0, 0, 0, 1, −12 , 0),

w := (0, −2, −1, 0, 0, 52 , 2);

t := e7 ∈ N(E), M(w) = (0, 0, 0, 0, 0, 0, 1),

w := (0, 0, 0, 0, 0, 0, 0),

w := (−3, 0, 0, 0, 1, −12 , 1),

w := (0, −2, −1, 0, 0, 52 , 2).

Example 21.2.18. Continuing the discussion begun in Example 21.1.1 in theset of univariate polynomials (see Example 21.2.3), where W := k[X ] and wefix V = ( f ) ⊂ k[X ] for a generic polynomial

f (X) :=n∑

i=0

ai Xn−i = a0 Xn + a1 Xn−1 + · · · + ai Xn−i + · · · + an−1 X + an,

such that a0 = 0, we have

• the Gauss basis E = Xi f : i ∈ N,• TE = TV = Xi : i ≥ n,• N(E) = N(V ) = Xi : i < n and• k[X ]/( f ) ∼= Spank(1, X, . . . , Xn−1).

In this setting both Gaussian reduction and complete Gaussian reductioncoincide with the Polynomial Division Algorithm (see Algorithm 1.1.3). Inparticular, for each g ∈ W

(Q, R) := PolynomialDivision(g, f )

and (w,

µ∑i=0

ci Xi f

):= CompleteGaussianReduction(g, E)

are related by Q = ∑µi=0 ci Xi and R = w.

21.3 Gauss and Euclid Revisited 63

Example 21.2.19. Continuing now the discussion of the multivariate case (seeExample 21.2.4), where 5 W := K0[X1, . . . , Xr ] and V is the ideal generatedby a sequence f1, . . . , fr ∈ K0[X1, . . . , Xr ] such that

• f1 ∈ K0[X1] is monic,• fi ∈ K0[X1, . . . , Xi−1][Xi ] is monic for each i ,• writing d j := deg j ( f j ) we have deg j ( fi ) < d j , j < i,

one has

• TV := Xa11 . . . Xar

r : there exists i : ai ≥ di ,• N(V ) := Xa1

1 . . . Xarr : ai < di for each i = B,

• E := t fi : t ∈ T, 1 ≤ i ≤ r is a Gauss generating set,• K ∼= K0[X1, . . . , Xr ]/( f1, . . . , fr ) ∼= K0[B] = SpanK0

(N(V )).

In this setting complete Gaussian reduction (but not Gaussian reduction)coincides verbatim with the Canonical Representation Algorithm (see Algo-rithm 8.3.1). In particular, for each g ∈ W

h := Reduction(g, f1, . . . , fr )and (

w,

m∑i=1

civi

):= CompleteGaussianReduction(g, E)

are related by h = w.

21.3 Gaussian Reduction and Euclidean Algorithm Revisited

While the algorithm of Figure 21.2, given a finite set B ⊂ W , allows us toproduce a finite Gauss basis E , we need a different approach to deal (as wewill need to in the next chapter) with a finite computation when B is infinite.

The informal approach we will follow requires us to alternate some finitecomputation with some recursive arguing; what we can do here is just set thenecessary notions and illustrate an informal scheme of computation using aconcrete example.

Definition 21.3.1. A set L ⊂ W is called an echelon set iff

for each w1, w2 ∈ L, w1 = w2 ⇒ T(w1) = T(w2).

Let B ⊂ W be a well-ordered generating set; a subset L ⊂ B such that

5 With the notation of Section 8.3.1

64 Gauss II

• L is an echelon set,• TL = TB,• for each v ∈ L, w ∈ B, T(v) = T(w) ⇒ v < w

will be called the canonical echelon set extracted from B.

In the definition of canonical echelon sets, the requirement that B be well-ordered is needed so that for each t ∈ TB a ‘canonical’ element w(t) ∈ Bsuch that T(w(t)) = t can be chosen to be inserted in L. Any well-ordering ofB can be used for this, which essentially means that each appropriate elementw(t) can be chosen as ‘canonical’.

Remark 21.3.2. Let B ⊂ W , V := Spank(B), and L be the canonical echelonset extracted from B.

Then B is a Gauss generating set of V iff L is a Gauss basis of V .In fact by construction TL = TB, so if one of these is equal to TV the

same holds for the other. Also for each t ∈ TV the uniqueness of the elementv ∈ L, such that T(v) = t , follows by construction.

Remark 21.3.3. We can now consider the difference between Gauss generatingsets and Gauss bases with respect to the notion of Gauss representation andstress the role of the requirement of the non-existence of elements v1, v2 ∈ Lsuch that T(v1) = T(v2).

Let w ∈ V = Spank(B), and let w = ∑ni=1 civi be any linear combination

of elements vi ∈ B.

This combination is not a Gauss representation in terms of B iff there existssome vi such that T(vi ) > T(w); of course this means that, if we denote τ :=maxT(vi ) and Λ := λ : T(vλ) = τ , we have

τ > T(w), #Λ > 1,∑λ∈Λ

cλ = 0.

Conversely, in any representation w = ∑ni=1 civi by elements vi belonging

to the echelon set L, still writing τ := maxT(vi ) and Λ := λ : T(vλ) = τ ,

#Λ = 1,∑λ∈Λ

cλ = 0, τ = T(w)

and the representation is a Gauss representation.

Corollary 21.3.4. Let B ⊂ W , V := Spank(B). Let B be ordered by a well-ordered ≺ such that T(w1) < T(w2) ⇒ w1 ≺ w2 and L be the canonicalechelon set extracted from it.


Then the following conditions are equivalent:

(1) L is a Gauss basis of V ;(2) B is a Gauss generating set of V ;(3) for each v ∈ B \ L and v ∈ B such that T(v) = T(v) and v ≺ v,

v − (lc(v)/ lc(v))v has a Gauss representation in terms of B;(4) for each v ∈ B \L and v ∈ B such that T(v) = T(v) and v ≺ v, v has

a Gauss representation v = (lc(v)/ lc(v))v + ∑mi=2 civi in terms of B

such that T(v) = T(v) > T(vi ) for i > 1;(5) for each v ∈ B \ L, denoting by v the unique element in L such that

T(v) = T(v), v has a Gauss representation v = (lc(v)/ lc(v))v +∑mi=2 civi in terms of B such that T(v) = T(v) > T(vi ) for i > 1;

(6) for each v ∈ B \ L, denoting by v the unique element in L such thatT(v) = T(v), v has a Gauss representation v = (lc(v)/ lc(v))v +∑m

i=2 civi in terms of L such that T(v) = T(v) > T(vi ) for i > 1;(7) each v ∈ V has a Gauss representation in terms of L.

Proof.

(1) ⇐⇒ (2) is Remark 21.3.2.(2) ⇒ (3) is just a reformulation of the algorithm of Figure 21.1.(3) ⇐⇒ (4) is obvious.(4) ⇒ (5) is obvious.(5) ⇒ (6) Assume this is false and let v ∈ B \L be the minimal counterex-

ample w.r.t. ≺, in the sense that the statement holds for each v′ ∈ B\Lsuch that v′ ≺ v.

Therefore in a Gauss representation

v =m∑

i=1

civi , v vi , for each i, (21.2)

whose existence is implied by (5), for each i either

• vi ∈ L or• vi has a Gauss representation

vi = lc(vi )

lc(vi )vi +

µi∑j=1

γi jvi j

in terms of L, where vi is the unique element in L such that T(vi ) =T(vi ).

Substituting in (21.2) for each vi ∈ L with its Gauss representation,we obtain a Gauss representation in terms of L also for v.

66 Gauss II

(6) ⇒ (7) Let w ∈ V = Spank(B), and let w = ∑ni=1 civi be any linear

combination of elements vi ∈ B. By assumption, each element vi ∈ Bhas a Gauss representation w = ∑ni

j=1 γi j vi j in terms of L.

Therefore, because of the argument of Remark 21.3.3

w =n∑

i=1

ni∑j=1

ciγi j vi j

is the required Gauss representation of w in terms of L.

(7) ⇒ (1) Corollary 21.2.7 implies that L is a Gauss generating set; theconstruction gives the non-existence of elements v1, v2 ∈ L such thatT(v1) = T(v2), thus implying that it is a Gauss basis.

Example 21.3.5. Completing Example 21.2.19, if we order the basis E setting

t fi ≺ τ f j ⇐⇒ tT( fi ) < τT( f j ) or tT( fi ) = τT( f j ), i < j,

we obtain the canonical echelon set

L :=r⋃

i=1

Xa1

1 . . . Xarr fi , a j < d j for each j < i

=r⋃

i=1

t fi , t ∈ T, t /∈ (T( f1), . . . , T( fi−i ))

which is then a Gauss basis.

Example 21.3.6. Let us again consider (see Example 21.2.18) W := k[X ]; thetwo polynomials

P0(X) := f (X) :=n∑

i=0

ai Xn−i = a0 Xn + · · · + ai Xn−i + · · · + an,

P1(X) := g(X) :=m∑

j=0

b j Xm− j = b0 Xm + · · · + b j Xm− j + · · · + bm,

with a0 = 0 = b0, d0 := n > m =: d1; and the ideal V := ( f, g).An obvious generating set is

B(1) := Xi P1(X) : i ∈ N ∪ Xi P0(X) : i ∈ Nwhich we consider well-ordered as

P1 ≺ X P1 ≺ · · · Xi P1 ≺ · · · ≺ P0 ≺ X P0 ≺ · · · ≺ Xi P0 ≺ · · · .Note that if we set d := n + m − 1 and consider the matrix whose rows

are the representation in terms of the basis 1, X, X2, . . . , Xd of the ordered


elements of the generating set B(1) whose degree is bounded by d we obtainexactly the Sylvester matrix (Definition 6.6.1).

Since TB(1) = Xi : i ≥ d1, the canonical echelon set extracted fromB(1) is L(1) := Xi P1(X) : i ∈ N.

In order to check whether L(1) is a Gauss basis of V by application of Corol-lary 21.3.4(4), we must check whether, for each i ∈ N, Xi P0 has a Gaussrepresentation

Xi P0 = a0b−10 Xn+i−m P1 +

n+i−m−1∑j=0

ci j X j P1

in terms of L(1) .Considering the first case (i = 0), as we remarked in Example 21.2.18, if

Q1(X) = a0b−10 Xn−m +

n−m−1∑j=0

c j X j

and P2(X) are such that

• P0 = Q1 P1 + P2,• deg(P2) =: d2 < d1,

we have that

• if P2 = 0 then T(P2) ∈ TB(1), and B(1) is not a Gauss generating set;• P2(X) = Can(P0,L(1));• P0 − P2 = a0b−1

0 Xn−m P1(X)+∑n−m−1j=0 c j X j P1(X) is the required Gauss

representation in terms of L(1).

Therefore

if P2(X) = 0, we have found the required Gauss representation of P0 interms of L(1), while

if P2(X) = 0 we have proved that B(1) is not a Gauss generating set,displaying an element P2(X) which belongs to V but not in the vec-torspace generated by L(1), since T(P2) ∈ TB(1), and which, there-fore, should be inserted in L(1).

The next computation (i = 1) is the most crucial. The computation alreadyperformed allows us to deduce the relation

X P0 − X P2 = a0b−10 Xn−m+1 P1(X) +

n−m∑j=1

c j−1 X j P1(X)

68 Gauss II

which

when P2(X) = 0, gives the required Gauss representation of X P0 in termsof L(1), while

if P2(X) = 0, we cannot reach any conclusion since generically it happensthat

• deg(X P2) ≥ d1,• T(X P2) ∈ TB(1),• X P2(X) = Can(X P0,L(1)),

• and the relation deduced is not sufficient to produce the requiredGauss representation of X P0.

The same happens in the general case (i ≥ 1): we have the relation

Xi P0 − Xi P2 = a0b−10 Xn−m+i P1(X) +

n−m+i−1∑j=i

c j−i X j P1(X)

which when P2(X) = 0, gives the required Gauss representation of Xi P0 interms of L(1),

while if P2(X) = 0, for i 0

• deg(Xi P2) ≥ d1,• T(Xi P2) ∈ TB(1),• Xi P2(X) = Can(Xi P0,L(1)),

• and the relation deduced is not sufficient to produce the requiredGauss representation of Xi P0.

As a consequence:

if P2(X) = 0, we conclude that L(1) is the required Gauss basis, while if P2(X) = 0, we can only define

B(2) := Xi P2(X) : i ∈ N ∪ Xi P1(X) : i ∈ Nand conclude that

• Spank(B(2)) = Spank(B(1)) = V,

• TV ⊇ TB(2) = Xi : i ≥ d2 TB(1)and we find ourselves with a better approximation but essentially inthe same situation as before, so that we can re-apply the same ap-proach.

Example 21.3.7. To complete the computation begun in the example above,we use the same notation as in Section 1.2, so that we consider the polynomial


remainder sequence P0, P1, . . . , Pλ, . . . , Pr , Pr+1 = 0 and the polynomialsQλ satisfying the relations

• d0 > d1 > d2 > · · · > dλ > · · · > dr ,• Pλ−1 = Qλ Pλ + Pλ+1,

where we write

• dλ := deg(Pλ), so that T(Pλ) = Xdλ and• Qλ := Qλ − lc(Pλ−1) lc(Pλ)

−1 Xdλ−1−dλ .

With this notation, we can interpret the Euclidean algorithm (Section 1.2) interms of Gaussian reduction (Figure 21.2) as follows.

Iteratively (1 ≤ λ ≤ r ) we define:

B(λ) := Xi Pλ(X) : i ∈ N ∪ Xi Pλ−1(X) : i ∈ Nwhich we consider well-ordered as

Pλ ≺ X Pλ ≺ · · · ≺ Xi Pλ ≺ · · · ≺ Pλ−1 ≺ X Pλ−1 ≺ · · · ≺ Xi Pλ−1 ≺ · · ·so that TB(λ) = Xi : i ≥ dλ, and the canonical echelon set extracted fromB(λ) is L(λ) := Xi Pλ(X) : i ∈ N.

The Polynomial Division Algorithm gives us, for each i , the relation

Xi Pλ−1 − Xi Pλ+1 = lc(Pλ−1) lc(Pλ)−1 Xdλ−1−dλ+i Pλ + Qλ Xi Pλ;

therefore

for λ < r ,

• Pλ+1(X) = 0,• and for i 0

• deg(Xi Pλ+1) ≥ dλ,

• T(Xi Pλ+1) ∈ TB(λ),• Xi Pλ+1(X) = Can(Xi Pλ−1,L(λ)),

• the relation deduced is not sufficient to produce the requiredGauss representation of Xi Pλ−1,

• but Spank(B(λ+1)) = Spank(B(λ)) = V,

• TV ⊇ TB(λ+1) = Xi : i ≥ dλ+1 TB(λ) · · ·

TB(01); for λ = r , since Pr+1 = 0, each polynomial Xi Pr−1 has the required Gauss

representation in terms of L(r) so that

• B(r) is a Gauss generating basis,• TV = TB(r) = Xi : i ≥ dr ,

70 Gauss II

Fig. 21.4. Gaussian Echelon Procedure

E := GaussianEchelon(B)where

W is a k-vectorspace,B ⊂ W ,V := Spank(B),E is a Gauss basis of V .

RepeatImpose a well-ordering < on B,Extract from B a canonical echelon set L,B′ := B \ L,Choose B′′ ⊂ B′,B′ := B′ \ B′′,For each w ∈ B′′ do

Compute N F(w) such thatT(N F(w)) < T(w),w − N F(w) has a Gauss representation in terms of L,

B∗ := N F(w) : w ∈ B′′ \ 0,B := L ∪ B′ ∪ B∗,

until B is an echelon set.E := B

• L(r) = Xi Pr (X) is a Gauss basis,• ( f, g) = V = (Pr ).

Algorithm 21.3.8. Our interpretation of the Euclidean algorithms in terms ofGaussian reduction leads us to mimic that ‘computation’ and to sketch inFigure 21.4 a ‘procedure’ which takes advantage of Corollary 21.3.4 in orderto extract from B a Gauss basis E of V := Spank(B).

Because the ‘procedure’ aims at the case in which B is infinite, thereis no guarantee of termination nor is it assumed that the procedure satis-fies Hermann’s endlichvielen Schritten (see the note on Algorithm 1.1.3.)assumption.

Throughout this chapter we have discussed two main examples: theEuclidean algorithms and canonical representation modulo an admissiblesequence within the Kronecker–Duval Model; since the latter is the multivari-ate extension of the former, it is worth investigating whether the computationdeveloped in Examples 21.3.6 and 21.3.7 and sketched in Figure 21.4 can beextended to the Kronecker–Duval Model.

It can and in doing so introduces Buchberger’s algorithm and the notion ofGrobner Bases.

Historical Remark 21.3.9. I am personally convinced that the route whichleads to Grobner bases and Buchberger’s algorithm is essentially the one which


is followed in this book: starting from the Euclidean algorithms as a tool forbuilding single extensions, Kronecker generalized it to the multivariate caseproducing his model for algebraic numbers. A crucial link is Macaulay’s re-search aimed at injecting effective linear algebra into Kronecker theory. Hisgoal was successfully attained by Grobner and Buchberger.

22

Buchberger

For each field k, denoting k its algebraic closure, the Euclidean algorithmsallow us to represent the roots α ∈ k of any set of univariate equations

f1(X) = f2(X) = · · · = fm(X) = 0, fi ∈ k[X ]

by means of the greatest common divisor, g := gcd( f1, . . . , fm) ∈ k[X ], sothat

for each α ∈ k, fi (α) = 0, for each i ⇐⇒ g(α) = 0

and each such root α is represented by the projection

k[X ]/g(X) k[α] ⊂ k.

The Kronecker–Duval model generalized this approach in order to deal withthe successive introduction of univariate roots expressed in terms of the previ-ous ones by successive application of Euclidean tools.

If we disregard the crucial problems of zero-testing and inverse computationof an algebraic expression, which were discussed in the previous volume andwhich led Kronecker to restrict the notion of admissible sequence to irreduciblepolynomials and Duval to relax this restriction to squarefree ones, we could tryto relax further this restriction and, keeping in mind Remark 20.4.5, give

Definition 22.0.1. A set

f1, . . . , fr ∈ k[X1, . . . , Xn] = k[Y1, . . . , Yd ][Z1, . . . , Zr ], n = d + r,

will be called a weak admissible sequence 1 if, for each i ,

deg j ( fi ) < d j := deg j ( f j ), for each j < i,

1 This informal definition is of course connected with the notions of complete intersection andregular sequence. For those, see Definition 36.1.1.

As regards the existence of such weak admissible sequences and the assumptions for this, seeChapter 34.

72

Buchberger 73

and fi (Y1, . . . , Yd , Z1, . . . , Zi ) ∈ k[Y1, . . . , Yd ][Z1, . . . , Zi−1][Zi ] has theshape

fi = qi (Y1, . . . , Yd)Zdii +

di −1∑j=0

pi j (Y1, . . . , Yd , Z1, . . . , Zi−1)Z ji

thus being monic in k(Y1, . . . , Yd)[Z1, . . . , Zi−1][Zi ].

This would allow us to restrict the duality given by Z and I to weak admis-sible sequences and ‘suitable’ algebraic varieties; it however requires strongassumptions, the most important being a preliminary application of a renum-bering of the variables.

Under this restriction we can then contemplate a computational model inwhich a ‘suitable’ set of roots Z ⊂ kn would be represented by giving a weakadmissible sequence such that each a := (α1, . . . , αn) ∈ Z satisfies fi (a) = 0,

for each i, and represent each such root a by means of the projection

A := k[X1, . . . , Xn]/( f1, . . . , fr ) k[α1, . . . , αn] ⊂ k.

The requirement imposed by Kronecker on admissible sequences that eachfi be irreducible over the field

Ai−1 := k(Y1, . . . , Yd)[Z1, . . . , Zi−1]/( f1, . . . , fi−1),

so that each ideal Ii := ( f1, . . . , fi ) is prime, was needed in order for A to bea field itself, allowing zero-testing and inverse computation.

Having relaxed the requirement of irreducibility of the fi s to be just square-free over Ai−1, Duval guaranteed that Ii is radical, and A is a Duval field,zero-testing and inverse computation being granted by Duval splitting.

Our hypothetical relaxed notion of weak admissible sequence, in which norequirement is imposed on the fi s except being monic, has the effect that

I = ( f1, . . . , fr ) = I(Z)

is just an ideal, A is just a ring and division of algebraic expressions cannot bedealt with by this model.

Moreover, even if the ideals I = ( f1, . . . , fr ) generated by a weak admissi-ble sequence ( f1, . . . , fr ) are naturally restricted to be zero-dimensional, thatis d = 0, n = r and Z(I) is a finite set, most zero-dimensional ideals I andcorresponding sets of roots Z(I) are not representable by such ‘admissible’sequences – the obvious example being the set of roots

Z := (0, 0), (1, 0), (0, 1) ∈ k2

74 Buchberger

whose corresponding ideal is

I(Z) := (X21 − X1, X1 X2, X2

2 − X2) ∈ k[X1, X2],

unless a splitting is forced, which however would be unnatural, unlikeKronecker factorization and Duval splitting.2

On the other hand, if we drop even the notion of weak admissible sequence,and just consider a generic ideal

I := ( f1, . . . , fs) ⊂ k[X1, . . . , Xn]

without imposing any condition 3 on its basis, we lose the crucial point of theKronecker–Duval Model, its ability to deal at least with multiplication: in factwe have dropped our interest in division. Addition and subtraction are in anycase given by the k-vectorspace structure of

A := k[X1, . . . , Xn]/I,

but multiplication in A was guaranteed in the univariate case by the DivisionAlgorithm (Algorithm 1.1.3) and, within the Kronecker–Duval Model, by itsgeneralization, the Canonical Representation Algorithm (Section 8.3.3).

Our next tasks, therefore, are to

• adapt the notion of admissible sequences in such a way as to perform thecomputation of canonical representations modulo I, in order to generalizeboth the Division and the Canonical Representation Algorithm, thus leadingto the notion of Grobner basis, and

• mimic the Gaussian algorithm interpretation of the Euclidean algorithm, dis-cussed in Section 21.3, thus leading to Buchberger’s algorithm.

In this way we can deal effectively with multiplication in A and represent eachroot

a := (α1, . . . , αn) ∈ Z(I)

2 An alternative and effective approach is to perform a ‘generic’ change of coordinates in order tohave I in ‘good position’ (see Section 35.6).

For instance, in the same example, if we perform the change of coordinates Y1 := X1 +cX2, Y2 := X2, for each c ∈ k \ 0, 1 we obtain

Z := (0, 0), (1, 0), (c, 1) ∈ k2

and

I(Z) :=(

Y 31 − (c + 1)Y 2

1 + cY1, Y2 − (c2 − c)−1Y 21 + (c2 − c)−1Y1

)∈ k[Y1, Y2].

3 Not even the requirement that I be zero-dimensional, that is that it have only finite solutions.

22.1 From Gauss to Grobner 75

by means of the ring projection

A := k[X1, . . . , Xn]/I k[α1, . . . , αn] ⊂ k.

22.1 From Gauss to Grobner

Let us therefore consider the polynomial ring P := k[X1, . . . , Xn] as a k-vectorspace generated by the basis

T := Xa11 . . . Xan

n : (a1, . . . , an) ∈ Nn.

In order to apply the notation and procedures discussed in the previous chap-ter, in particular to define T( f ), for any f ∈ P , we need T to be well-ordered.A further requirement on the well-ordering < is to be imposed: since we willdeal with the linear algebra structure of ideals, let us consider what distin-guishes an ideal from a generic vectorspace I ⊂ P:

Corollary 22.1.1. Let I ⊂ P be a k-subvectorspace. Then I is an ideal iff foreach f (X1, . . . , Xn) ∈ I and j ≤ n, X j f (X1, . . . , Xn) ∈ I.

It is therefore natural 4 to require that the definition of T( · ) will be pre-served by multiplication by variables:

for each f ∈ P, and i ≤ n, T(Xi f ) = Xi T( f );as a consequence we will introduce

Definition 22.1.2. An ordering < on T will be called

• a semigroup ordering if for each t, t1, t2 ∈ T :

t1 < t2 ⇒ t t1 < t t2,

• a term ordering if it is a well-ordering and a semigroup ordering.

4 Notwithstanding that I have often challenged the assumptions of Grobner theory in orderto generalize them as much as possible, I have always considered this assumption, that <

must be a semigroup ordering, as essential until a Grobner basis theory for group algebraswas independently provided in K. Madlener and B. Reinert, Computing Grobner bases inmonoid and group rings, Proc. ISSAC ’93, ACM (1993), 254–263 and A. Rosenmann, Analgorithm for constructing Grobner and free Schreier bases in free group algebras, J. Symb.Comp. 16 (1993), 523–549, simply by not assuming that the orderings were compatible withthe product and performing elementary modifications to the theory: for knowledgeable read-ers, they just assumed that a Grobner basis element could have more than a single leadingterm.

Their result, which can be easily presented in the same way as in this chapter, will probablybe discussed in the next volume.

76 Buchberger

Once a term ordering < is fixed, each polynomial f (X1, . . . , Xn) ∈ P hasa unique ordered representation as an ordered linear combination of the termst in T with coefficients in k:

f =s∑

i=1

c( f, ti )ti : c( f, ti ) ∈ k \ 0, ti ∈ T , t1 > · · · > ts .

Then we will denote by

• T( f ) := t1, the maximal term of f ,• lc( f ) := c( f, t1), the leading cofficient of f ,• M( f ) := c( f, t1)t1, the maximal monomial of f .

Lemma 22.1.3. Let < be a semigroup ordering.Then for each f (X1, . . . , Xn) ∈ I and j ≤ n, T(X j f ) = X j T( f ).

Proof. Let f = ∑si=1 c( f, ti )ti , t1 > · · · > ts; then

X j f =s∑

i=1

c( f, ti )X j ti =s∑

i=1

c(X j f, X j ti )X j ti , X j t1 > · · · > X j ts .

An essential tool in the development of Grobner theory is (Gordan’s)Dickson’s Lemma (Corollary 20.8.4), which, in this context,

• proves that Grobner bases are finite,• explicitly provides the finite basis of an ideal whose existence is implied by

Hilbert’s Basissatz,• guarantees termination of Buchberger’s algorithm, and• guarantees the existence and computability of canonical forms.

It is worth noting that in the non-commutative case the corresponding theoryis haunted by the insolvability of the Word Problem, which implies that, ingeneral,

• Grobner bases are infinite,• Buchberger’s algorithm terminates and canonical forms are computable only

when they are finite;• moreover the existence for a given ideal of a finite Grobner basis w.r.t. any

term ordering is an insolvable problem.

For our development we also need the following corollary of Corol-lary 20.8.4:

Corollary 22.1.4. Let < be a semigroup ordering; then the following condi-tions are equivalent:

22.1 From Gauss to Grobner 77

(1) < is a term ordering;(2) < is a well-ordering;(3) for each j , X j > 1;(4) for each t ∈ T , t ≥ 1;(5) for each t1, t2 ∈ T , t1 = t2, t1 | t2 ⇒ t1 < t2.

Proof.

(2) ⇒ (3) Assume the existence of j : X j < 1; then

1 > X j > X2j > · · · > Xρ

j > Xρ+1j > · · ·

would be an infinite decreasing sequence, contradicting the assump-tion.

(3) ⇒ (4) Obvious.(4) ⇒ (5) By assumption there is t ∈ T \ 1 such that t t1 = t2; since <

is a semigroup ordering 1 < t ⇒ t1 = 1t1 < t t1 = t2.(5) ⇒ (2) Assume < is not a well-ordering; then there is an infinite se-

quence

t1 > t2 > · · · > ti > · · · ,contradicting Corollary 20.8.4 which implies the existence of N ∈ N

such that for each i > N there is j ≤ N ti .

For any set F ⊂ P let us write

• TF := T( f ) : f ∈ F;• T(F) := τT( f ) : τ ∈ T , f ∈ F;• N(F) := T \ T(F);• k[N(F)] := Spank(N(F)).

Lemma 22.1.5. Let I ⊂ P be an ideal, then:

(1) TI = T(I);(2) T(I) ⊂ T is a monomial ideal;(3) N(I) ⊂ T is an order ideal that is

t1t2 ∈ N(I) ⇒ t1 ∈ N(I).

Proof.

(1) Let t ∈ TI, τ ∈ T ; by definition there is f ∈ I : T( f ) = t. Since < is aterm ordering, τ t = T(τ f ) ∈ TI.

(2) This is a reformulation of the previous statement.(3) t1 ∈ T(I) would imply t1t2 ∈ T(I) by (1).

78 Buchberger

22.2 Grobner Basis

Let us fix a term ordering < and let I ⊂ P be an ideal, A := P/I.

Definition 22.2.1 (Buchberger). A subset G ⊂ I will be called a Grobnerbasis of I if

T(G) = TI,that is TG generates the monomial ideal T(I) = TI.

We say that f ∈ P \ 0 has

• a Grobner representation in terms of G if it can be written as

f =m∑

i=1

pi gi ,

with pi ∈ P, gi ∈ G and T(pi )T(gi ) ≤ T( f ) for each i;• a (strong) Grobner representation in terms of G if it can be written as

f =µ∑

i=1

ci ti gi ,

with ci ∈ k \ 0, ti ∈ T , gi ∈ G and

T( f ) = t1T(g1) > · · · > ti T(gi ) > · · · .

Lemma 22.2.2. For G ⊂ I, the following conditions are equivalent:

G1 G is a Grobner basis of I;G2 tg : g ∈ G, t ∈ T is a Gauss generating set.

Proof. Both statements are equivalent to

TI = T(tg) : g ∈ G, t ∈ T = T(G).

Remark 22.2.3. In connection with Corollary 21.2.7 and Remark 21.2.8, notethat, as the notion of Grobner representation coincides with that of Gauss repre-sentation (condition (1)), the notion of strong Grobner representation coincideswith that of condition (3).

Algorithm 22.2.4. If we reformulate Gaussian reduction (Figure 21.1) using

B := tg : g ∈ G, t ∈ T we obtain Buchberger’s Normal Form Algorithm which is a crucial tool in thealgorithmical approach to Grobner bases (Figure 22.1).

Let us formalize the output of this algorithm, by the following definition

22.2 Grobner Basis 79

Fig. 22.1. Buchberger Normal Form Algorithm

(g,∑m

i=1 ci ti gi ) := NormalForm( f, F)where

F ⊂ P ,f ∈ P ,g ∈ P ,ci ∈ k \ 0, ti ∈ T , gi ∈ F ,f − g = ∑m

i=1 ci ti gi is a strong Grobner representation in terms of F ,T( f ) ∈ T(F) ⇒ T( f ) = t1T(g1) > t2T(g2) > · · · > tmT(gm) > T(g),T( f ) /∈ T(F) ⇒ f = g, m = 0,

∑mi=1 ci ti gi = 0,

g = 0 ⇒ T(g) /∈ T(F),g := f, i := 0,While T(g) ∈ T(F) do

Let t ∈ T , γ ∈ F : tT(γ ) = T(g),i := i + 1, ci := lc(g)/ lc(γ ), ti := t, gi := γ,g := g − ci ti gi .

m := i

Definition 22.2.5. Given f ∈ P \ 0, F ⊂ P , an element g ∈ P is called anormal form of f w.r.t. F, if

f − g ∈ (F) has a strong Grobner representation in terms of F andg = 0 ⇒ T(g) /∈ T(F).

Then the algorithm of Figure 22.1 proves that

Proposition 22.2.6. For each f ∈ P \ 0, F ⊂ P , there is a normal formg := N F( f, F) of f w.r.t. F.

The importance of normal forms is explained by

Theorem 22.2.7. Let I ⊂ P be an ideal and

G := g1, . . . , gm ⊂ I \ 0.The following conditions are equivalent:

G1 G is a Grobner basis of I;G3 f ∈ I ⇐⇒ it has a Grobner representation in terms of G;G4 f ∈ I ⇐⇒ it has a strong Grobner representation in terms of G;G5 for each f ∈ P \ 0 and any normal form h of f w.r.t. G, we have

f ∈ I ⇐⇒ h = 0.

Proof.

G1 ⇒ G5 Let f ∈ P \0 and h be a normal form of f w.r.t. G. Then either

• h = 0 and f = f − h ∈ (G) ⊂ I, or• h = 0, T(h) /∈ T(G) = TI, h ∈ I and f ∈ I.

80 Buchberger

G5 ⇒ G4 If f has a strong Grobner representation in terms of G, then f ∈(G) ⊂ I.

Conversely, if f ∈ I and h is a normal form of f w.r.t. G, thenh = 0 and f = f − h has a strong Grobner representation in termsof G.

G4 ⇒ G3 If f has a Grobner representation in terms of G, f ∈ (G) ⊂ I.Conversely, if f ∈ I then, by G4 it has a strong Grobner representa-

tion f = ∑µj=1 c j t j gi j ; for each i, 1 ≤ i ≤ m, let Ii := j : i j = i

and let pi := ∑j∈Ii

c j t j ; then

f =µ∑

j=1

c j t j gi j =m∑

i=1

∑j∈Ii

c j t j gi =m∑

i=1

pi gi

and

maxi

T(pi )T(gi ) = maxj

t j T(gi j ) ≤ T( f ).

G3 ⇒ G1 Let τ ∈ TI; then there is f ∈ I such that T( f ) = τ.

Let f = ∑mi=1 pi gi be a Grobner representation.

Then, for some i , τ = T( f ) = T(pi )T(gi ), that is τ ∈ T(G).

Corollary 22.2.8 (Gordan). Let G be a Grobner basis of I; then G is a (finite)basis of I.

Proof. If G is a Grobner basis of I, Theorem 22.2.7 implies that each f ∈ Ihas a Grobner representation in terms of G, so that I = (G).

Remark 22.2.9. This theorem explains the crucial role of the notion of normalform and Buchberger’s Normal Form Algorithm in Grobner theory; in fact,

• if G is a Grobner basis, it allows us to check, for any f ∈ P whether f ∈ I,• and, when f ∈ I, it produces a normal form g := N F( f, F) of f which,

while not unique, has an important uniqueness property: T(g) depends onlyon f and G (see Proposition 22.2.10 below),

• it therefore allows us to devise an effective test to check whether G is aGrobner basis; it is in fact possible, as we will show later, to produce, as a fu-nction of G, a finite set of elements, the ‘S-polynomials’ Σ(G) ⊂ (G) = I,whose normal forms are therefore all 0 if G is a Grobner basis, but whichhas the important property that the converse holds, that is the conditions

• G is a Grobner basis,• N F(σ, G) = 0, for each σ ∈ Σ(G),

are equivalent.

22.2 Grobner Basis 81

Proposition 22.2.10. If F is a Grobner basis for the ideal I ⊂ P , then thefollowing hold.

(1) Let g ∈ P be a normal form of f w.r.t. F. If g = 0, then

T(g) = minT(h) : h − f ∈ I.

(2) Let f, f ′ ∈ P \ I be such that f − f ′ ∈ I. Let g be a normal form of fw.r.t. F and g′ be a normal form of f ′ w.r.t. F. Then

M(g) = M(g′).

Proof.

(1) Let h ∈ P be such that h − f ∈ I; then h − g ∈ I and T(h − g) ∈ T(I).If T(g) > T(h) then T(h − g) = T(g) ∈ TI, giving a contra-diction.

(2) The assumption implies that f − g′ ∈ I so that, by the previous result,T(g) ≤ T(g′). Symmetrically, f ′ − g ∈ I and T(g′) ≤ T(g). There-fore T(g) = T(g′) and either

• T(g −g′) = T(g) = T(g′) and M(g −g′) = M(g)−M(g′), whichis impossible since g − g′ ∈ I and T(g − g′) ∈ TI, or

• T(g − g′) < T(g) and M(g) = M(g′).

Algorithm 22.2.11. As the reformulation of Gaussian reduction (Figure 21.1)produced Buchberger’s Normal Form Algorithm (Figure 22.1) and the notionof normal form and its applications to Grobner theory, by reformulating com-plete Gaussian reduction (Figure 21.3) we will in the same way obtain a toolfor performing arithmetical operations within A = P \ I by means of the notionof canonical form, and Buchberger’s Canonical Form Algorithm (Figure 22.2)for computing it.

Lemma 22.2.12 (Buchberger). We have:

(1) P ∼= I ⊕ k[N(I)];(2) A ∼= k[N(I)];(3) for each f ∈ P , there is a unique

g := Can( f, I) =∑

t∈N(I)

γ ( f, t, <)t ∈ k[N(I)]

such that f − g ∈ I.

82 Buchberger

Fig. 22.2. Buchberger Canonical Form Algorithm

(g,∑m

i=1 ci ti gi ) := CanonicalForm( f, G)where

I ⊂ P is an ideal,G is a Grobner basis of I,f ∈ P ,g ∈ k[N(I)],ci ∈ k \ 0, ti ∈ T , gi ∈ G,f − g = ∑m

i=1 ci ti gi is a strong Grobner representation in terms of G,T( f − g) = t1T(g1) > t2T(g2) > · · · > tmT(gm).

h := f, i := 0, g := 0,While h = 0 do%% f = g + ∑m

i=1 ci ti gi + h;%% T( f − g) ≥ T(h);%% i > 0 ⇒ T( f − g) = t1T(g1) > t2T(g2) > · · · > ti T(gi ) > T(h);

If T(h) ∈ T(G) doLet t ∈ T , γ ∈ G : tT(γ ) = T(h),i := i + 1, ci := lc(h)/ lc(γ ), ti := t, gi := γ,h := h − ci ti gi .

Else%% T(h) ∈ N(I)h := h − M(h), g := g + M(h)

m := i

Moreover:

(a) Can( f1, I) = Can( f2, I) ⇐⇒ f1 − f2 ∈ I;(b) Can( f, I) = 0 ⇐⇒ f ∈ I.

(4) for each f ∈ P, f − Can( f, I) has a strong Grobner representation interms of any Grobner basis.

Proof. This is essentially a reformulation of Corollary 21.2.16 and a directconsequence of the algorithm of Figure 22.2.

Definition 22.2.13. For each f ∈ P the unique element

g := Can( f, I) ∈ k[N(I)]

such that f − g ∈ I will be called the canonical form of f w.r.t. I.

While the existence of a finite Grobner basis of any ideal is a direct conse-quence of Gordan’s Lemma, Lemma 22.2.12 allows us to exhibit one:5

Corollary 22.2.14. There is a unique set G ⊂ I such that

5 The result, of course, is just theoretical: the exhibition of a reduced Grobner basis requiresthe computation of canonical forms, which, in general, requires a preliminary knowledge of aGrobner basis.

22.3 Toward Buchberger’s Algorithm 83

• TG is an irredundant basis of T(I);• for each g ∈ G, lc(g) = 1;• for each g ∈ G, g = T(g) − Can(T(g), I).

The set G is called the reduced Grobner basis of I.

Corollary 22.2.15. Let I ⊂ P be an ideal and

G := g1, . . . , gm ⊂ I \ 0.The following conditions are equivalent:

G1 G is a Grobner basis of I;G6 for each f ∈ P \ 0, f − Can( f, I) has a strong Grobner representation

in terms of G;

Proof. G1 ⇒ G6 follows from Lemma 22.2.12(4).Conversely, since for each f ∈ I, Can( f, I) = 0 and therefore f has a strong

Grobner representation in terms of G, then G6 implies condition G4 of Theo-rem 22.2.7.

22.3 Toward Buchberger’s Algorithm

If the ideal I ⊂ P is given by a basis F , a generating set of I as a k-vector-space is

B := tg : g ∈ F, t ∈ T and F is a Grobner basis iff B is a Gauss generating set.

A ‘procedure’ to test whether B is a Gauss generating set – so that F is aGrobner basis – and, in the negative case, to extend B to a Gauss generatingset was outlined in Section 21.3: it consists of repeatedly

• extracting an echelon set L ⊂ B,• computing a normal form N F(v) for each element v ∈ B \ L in order to

check whether

• N F(v) = 0 for each v, implying that each v has a Gauss representationin terms of L and F is a Grobner basis, or

• there are some v : N F(v) = 0,

• in which case, updating B as

B := L ∪ N F(v) : v ∈ B \ L \ 0.Our aim is to describe a finite computation (Buchberger’s algorithm) which

performs this ‘procedure’ in order to extend the given basis F to a Grobner

84 Buchberger

basis G of I. Let us begin by remarking that the discussion in Section 21.3has pointed to some aspects which will be crucial in the application of this‘procedure’:

• The computation of the normal forms of the (infinite) elements v ∈ B \ Lcan be reduced to a finite computation scheme which will be performed by

• extracting a suitable finite set 6 Σ(F) ⊂ B \ L such that

B \ L ⊆ tv, t ∈ T , v ∈ Σ(F),• computing for each v ∈ Σ(F) a normal form N F(v) and• lifting the result in order to produce a partial reduction t N F(v) for each

element tv, t ∈ T .

• The upgrade of B must be performed by producing G ⊃ F such that

tg : g ∈ G, t ∈ T = L ∪ N F(v) : v ∈ B \ Lin order to produce a new ideal basis G which can be tested to see whetherit is a Grobner basis; the obvious choice 7 is

G := F ∪ N F(v), v ∈ Σ(F).• With such a construction, since

tg : g ∈ G, t ∈ T ⊃ tg : g ∈ F, t ∈ T ,the set B will contain elements already treated and which, thanks to theinclusion of the elements N F(v), v ∈ Σ(F), have a Gauss representationin terms of

L ∪ t N F(v), v ∈ Σ(F), t ∈ T ;this information will be used in order to avoid unnecessary computations.

For the application of this ‘procedure’, in order to have a strategy for extractingthe canonical echelon set from B(G), we need, for any basis G ⊂ I, to imposea well-ordering ≺ on B(G) := tg : g ∈ G, t ∈ T such that

T(w1) < T(w2) ⇒ w1 ≺ w2;we therefore impose an enumeration on the elements of G as G :=g1, . . . , gs and the following ordering on B(G):

t1g j1 ≺ t2g j2 ⇐⇒

t1T(g j1) < t2T(g j2)

t1T(g j1) = t2T(g j2), j1 < j2.

6 Compare Remark 22.2.9 and the discussion there on the properties of normal forms.7 Again compare Remark 22.2.9.


To avoid cumbersome notation in the following, let us assume wlog that foreach i, lc(gi ) = 1, and let us define for each set i1, . . . , i j ⊂ 1, . . . , s

T(i1, . . . , i j ) := lcm(T(gi ) : i ∈ i1, . . . , i j ),and

T(i1, . . . , i j ) := t : T(gi ) | t ⇐⇒ i ∈ i1, . . . , i j

;in particular, for i, j, k, 1 ≤ i, j, k ≤ s:

T(i) := T(gi ),

T(i, j) := lcm(T(gi ), T(g j )),

T(i, j, k) := lcm(T(gi ), T(g j ), T(gk)).

Example 22.3.1. We will undertake this informal introduction of Buchberger’salgorithm by computing a Grobner basis, with respect to the lexicographicalorder < induced by X < Y , for the ideal I generated by (G) = (g1, g2, g3) ⊂k[X, Y ] where

g1 := Y 5 − Y 3, g2 := X2Y 2 − X2, g3 := X5 − X.

As in Example 21.2.4 the monomial structure of B(G) can be pictured on thepoints of the lattice of the positive coordinates in the plane as

......

......

......

......

• • + + + × × × · · ·• • + + + × × × · · ·Y 5 • X2Y 5 + + X5Y 5 × × · · · · · · · · · X2Y 2 X5Y 2 · · · ∗ ∗ ∗ · · · X5 ∗ ∗ · · ·

where

represents the terms t ∈ N(G).• represents the terms t ∈ T(1), represents the terms t ∈ T(2),∗ represents the terms t ∈ T(3),+ represents the terms t ∈ T(1, 2), represents the terms t ∈ T(2, 3),× represents the terms t ∈ T(1, 2, 3).

86 Buchberger

Let us also define:

T1(G) := ∅,

T j (G) := t ∈ T : tT( j) ∈ (T(1), . . . , T( j − 1))

,

N j (G) := t ∈ T : tT( j) ∈ (T(1), . . . , T( j − 1))

= T \ T j (G),

L j (G) := tT( j) : t ∈ N j (G),which satisfy

Lemma 22.3.2. We have

(1) T is the disjoint union of N(G), L1(G), . . . Ls(G):

T = N(G) L1(G) · · · Ls(G).

(2) N j (G) is an order ideal of T .(3) T j (G) is an ideal of T generated by (T(i, j)/T( j)) : 1 ≤ i < j .

(4) L(G) := tgi : 1 ≤ i ≤ s, t ∈ Ni (G) is the canonical echelon setextracted from B(G).

Corollary 22.3.3. The following conditions are equivalent:

(1) L(G) is a Gauss basis of I;(2) G is a Grobner basis of I;(3) for each j , t ∈ T j (G), i < j , t ∈ T such that tT(g j ) = tT(gi ),

tg j − tgi has a Grobner representation in terms of G;(4) for each j , for each t ∈ T j (G), denoting for i < j, by t ∈ Ni (G) the

unique elements such that tT(g j ) = tT(gi ), tg j − tgi has a Grobnerrepresentation in terms of G;

(5) for each j , for each t ∈ T j (G), tg j has a Gauss representation interms of L(G).

Proof. This is a restatement of Corollary 21.3.4.

Example 22.3.4. Continuing Example 22.3.1 we have

T1(G) := ∅, N1(G) := T ,

T2(G) := (Y 3), N2(G) := XaY b : b ≤ 2,T3(G) := (Y 2), N3(G) := XaY b : b ≤ 1,

and

L1(G) := T(1) ∪ T(1, 2) ∪ T(1, 2, 3),

L2(G) := T(2) ∪ T(2, 3),

L3(G) := T(3),


so that...

......

......

......

...

• • • • • • • • · · ·• • • • • • • • · · ·•Y 5 • •X2Y 5 • • •X5Y 5 • • · · · · · · · · · X2Y 2 X5Y 2 · · · ∗ ∗ ∗ · · · ∗X5 ∗ ∗ · · ·

where

represents the terms t ∈ N(G),• represents the terms t ∈ L1(G), represents the terms t ∈ L2(G),∗ represents the terms t ∈ L3(G).

As a consequence we choose the canonical echelon set

L(G) := tg1 : t ∈ N1(G) ∪ tg2 : t ∈ N2(G) ∪ tg3 : t ∈ N3(G)and we have to prove that the elements

tg2 : tT(g2) ∈ T(1, 2) ∪ T(1, 2, 3) ∪ tg3 : tT(g3) ∈ T(2, 3) ∪ T(1, 2, 3)have a Gauss representation in terms of L(G).

Remark 22.3.5. Let j ≤ m, t j ∈ T j (G), i < j, ti ∈ T be such that

t j T( j) = ti T(i) =: t;then T(i, j) = lcm (T(i), T( j)) | t and there is τ ∈ T such that t = τT(i, j).

IfT(i, j)

T( j)g j − T(i, j)

T(i)gi =

m∑k=1

pk gk

is a Grobner representation in terms of G, then

t j g j − ti gi = τT(i, j)

T( j)g j − τ

T(i, j)

T(i)gi =

m∑k=1

τpk gk

and

t j g j = ti gi −s∑

k=1

τpk gk

is a Grobner representation in terms of G and also satisfies, as a Gauss repre-sentation, the condition of Corollary 21.2.7(2).

88 Buchberger

As a consequence, if we write, for each i, j, 1 ≤ i < j ≤ m

S(i, j) := T(i, j)

T( j)g j − T(i, j)

T(i)gi ,

and

Σ(G) := S(i, j) : 1 ≤ i < j ≤ mwe have


(1) Each S(i, j) ∈ Σ(G) has a Grobner representation in terms of G.(2) For each j, t ∈ T j (G), i < j, t ∈ T such that tT(g j ) = tT(gi ), the

element tg j has a Grobner representation tg j = tgi + ∑mk=1 pk gk in

terms of G where tT(g j ) = tT(gi ) > T(pk)T(gk), for each k.

(3) G is a Grobner basis of I.

Example 22.3.7. Continuing Example 22.3.1, we have

S(1, 2) := Y 3g2 − X2g1 = 0,

S(1, 3) := Y 5g3 − X5g1 = X5Y 3 − XY 5,

S(2, 3) := Y 2g3 − X3g2 = X5 − XY 2.

Since S(1, 2) = 0 we know that N F(S(1, 2)) = 0 so that for each τ ∈T2(G) = (Y 3), τg2 has the Gauss representation

τg2 = τ

Y 3X2g1

in terms of L(G).We have therefore concluded that all elements

tg2 : tT(g2) ∈ T(1, 2) ∪ T(1, 2, 3)have a Gauss representation in terms of L(G) , since we have explicitly pro-duced such a representation.

Example 22.3.8. The conclusions of the computation related to S(1, 3) area bit more subtle and explain the cryptic remark at the end of the previousexample.

Since

S(1, 3) := Y 5g3 − X5g1

= X5Y 3 − XY 5

= Y 3g3 − XY 5 + XY 3

= Y 3g3 − Xg1,


so that NormalForm(S(1, 3), F) = (0, Y 3g3 − Xg1), then for each τ ∈ (Y 5),N F((τ/Y 5)S(1, 3)) = 0 and τg3 has the Gauss representation

τg3 = τ

Y 5X5g1 + τ

Y 5Y 3g3 − τ

Y 5Xg1

in terms of B.However, we cannot conclude directly that it has a Gauss representation in

terms of L(G); in fact already the Gauss representation of the element Y 5g3

which is

Y 5g3 = X5g1 + Y 3g3 − Xg1,

involves not only the elements in L(G) but also

Y 3g3 ∈ tg3 : tT(g3) ∈ T(2, 3) ∈ B(G) \ L(G).

In the proof of Corollary 21.3.4, (5) ⇒ (6) is argued by induction and(in this case) would require that the elements in tg3 : tT(g3) ∈ T(2, 3)already have a representation in terms of L(G).

So far therefore we have only proved that the elements

tg2 : tT(g2) ∈ T(1, 2) ∪ T(1, 2, 3) ∪ tg3 : tT(g3) ∈ T(1, 2, 3)have a Gauss representation in terms of

L(G) ∪ tg3 : tT(g3) ∈ T(2, 3).I wanted to stress this obvious remark because we will have to return later

to this computation of the normal form of S(1, 3) when explaining some moresubtle aspects of Buchberger’s algorithm.

Example 22.3.9. The last computation to be performed is therefore the com-putation of a normal form of S(2, 3) which should dispose of the elementstg3 : tT(g3) ∈ T(2, 3) and, indirectly, of those in tg3 : tT (g3) ∈ T(1, 2, 3).

The computation gives:

S(2, 3) := Y 2g3 − X3g2

= X5 − XY 2

= g3 − XY 2 + X,

so that N F(S(2, 3)) = −XY 2+ X =: −g4, and T(g4) ∈ T(I)\T(G) implyingthat G is not a Grobner basis.

Lemma 22.3.10. If N F(σ ) : σ ∈ Σ(G) \ 0 =: S(G) = ∅ then, writingG ′ := G ∪ S(G), we have

90 Buchberger

• G is not a Grobner basis of I,• S(G) ⊂ (G) = I,• I = (G ′),• T(G) T(G ′) ⊂ T(I).

Proof. For each element in σ ∈ Σ(G), by the definition of normal forms,T(N F(σ )) ∈ T(G), while N F(σ ) ∈ (G) since σ ∈ (G), implying thatT(N F(σ )) ∈ T(I). This is sufficient to prove all the claims.

Example 22.3.11. We therefore deduce that each element

tg3 : tT(g3) ∈ T(2, 3) ∪ T(1, 2, 3) = τY 2g3, τ ∈ T can be expressed as

τY 2g3 = τ X3g2 + τg3 − τg4

and, arguing by induction on the <-ordered elements τ ∈ T , has a Gaussrepresentation in terms of

B′ := L ∪ tg4 : t ∈ T.Therefore we can conclude that all the elements in

tg2 : tT(g2) ∈ T(1, 2) ∪ T(1, 2, 3) ∪ tg3 : tT(g3) ∈ T(2, 3) ∪ T(1, 2, 3)have a Gauss representation in terms of B′.

If we consider the new basis G ′ := g1, g2, g3, g4 the set T can now bepartitioned as

T = N(G ′) T(1) U(1, 4) U(2, 4) T(3) T(4)

where

U(1, 4) := T(1, 4) ∪ T(1, 2, 4) ∪ T(1, 2, 3, 4),

U(2, 4) := T(2, 4) ∪ T(2, 3, 4);consequently B′ can then be partitioned as

B′ = tg1 : tT(1) ∈ T(1) ∪ tg1 : tT(1) ∈ U(1, 4)∪ tg2 : tT(2) ∈ U(2, 4) ∪ tg3 : tT(3) ∈ T(3)∪ tg4 : t ∈ T .

From this we can extract the canonical echelon set

L′ := tg1 : tT(1) ∈ T(1) ∪ tg1 : tT(1) ∈ U(1, 4)∪ tg2 : tT(2) ∈ U(2, 4) ∪ tg3 : tT(3) ∈ T(3)∪ tg4 : t ∈ T(4)


so that we have to check whether the elements in

tg4 : tT(4) ∈ U(1, 4) ∪ tg4 : tT(4) ∈ U(2, 4)have a Gauss representation.

The corresponding monomial structure can be pictured as...

......

......

......

...

• · · ·• · · ·•Y 5

XY 5 · · ·

+ · · · + · · · +XY 2 X2Y 2 · · · ∗ ∗ ∗ · · · ∗X5 ∗ ∗ · · ·

where

represents the terms t ∈ N(G ′),• represents the terms t ∈ T(1), represents the terms t ∈ U(1, 4), represents the terms t ∈ U(2, 4),∗ represents the terms t ∈ T(3),+ represents the terms t ∈ T(4).

We have therefore to compute the normal forms of

• S(1, 4) := Y 3g4 − Xg1 = 0, proving that the elements in

tg4 : tT(4) ∈ U(1, 4)have a Gauss representation in terms of L′;

• S(2, 4) := Xg4 − g2 = 0, proving that the elements in

tg4 : tT(4) ∈ U(2, 4)also have a Gauss representation in terms of L′.

As a consequence we have shown that each element in

B(G ′) := tg : g ∈ G ′, t ∈ T has a Gauss representation in terms of

L′ := tg1 : t ∈ T ∪ tg2 : tT(2) ∈ U(2, 4)∪ tg3 : tT(3) ∈ T(3) ∪ tg4 : t ∈ T(4)

92 Buchberger

= tg1 : t ∈ L′1(G

′) ∪ tg2 : t ∈ L′2(G

′)∪ tg3 : t ∈ L′

3(G′) ∪ tg4 : t ∈ L′

4(G′).

whereL′

1(G′) := T ,

L′2(G

′) := t : tT(2) ∈ (T(1)),L′

3(G′) := t : tT(3) ∈ (T(1), T(2)),

L′4(G

′) := t : tT(4) ∈ (T(1), T(2), T(3))whose corresponding monomial structure can be pictured as

......

......

......

......

• • • • • • • • · · ·• • • • • • • • · · ·•Y 5 • • • • • • • · · · + · · · + · · · +XY 2 X2Y 2 · · · ∗ ∗ ∗ · · · ∗X5 ∗ ∗ · · ·

where

represents the terms t ∈ N(G ′),• represents the terms t ∈ L′

1(G′),

represents the terms t ∈ L′2(G

′),∗ represents the terms t ∈ L′

3(G′),

+ represents the terms t ∈ L′4(G

′).

We can therefore conclude that G ′ = g1, g2, g3, g4 is a Grobner basis ofthe ideal I.

Remark 22.3.12. This computation requires some remarks:

(1) The statement of Corollary 22.3.6 (and that of the corresponding The-orem 22.4.3) requires the computation of the normal form S(3, 4) inorder to conclude, while we are able to reach the conclusion withoutusing S(3, 4).

The corresponding computation, in fact, which gives

S(3, 4) = X4g4 − Y 2g3 = −X5 + XY 2 = −g3 + g4,

is useless: there is no need to prove that Y 2g3 has a Gauss repre-sentation in terms of X4g4 and of elements g ∈ B(G) such thatT(g) < Y 2T(g3) since we already know that


• Y 2g3 has the Gauss representation Y 2g3 = X3g2 − g3 − g4 in termsof X3g2 and of elements g ∈ B(G) such that T(g) < Y 2T(g3) and

• X3g2 has the Gauss representation X3g2 = X4g4 in terms of X4g4

and of elements g ∈ B(G) such that T(g) < X3T(g2) = Y 2T(g3).

In fact, from

S(2, 3) := Y 2g3 − X3g2 = g3 − g4,

S(2, 4) := Xg4 − g2 = 0,

we could have directly deduced

S(3, 4) = X4g4 − Y 2g3

= (X4g4 − X3g2) − (Y 2g3 − X3g2)

= X3S(2, 4) − S(2, 3)

= X30 − (g3 − g4)

= −g3 + g4.

Since the computation of normal form is quite time–space consuming,the efficiency of an implementation of Buchberger’s algorithm stronglydepends on being able to deduce such ‘useless’ computations.

(2) Our computation started with the basis G := g1, g2, g3 generatingthe ideal I and aimed to check whether it was a Grobner basis. Theconclusion was that it was not, since XY 2 ∈ T(I) \ T(G) and that G ′

was a Grobner basis.However, our computations allow us to conclude also that all

elements in B(G) can be Gauss represented using only G ′′ :=g4, g1, g3; in fact G ′′ is a Grobner basis of I since G ′′ ⊂ I and

T(G ′′) = (XY 2, Y 5, X5) = T(I).

In fact, if we order G ′ as g4, g1, g2, g3, we obtain as the canonicalechelon set

L′′ := tg4 : t ∈ T ∪ tg1 : tT(1) ∈ T(1) ∪ tg3 : tT(3) ∈ T(3)= tg4 : t ∈ L′′

4(G′) ∪ tg1 : t ∈ L′′

1(G′)

∪ tg2 : t ∈ L′′2(G

′) ∪ tg3 : t ∈ L′′3(G

′)where

L′′4(G

′) := T ,

L′′1(G

′) := t : tT(1) ∈ (T(4)),L′′

2(G′) := t : tT (2) ∈ (T(4), T(1)) = ∅,

L′′3(G

′) := t : tT (1) ∈ (T(4), T(1), T(2));

94 Buchberger

since L′′2(G

′) = ∅, the same computation S(1, 4) = S(2, 4) = 0 showsthat G ′′ is a Grobner basis and produces the monomial structure

......

......

......

......

• + + + + + + + · · ·• + + + + + + + · · ·•Y 5

+ + + + + + + · · · + + + + + + + · · · + + + + + + + · · · + XY 2

+ X2Y 2+ + + + + · · ·

∗ ∗ ∗ · · · ∗X5 ∗ ∗ · · ·

where

represents the terms t ∈ N(G ′).+ represents the terms t ∈ L′′

4(G′).

• represents the terms t ∈ L′′1(G

′),∗ represents the terms t ∈ L′′

3(G′).

The ‘moral’ – as the Countess said – is that the ordering of the elementsand of the computation can dramatically change the computation pat-tern and therefore that such aspects must be taken into consideration;this will be discussed in Chapter 25.

(3) On the other hand, once a Grobner basis of I is produced, as we didwhen computing G ′ := g4, g1, g2, g3, and therefore T(I) =(Y 5, X5, XY 2) is known, it is often most efficient to use this knowl-edge in order to produce a better-shaped basis, like the reduced Grobnerbasis, by performing CanonicalForm(t, G ′) on the minimal basis ofT(I).

Remark 22.3.13. It may have been noted that in the computation outlined inExample 22.3.9 we were required to perform normal form computation notfor all elements tg3 : tT(g3) ∈ T(2, 3) ∪ T(1, 2, 3) but only for those intg3 : tT(g3) ∈ T(2, 3) and therefore we needed to add to L not all theelements tg4 : t ∈ T but only the elements

tg4 : tT(g4) ∈ T(4) ∪ U(2, 4) = tg4 : t ∈ T ∗where T ∗ := Xa1

1 Xa22 : a2 < 3.

So we would only have had to deal with

B∗ = tg1 : tT(1) ∈ T(1) ∪ tg1 : tT(1) ∈ U(1, 4)∪ tg2 : tT(2) ∈ U(2, 4) ∪ tg3 : tT(3) ∈ T(3)∪ tg4 : t ∈ T(4) ∪ tg4 : t ∈ U(2, 4).


from which we would have extracted the same canonical echelon set

L′ := tg1 : tT(1) ∈ T(1) ∪ tg1 : tT(1) ∈ U(1, 4)∪ tg2 : tT(2) ∈ U(2, 4) ∪ tg3 : tT(3) ∈ T(3)∪ tg4 : t ∈ T(4)

but we would have had to check only whether the elements in

tg4 : tT(4) ∈ U(2, 4)have a Gauss representation and not also what happens for the elements in

tg4 : tT(4) ∈ U(1, 4).The corresponding monomial structure can be pictured as...

......

......

......

...

• • • • • • • • · · ·• • • • • • • • · · ·•Y 5 • • • • • • • · · · + · · · + · · · +XY 2 X2Y 2 · · · ∗ ∗ ∗ · · · ∗X5 ∗ ∗ · · ·

where

represents the terms t ∈ N(G ′),• represents the terms t ∈ T(1) ∪ U(1, 4), represents the terms t ∈ U(2, 4),∗ represents the terms t ∈ T(3),+ represents the terms t ∈ T(4).

As a consequence, to check whether G ′ was a Grobner basis it would onlyhave been necessary to compute the normal forms of

S(2, 4) := Xg4 − g2 = 0

but we would have avoided the useless computation

S(1, 4) := Y 3g4 − Xg1 = 0.

This line of research, which has been pursued by Gebauer and Moller underthe notion of staggered linear bases and recently refined by Faugere,8 will bediscussed in Chapter 25.

8 In R. Gebauer and H. M. Moller, Buchberger’s algorithm and staggered linear bases. Proc.SYMSAC 1986, pp. 218–221 and J.-C. Faugere, A new efficient algorithm for computing Grobnerbases without reduction to zero (F5). Proc. ISSAC ’02, ACM (2002).

96 Buchberger

22.4 Buchberger’s Algorithm (1)

Definition 22.4.1 (Buchberger). For each f, g ∈ P such that lc( f ) = 1 =lc(g), the polynomial

S(g, f ) := lcm(T( f ), T(g)

T( f )f − lcm(T( f ), T(g)

T(g)g

is called the S-polynomial of f and g.

Definition 22.4.2. Let f, g ∈ P be such that lc( f ) = 1 = lc(g). We say thatthe S-polynomial of f and g has a weak Grobner representation in terms ofG if it can be written as S(g, f ) = ∑m

k=1 pk gk, with pk ∈ P, gk ∈ G andT(pk)T(gk) < lcm(T( f ), T(g)), for each k.

The difference between a Grobner representation and a weak Grobner rep-resentation is that in the second we require that

T(pk)T(gk) < lcm(T( f ), T(g)),

while in the first we require that

T(pk)T(gk) ≤ T(S(g, f ));since T(S(g, f )) < lcm(T( f ), T(g)) a Grobner representation is a weakGrobner representation.

The reader should keep in mind that true weak Grobner representations donot exist: they are just a fiction; as with unicorns, I cannot provide a singleexample of a weak Grobner representation which is not itself a Grobner repre-sentation.

Theorem 22.4.3 shows that it is sufficient to assume that each S-polynomialhas a weak Grobner representation in order to deduce that each polynomial inI – and therefore each S-polynomial – has a Grobner representation.

These fictional objects however have a role in the implementation ofBuchberger’s algorithm: they are used in Lemma 22.5.3 to show that it issufficient to check that a suitable subset of S-polynomials has weak Grobnerrepresentations in order to deduce that all of them have a Grobner represen-tation.

Theorem 22.4.3 (Buchberger). For a basis G := g1, . . . , gm ⊂ P \ 0generating the ideal I and such that lc(gi ) = 1, for each i, the followingconditions are equivalent:

G3 f ∈ I iff it has a Grobner representation in terms of G;G7 for each i, j, 1 ≤ i < j ≤ m, the S-polynomial S(i, j) has a weak

Grobner representation in terms of G.

22.4 Buchberger’s Algorithm (1) 97

Proof. (See Corollary 21.3.4) Since, for each i, j, 1 ≤ i < j ≤ m, S(i, j) ∈(G) = I, then, as a consequence of G3, it has a Grobner representation

S(i, j) =m∑

k=1

pk gk,

where T(pk)T(gk) ≤ T(S(i, j)) for each k; since T(S(i, j)) < lcm(T( f ),

T(g)) this is also a weak Grobner representation in terms of G and G7 holds.Conversely, let us consider an element h ∈ I; since G is a basis of I there

is a representation h = ∑mk=1 pk gk . If γ1 := maxkT(pk)T(gk) ≤ T(h) the

representation is a Grobner one, and we are through.Otherwise, writing J := k : T(pk)T(gk) = γ1 we have∑

j∈J

M(p j )T(g j ) =∑j∈J

lc(p j )T(p j )T(g j ) = 0, and∑j∈J

lc(p j ) = 0.

In this case, we intend to show that there is another representation

h =m∑

k=1

p′k gk : γ2 := max

kT(p′

k)T(gk) < γ1.

Then the thesis follows from an inductive argument, since < is a well-orderingand we cannot have an infinite decreasing sequence

γ1 > γ2 > · · · > γν > · · · > T(h).

Let us write ι := min(J ). For each j ∈ J, j = ι, since T( j) | γ1, there isτ j ∈ T such that

τ j T(ι, j) = γ1 = T(p j )T(g j ), and T(p j ) = τ jT(ι, j)

T( j).

Therefore

∑j∈J

lc(p j )T(p j )g j =∑j∈J

lc(p j )τ jT(ι, j)

T( j)g j

=∑j∈J

lc(p j )τ j

(T(ι, j)

T( j)g j − T(ι, j)

T(ι)gι

)

+(∑

j∈J

lc(p j )

)γ1

T(ι)gι

=∑j∈J

lc(p j )τ j S(ι, j).

98 Buchberger

By assumption, each S(ι, j) has a weak Grobner representation

S(ι, j) =m∑

i=1

pi j gi : τ j T(pi j )T(gi ) < τ j T(ι, j) = γ1.

Therefore if we define, for each j ∈ J , q j := p j − M(p j ), since T(q j ) <

T(p j ) we have

h =m∑

i=1

pi gi

=∑j∈J

lc(p j )T(p j )g j +∑j∈J

q j g j +∑i ∈J

pi gi

=∑j∈J

lc(p j )τ j S(ι, j) +∑j∈J

q j g j +∑i ∈J

pi gi

=m∑

i=1

∑j∈J

lc(p j )τ j pi j gi +∑j∈J

q j g j +∑i ∈J

pi gi

which is the required Grobner representation.

Algorithm 22.4.4 (Buchberger). Through the introduction of the S-polynomi-als, Theorem 22.4.3 gives an effective condition for testing whether G is aGrobner basis: given G, one has to compute the S-polynomials among its ele-ments, and check whether the normal form of each of them is zero. If this is thecase, then G is a Grobner basis. In the negative case, the computation of thenormal forms produces elements g ∈ I such that T(g) ∈ T(G); enlarging Gwith these new elements produces a basis G ′ such that T(G) T(G ′) ⊂ T(I)on which the test can again be applied.

This algorithm is sketched in Figure 22.3.

22.5 Buchberger’s Criteria

The discussion in Remark 22.3.12(1) introduces an improvement to Buch-berger’s algorithm: if an S-pair σ can be expressed as a term-bounded com-bination of other S-pairs, whose normal forms w.r.t. G are 0, it is possible toprove that the same happens for σ (see Lemma 22.5.3 below) and it is thereforeuseless to compute its normal form.

More generally, since normal form computation is often a costly computa-tion, if it is possible to detect easily that the normal form of an S-pair σ is zero,N F(σ, G) = 0, computing it is not only useless, but even dangerous in its useof time and space.

The main criteria for detecting useless pairs were already introduced byBuchberger in his original algorithm; the most easy is

22.5 Buchberger’s Criteria 99

Fig. 22.3. Buchberger Algorithm (sketch)

G := GrobnerBasis(F)where

F := g1, . . . , gs ⊂ P ,lc(gi ) = 1,for each i,I is the ideal generated by F ,G is a Grobner basis of I;

G := FB := i, j, 1 ≤ i < j ≤ sWhile B = ∅ do

Choose i, j ∈ B,B := B \ i, j,h := S(i, j),(h,

∑mi=1 ci ti gi ) := NormalForm(h, G),

If h = 0 thens := s + 1, gs := lc(h)−1h, G := G ∪ gs,B := B ∪ i, s, 1 ≤ i < s

Lemma 22.5.1 (Buchberger’s First Criterion).

T(i)T( j) = T(i, j) ⇒ N F(S(i, j), G) = 0.

Proof. Write pi := gi − T(i), p j := g j − T( j) and note that T(pi ) < T(gi ),

T(p j ) < T(g j ).

Then we have:

0 = gi g j − g j gi = T(i)g j + pi g j − T( j)gi − p j gi ,

and

S(i, j) := T(i, j)

T( j)g j − T(i, j)

T(i)gi = T(i)g j − T( j)gi = p j gi − pi g j .

There are then two possibilities:

• either M(p j )T(gi ) = M(pi )M(g j ) in which case

T(S(i, j)) = max(T(p j )T(gi ), T(pi )T(g j ))

and S(i, j) = p j gi − pi g j is a Grobner representation;• or M(p j )T(gi ) = M(pi )T(g j ), T(S(i, j)) < T(p j )T(gi ) = T(pi )T(g j ),

in which case S(i, j) = p j gi − pi g j would not be a Grobner representation.

But the latter case is impossible: from

T(g j )T(gi ) > T(p j )T(gi ) = T(pi )T(g j )

we deduce lcm(T(gi ), T(g j )) = T(g j )T(gi ), contradicting the assumptionT(i, j) = T(i)T( j).

100 Buchberger

Example 22.5.2. This result has already been illustrated in Example 22.3.8,where T(1)T(3) = T(1, 3) and we found out that S(1, 3) = Y 3g3 − Xg1.

The second criterion introduced by Buchberger is that illustrated by Re-mark 22.3.12.(1):

Lemma 22.5.3 (Buchberger’s Second Criterion). For i, j, 1 ≤ i < j ≤ s,if there is k, 1 ≤ k ≤ s : T(k) | T(i, j), and S(i, k) and S(k, j) have a weakGrobner representation in terms of G, then S(i, j) also has a weak Grobnerrepresentation.

Proof. Since T(k) | T(i, j), then there exist ti , t j ∈ T such that

t j T(k, j) = T(i, j) = ti T(i, k);therefore

S(i, j) = T(i, j)

T( j)g j − T(i, j)

T(i)gi

= t jT(k, j)

T( j)g j − t j

T(k, j)

T(k)gk + ti

T(i, k)

T(k)gk − ti

T(i, k)

T(i)gi

= t j S(k, j) − ti S(i, k).

By assumption we have weak Grobner representations S(k, j) = ∑l pl gl and

S(i, k) = ∑ pg such that, for each l, ,

t j T(pl)T(gl) < t j T(k, j) = T(i, j) = ti T(i, k) > ti T(p)T (g),

so that

S(i, j) = t j S(k, j) − ti S(i, k) =∑

l

t j pl gl −∑

ti pg

is the required weak Grobner representation.

Corollary 22.5.4 (Buchberger). The following conditions are equivalent:

G7 for each i, j, 1 ≤ i < j ≤ s, the S-polynomial S(i, j) has a weak Grobnerrepresentation in terms of G;

G8 for each i, j, 1 ≤ i < j ≤ s, there exist i = i0, i1, . . . , iρ, . . . ir = j, 1 ≤iρ ≤ s:

• lcm(T(iρ)) = T(i, j),• each S-polynomial S(iρ−1, iρ) has a weak Grobner representation

in terms of G.


Fig. 22.4. Buchberger Algorithm with Criteria (sketch)


F := g1, . . . , gs ⊂ P ,lc(gi ) = 1, for each i,I is the ideal generated by F ,G is a Grobner basis of I;

G := F ,B := i, j, 1 ≤ i < j ≤ s,While B = ∅ do

Choose i, j ∈ B,B := B \ i, j,If T(i, j) = T(i)T( j) or there is no k:

T(k) | T(i, j),i, k ∈ B,k, j ∈ B,

thenh := S(i, j),(h,

∑mi=1 ci ti gi ) := NormalForm(h, G),

If h = 0 thens := s + 1, gs := lc(h)−1h, G := G ∪ gs,B := B ∪ i, s1 ≤ i < s

Algorithm 22.5.5 (Buchberger). Buchberger’s criteria allow us to improveBuchberger’s algorithm, avoiding useless normal form computation: any timea new S-polynomial S(i, j) is considered, it is first tested to see whether itsatisfies Lemmata 22.5.1 and 22.5.3. This improvement of the algorithm issketched in Figure 22.4.

This algorithm is correct since, each time a new pair i, j is taken intoconsideration,

• either T(i, j) = T(i)T( j) and S(i, j) satisfies Buchberger’s First Criterionand is useless;

• or there is k, 1 ≤ k ≤ s such that

• T(k) | T(i, j),• i, k is not in B, so that it has been already tested and we recursively

know that S(i, k) has a weak Grobner representation in terms of G,• k, j is not in B, so that S(k, j) has a weak Grobner representation in

terms of G,

so that Lemma 22.5.3 allows us to conclude that S(i, j) has a weak Grobnerrepresentation in terms of G;

• both cases are not satisfied, and the normal form of S(i, j) is to be computed.

102 Buchberger

The introduction of the fictional notion of weak Grobner representation isnow justified by Lemma 22.5.3 and Corollary 22.5.4: both cannot hold if westate them for the notion of ‘Grobner representation’. What is hidden in the no-tion of weak Grobner representation is the ability to apply it recursively: bothAlgorithm 22.5.5 and Corollary 22.5.4 recursively argue that an S-polynomialS(i, j) is useless by means of Lemma 22.5.3, because they assume – eitherby a recursive argument or by a normal form computation – that both S(i, k)

and S(k, j) have a weak Grobner representation in terms of G; the boot-strap needed by this recursive application of Lemma 22.5.3, as it is explicitlystressed by Corollary 22.5.4, is a sequence of previous explicit computationsof normal forms of (useful) S-polynomials; for such pairs a strong Grobnerrepresentation is explicitly produced. The recursive argument then deducesthat all the other (useless) S-polynomials have a weak Grobner representa-tion.

In this recursive argument one must be very careful to avoid aporetic loopslike the one illustrated in the following example.

Example 22.5.6. Let us consider the example

G := g1, g2, g3, g4 ∈ k[X1, X2, X3, X4]

where

g1 := X21 X2

2 X23 X4, g2 := X2

1 X22 X3 X2

4,

g3 := X21 X2 X2

3 X24, g4 := X1 X2

2 X23 X2

4 − 1,

which, for each i, j, k, satisfies

T(k) | X21 X2

2 X23 X2

4 = T(i, j, k) = T(i, j).

The application of Lemma 22.5.3 in order to deduce that

1, 3: S(1, 3) has a weak Grobner representation in terms of G because

T(2) | T(1, 3) = T(1, 2, 3),


T(2) | T(1, 4) = T(1, 2, 4),


T(1) | T(3, 4) = T(1, 3, 4),


T(3) | T(2, 4) = T(2, 3, 4),


and to conclude that each S-polynomial has a weak Grobner representation interms of G and that G itself is a Grobner basis, since

1, 2: S(1, 2) = 0 has a strong Grobner representation in terms of G,2, 3: S(2, 3) = 0 has a strong Grobner representation in terms of G,

is wrong and leads to a wrong conclusion.For each i, 1 ≤ i ≤ 3,

0 = S(i, 4) = X1g4−X5−i gi = −X1 ∈ (G) and T(S(i, 4)) = X1 ∈ T(G).

The aporetic loop which leads to this wrong deduction from Lemma 22.5.3is based on the correct statements that

1, 3: S(1, 3) has a weak Grobner representation in terms of G if S(1, 2) andS(2, 3) have such representation,




and the wrong application of the loop argument that

S(1, 4) has a weak Grobner representation

⇐ S(1, 2), S(2, 4) have a weak Grobner representation

⇐ S(1, 2), S(2, 3), S(3, 4) have a weak Grobner representation

⇐ S(1, 2), S(2, 3), S(1, 3), S(1, 4) have a weak Grobner representation.

Of course, this mistake can pollute neither Corollary 22.5.4 – where one mustexplicitly provide a series of S-pairs for which the existence of a Grobner rep-resentation is known – nor Algorithm 22.5.5, which, in this example wouldhave given the correct deduction:

1, 2: S(1, 2) = 0 has a strong Grobner representation in terms of G;2, 3: S(2, 3) = 0 has a strong Grobner representation in terms of G;1, 3: S(1, 3) has a weak Grobner representation in terms of G because T(2) |

T(1, 3) and S(1, 2) and S(2, 3) have such representation;3, 4: S(3, 4) = −X =: −g5 has a weak Grobner representation in terms of

G ′ := G ∪ g5;2, 4: S(2, 4) has a weak Grobner representation in terms of G because T(3) |

T(2, 4) and S(2, 3) and S(3, 4) have such representation;

104 Buchberger

1, 4: S(1, 4) has a weak Grobner representation in terms of G ′ becauseT(3) | T(1, 4) and S(1, 3) and S(3, 4) have such representation;

i, 5: S(i, 5) = 0, has a strong Grobner representation in terms of G ′, foreach i .

The moral of this example is that the recursive application of Lemma 22.5.3cannot be performed if it is applied before the normal form computation; thisargument is safe only if applied after the related normal form computationsare performed.


Before I present Buchberger’s algorithm there are some more elementary im-provements which have been introduced since the first implementation.

Definition 22.6.1. A set G ⊂ P is called autoreduced if for each f ∈ G, f =Can( f, (G \ f )).

It should be clear that writing

S∗(i, j) := T(i, j)

T( j)Can(g j , (G \ g j )) − T(i, j)

T(i)Can(gi , (G \ gi )),

one has

N F(S(i, j), G) = N F(S∗(i, j), G))

and the first is slower to be computed.Therefore it is advantageous to autoreduce the input basis before applying

Buchberger’s algorithm to it.Moreover, this preliminary autoreduction would in any case give a better

basis G ′, in the sense that T(G) ⊂ T(G ′) ⊂ T(I).

Remark 22.6.2. In the same mood, it often happens that a basis element fi

becomes redundant when the algorithm produces a new element fs suchthat T(s) | T(i). It is then space-saving to remove fi from G and all thepairs i, j, j < s, from B, of course after having computed N F( fi , G) =N F(S(i, s), G). The only other thing to take care of is to avoid inserting otherpairs S(i, t), t > s, in further computations. The introduction of the subset Jin Algorithm 22.6.3 aims to do that.

Algorithm 22.6.3. We can now present in detail Buchberger’s algorithm in Fig-ure 22.5.


Fig. 22.5. Buchberger Algorithm


F ⊂ P \ 0,I is the ideal generated by F ,G is a Grobner basis of I;

While exist g, h ∈ F : T(g) | T(h) doF := F \ h ∪ S(h, g)

G := F \ 0Re-order G =: g1, . . . , gs so that T(i) < T( j) ⇐⇒ i < j.For each i, 1 ≤ i ≤ s do

G := G \ gi , h := gi , gi := 0,While h = 0 do

If exist t ∈ T , γ ∈ G : tT(γ ) = T(h) doh := h − (lc(h)/ lc(γ ))tγ

Elseh := h − M(h), gi := gi + M(h)

gi := lc(gi )−1gi , G := G ∪ gi ,

B := i, j, 1 ≤ i < j ≤ sJ := r, 1 ≤ r ≤ s

o While B = ∅ doo Choose i, j ∈ B

B := B \ i, j,If T(i, j) = T(i)T( j) or there is no k:

T(k) | T(i, j),i, k ∈ B,k, j ∈ B,

thenh := S(i, j)

i While T(h) ∈ T(G) doi Choose t ∈ T , γ ∈ G : tT(γ ) = T(h)

h := h − lc(h)tγIf h = 0 then

s := s + 1, gs := lc(h)−1h, G := G ∪ gsB := B ∪ i, s, i ∈ J ,For each i ∈ J do

If T(s) | T(i) doJ := J \ i, G := G \ gi , B := B \ i, j, j < s,

J := J ∪ s

In order to prove termination of Buchberger’s algorithm, let us remark thatit consists of two While-loops: an inner loop (i ) and an outer loop (o), bothcontrolled by a Choose instruction, an inner choice (i ) and an outer choice(o). Our termination proof is based on indexing these choices:

i : Each choice is indexed by T(h) and, in each loop, h is replaced byhnew := h − lc(h)tγ. Since T(h) > T(hnew) non termination ofthe inner loop would imply the existence of an infinite decreasing

106 Buchberger

sequence of elements

γ1 > γ2 > · · · > γν > · · ·in T and this would contradict Gordan’s Lemma (Proposition 20.8.3).

o : Each choice is performed in B and the total set of the loops is indexedby i, j : 1 ≤ i < j ≤ s which is finite if and only if the set G isfinite. Note that the proof of the existence of finite Grobner bases as aconsequence of Gordan’s Lemma (Corollary 22.2.8), is not sufficientto prove the finiteness of the explicit Grobner bases produced by thealgorithm. However, a more subtle application of Gordan’s Lemma issufficient: if the algorithm does not terminate, it produces an infinitesequence

T(g1), T(g2), . . . , T(gn), . . . such that T(g j ) T(gi ) if i > j,

contradicting Corollary 20.8.4 which states the existence of N ∈ N

such that for each i ≥ N exists j ≤ N : T(g j ) | T(gi ).

Termination of Buchberger’s algorithm therefore depends in two ways onGordan’s Lemma.

Among the generalizations of Buchberger’s algorithm, there are two whichexplicitly challenged Gordan’s termination proof of the algorithm:

i The notion of (Hironaka) standard bases was introduced both in the seriesring k[[X1, . . . , Xn]] and in the polynomial ring k[X1, . . . , Xn] and ithas application in local algebra; essentially the definition is the sameas that of Grobner bases 9 except that < is not necessarily a well-ordering; actually it is required that 1 > Xi , for each i, thereforemaking the application of Gordan’s Lemma impossible; however,Grobner bases theory can be verbatim adapted mutatis mutandis 10

and standard bases in k[X1, . . . , Xn] can be computed in a finitenumber of steps by an appropriate modification of Buchberger’s al-gorithm, the tangent cone algorithm; the only requirement needed on< is that it is inf-limited, that is for any t ∈ T there is no infinitesequence of terms γν ∈ T such that

γ1 > γ2 > · · · > γν > · · · > t.

9 For any element f , T( f ) defines the maximal term in its expansion w.r.t. a semigroup ordering<; and G is a standard basis of the ideal I if T(g) : g ∈ G generates T( f ) : f ∈ I.

10 For instance the corresponding notion of standard representation in terms of G is a represen-tation f = ∑s

i=1(pi /1 + qi )gi , where gi ∈ G, pi , qi ∈ k[X1, . . . , Xn ], T(qi )(0) = 0,

T(pi )T(gi ) ≤ T( f ).


o In the non-commutative case – in which one considers the free semigroupS generated by the n symbols X1, . . . , Xn , the ring k[S] := Spank(S)

and a well-ordering < – Grobner theory can again be elementarilygeneralized. However, as a consequence of the insolvability of theWord Problem, there are finitely generated two-sided ideals I suchthat T( f ) : f ∈ I is not finitely generated. Notwithstanding that, inthis setting also Buchberger’s algorithm can easily be adapted in sucha way that it terminates if and only if I has a finite Grobner basis G, inwhich case G is returned; the modification consists only of restrictingthe Choose instruction o, in order to choose in each step an optimalS-polynomial to be treated.

The complexity of the algorithm is much less trivial.In the discussion we will restrict ourselves to the case of an ordering <

which is degree-compatible,11 that is such that, for each t1, t2 ∈ T

deg(t1) < deg(t2) ⇒ t1 < t2.

If, at some time, we will have to compute the normal form of a polyno-mial f , then, in principle, the inner-loop i in the step i could chooseeach term τ < T( f ); this suggests that we consider as a preliminarymeasure γ , the cardinality of all terms of degree bounded by the maxi-mal degree of the polynomials produced by the algorithm; such a polyno-mial is the largest S-polynomial between two elements of the output Grobnerbasis.12

The cardinality of the reduced Grobner basis of I is bounded by (much lessthan) γ , so that the number of S-polynomials to be tested is bounded by γ 2;each such S-polynomial could have as many terms as (but usually many lessthan) γ , implying that the complete Gaussian reductions, needed to test that itsnormal form is 0, would require γ 2 operations.

The good news from this analysis is that Buchberger’s algorithm has polyno-mial complexity (and of low degree, actually γ 4!) in terms of γ ; the bad newsis that γ is huge itself. In fact the cardinality of all terms in k[X1, . . . , Xn] ofdegree bounded by R is

(R+nn

) ≈ Rn . While it is true that our analysis has beenquite casual, there is no advantage in trying a deeper analysis: assuming thatour input is nothing more than the set of all terms of degree R, their number isstill

(R+n−1n−1

) ≈ Rn−1. In conclusion, there is no way of getting a better valuethan γ for measuring the complexity, and, on the basis of that parameter, thecomplexity is as good as γ 4.

11 But we will show in the next chapter (see Proposition 23.2.7) that this restriction is wlog.12 In principle, each S-polynomial should be tested for the existence of a Grobner representation.

108 Buchberger

But the bad news is not yet over: clearly the highest degree of the polyno-mials produced by the algorithm can be measured by

G(I) := maxdeg(g) : g ∈ Gwhere G denotes the output basis. This value can be evaluated in terms of

• n, the number of variables,• D := maxdeg( f ) : f ∈ F, the maximal degree of the elements of the

input basis,• d, the dimension 13 of I;

under strong assumptions14 and the result is

G ≤ (D + 1)(n−d)2d.

Unlike the previous evaluation, this is a quite careful one, and there areexplicit examples (Mayr–Meyer examples), for each (δ, ν) ∈ N

2, δ ≥ 2 of anideal for which we have

G := δ2ν−1, D = δ + 2, n = 10ν + 2.

13 For the definition see Section 27.11.14 The ideal must be homogeneous and in generic position; the term ordering must be the de-

grevlex ordering.

23

Macaulay I

When Buchberger’s algorithm (1965) became available within the algebraicgeometry community, two unrelated results by Macaulay were seen in a differ-ent perspective. They are

• Macaulay’s remark (Lemma 23.3.1 and Corollary 23.3.2) that an ideal I andits monomial ideal T(I) have the same Hilbert function, thus combinatoriallyallowing us to deduce information on H(T ; I);

• the notion of H -basis (Definition 23.2.1) which mimics the notion ofGrobner bases using linear forms in place of maximal terms and whose com-putation was performed by Macaulay (Example 23.7.1) a la Buchberger bycomputing the syzygies among the leading forms of the bases and liftingthem to relations between the basis elements.

The earliest research aimed at computing ideal theoretical problems by ap-plying the Grobner technology introduced by Buchberger was strongly influ-enced by these ideas of Macaulay; they provided a specific paradigm, whichreduced the computational problems for ideals to the corresponding combina-torial problems over monomials. For instance:

• the problem of computing the Hilbert function of an ideal I, follow-ing Hilbert’s argument, is easily reduced to a combinatorical inclusion–exclusion counting of monomials (Corollary 23.4.3);

• a deeper analysis and generalization of Macaulay’s H-basis computationled Spear and Schreier to formulate and prove the Lifting Theorem (The-orem 23.7.3) which is the basis of the algorithms for computing resolutions.

This paradigm of Macaulay is behind all the applications of Grobner tech-nology in computational algebraic geometry and can be considered, jointlywith Buchberger theory, as responsible for the successful introduction of ef-fective methods in algebraic geometry. This chapter will illustrate Macaulay’s

109

110 Macaulay I

paradigm, applying it to the computation of a Hilbert function and resolutionof a polynomial ideal.

After a preliminary section discussing homogenization and affinization ofideals (Section 23.1), I introduce Macaulay’s notion of H-bases (Section 23.2)and Macaulay’s lemma relating the Hilbert function of an ideal with that of itsmonomial ideal (Section 23.3).

I then discuss how the central ideal theoretical problem of computing theHilbert function and resolution of an ideal can be combinatorially solved formonomial ideals by means of the Taylor resolution (Section 23.4) and presentthe best available algorithm for Hilbert function computation, the ‘Divide-and-Conquer’ Algorithm (Section 23.5).

After an explanatory discussion on the relation between Grobner basesand H-bases of a module (Section 23.6), I will discuss the Lifting Theorem(Section 23.7) and its direct application to the computation of resolutions(Section 23.8).

Finally (Section 23.9) I will present Macaulay’s criticism of Kronecker’ssolver (Theorem 20.4.1) whose double exponentiality is proved by Macaulay;in this section I will also present the well-known Grete Hermann bound. In anappendix (Section 23.10) I will refer to the recent results on the Nullstellensatzbound.

23.1 Homogenization and Affinization

If we associate to each point (x1, . . . , xn) in the affine space kn the projectivepoint in P

n(k) whose homogeneous coordinates are (1, x1, . . . , xn), we definean immersion kn → P

n(k) whose image is

(x0, x1, . . . , xn) ∈ Pn(k) : x0 = 0,

that is the complement of the improper hyperplane or hyperplane at infinityX0 = 0, and whose inverse is the map which associates to each projective point(x0, x1, . . . , xn) ∈ P

n(k), x0 = 0, the affine point (x1/x0, . . . , xn/x0) ∈ kn .Before discussing the relation between affine and projective varieties im-

plied by this identification, it is better to discuss the relation between non-homogeneous and homogeneous ideals. The basis of that is of course therelation between k[X1, . . . , Xn] and k[X0, X1, . . . , Xn] which mimics the onebetween the spaces.

Let us consider the maps

h– : k[X1, . . . , Xn] → k[X0, X1, . . . , Xn]

23.1 Homogenization and Affinization 111

anda– : k[X0, X1, . . . , Xn] → k[X1, . . . , Xn]

defined by

h f (X1, . . . , Xn) := Xdeg( f )

0 f

(X1

X0, . . . ,

Xn

X0

),

a f (X0, X1, . . . , Xn) := f (1, X1, . . . , Xn).

The image of h– is the set of all the forms, that is homogeneous polynomials,in k[X0, X1, . . . , Xn] and, as a consequence, a– is to be considered restricted toforms only. Within this restriction, both h– and a– are polynomial morphisms.

Note that, while h is injective, this is not true for a since, for each formf ∈ k[X0, X1, . . . , Xn] and each t ∈ N we have a f = a(Xt

0 f ).Therefore, while ah f = f for each f ∈ k[X1, . . . , Xn], in general ha f = f.More precisely, each form f ∈ k[X0, X1, . . . , Xn] can be uniquely writ-

ten as f = X δ0g, with g homogeneous and g ∈ (X0), so that a f = ag and

ha f = g. Therefore ha– is the identity only on the forms g ∈ (X0),1 while in

general the effect of applying ha– to a form is the removal of each factor X δ0

from it.In this context note that, while deg(h f ) = deg( f ), we have

δ := deg( f ) − deg(a f ) ≥ 0 and f = X δ0

ha f.

As a consequence, we have to take care when extending h– and a– to ideals;in fact, while, for any homogeneous ideal I,

aI := a f : f a form in Iis an ideal, for an ideal I ⊂ k[X1, . . . , Xn], the set of forms h f : f ∈ I isnot the set of all forms belonging to the ideal generated by it, since it does notcontain the forms Xt

0h f , therefore the correct definition is

hI := SpankXt0

h f : f ∈ I, t ∈ N.With this definition:

Lemma 23.1.1. We have:

(1) ahI = I, for any ideal I ⊂ k[X1, . . . , Xn];(2) for any homogeneous ideal I, there is m ≥ 1 such that

haI ⊃ I ⊃ Xm0

haI.

1 This of course, parallels the necessary removal of the improper hyperplane X0 = 0 in order toidentify projective and affine points.

112 Macaulay I

Moreover, while for a homogeneous ideal

( f1, . . . , fs) =: I ⊂ k[X0, X1, . . . , Xn]

given through a basis, we have aI = (a f1, . . . ,a fs), for any ideal

( f1, . . . , fs) =: I ⊂ k[X1, . . . , Xn]

the relation between the ideals hI and I := (h f1, . . . ,h fs), is a bit more com-

plex.

Example 23.1.2 (Macaulay). Let I := ( f1, f2) ∈ k[X1, X2, X3] where

f1 := X21, f2 := X2 + X1 X3.

Then, both

f3 := X1 X2 = −X3 f1 + X1 f2,

f4 := X22 = X2

3 f1 + (X2 − X1 X3) f2

belong to I. Therefore h f3,h f4 ∈ hI but, while

X0h f3 = X0 X1 X2 = −X0 X3

h f1 + X1h f2

and

X20

h f4 := X20 X2

2 = X20 X2

3h f1 + (X0 X2 − X1 X3)

h f2,

belong to (h f1,h f2), it is easy to verify that h f3,

h f4 and X0h f4 are not in

(h f1,h f2).

Remark 23.1.3. Let f ∈ I, d := deg( f ), and choose a representation

f =∑

i

gi fi

which minimizes γ := maxdeg(gi ) + deg( fi ).Then, while Xγ−d

0h f ∈ I, Xe

0h f ∈ I for each e < γ − d.

As a consequence, using the notation above, writing, for any polynomial fand any (not necessarily homogeneous) ideal I ⊂ k[X0, . . . , Xn],

(I : f ) := g ∈ k[X0, X1, . . . , Xn] : f g ∈ I,(I : f ∞)

:= g ∈ k[X0, X1, . . . , Xn] : ∃ρ ∈ N, f ρg ∈ I,and remarking that

I ⊂ (I : f ) ⊂ (I : f 2) ⊂ · · · ⊂ (I : f ∞)

,

we have

23.1 Homogenization and Affinization 113


(1) if f ∈ I, exists δ : X δ0

h f ∈ I;(2) hI = (

hI : X∞0

) = (hI : X0

);

(3) I = (I : X0) ⇐⇒ I = hI;(4) if each f ∈ I has a representation f = ∑

i gi fi in terms of ( f1, . . . , fs)

which satisfies deg( f ) ≥ deg(gi ) + deg( fi ) for each i, then hI = I.

Proof. All the statements are elementary; the only one which may need a com-ment is the proof of I = (I : X0) ⇒ I ⊃ hI : if g ∈ hI, there is f ∈ I suchthat g = h f ; by (1) we deduce that g ∈ (

I : X∞0

) = I.

Corollary 23.1.5. For any homogeneous ideal I we have

haI = I ⇐⇒ I = (I : X0) .

Corollary 23.1.6. The maps h− and a− between ideals in k[X1, . . . , Xn] andhomogeneous ideals I ⊂ k[X0, X1, . . . , Xn] satisfying I = (I : X0) are inverseto each other and preserve inclusion and usual ideal-theoretical operations.

Under the natural identification of the affine space kn with its image inP

n(k), we can associate to each projective variety Z ⊂ Pn(k), the set

aZ := Z ∩ kn .

Recalling that a projective variety is, by definition, the set of roots of a ho-mogeneous ideal,

Lemma 23.1.7. For each homogeneous ideal I ⊂ k[X0, X1, . . . , Xn] we have

a(Z(I)) = Z(aI).

Proof. One has

(1, x1, . . . , xn) ∈ Z(I) ⇐⇒ F(1, x1, . . . , xn) = 0 for each form F ∈ I,

⇐⇒ aF(1, x1, . . . , xn) = 0 for each form F ∈ I,

⇐⇒ G(x1, . . . , xn) = 0 for each G ∈ aI,

⇐⇒ (x1, . . . , xn) ∈ Z(aI)

proves that aZ is an affine variety.To any affine variety Z ⊂ kn one can associate its projective closure hZ,

which can be defined as the smallest projective variety containing it, or, in

114 Macaulay I

ideal-theoretical terms, as

hZ = Z(h(I(Z))).

There is a result similar to Corollary 23.1.6 but its enunciation and proofrequire technology outside the scope of this book. We limit ourselves thereforeto recording it with no comment:2

Fact 23.1.8. The maps h– and a– between affine varieties in kn and projectivevarieties in P

n(k) having no irreducible component at infinity are inverse toeach other and preserve inclusion.

Moreover we have

• h(Z(I)) = Z(hI) for each ideal I ⊂ k[X1, . . . , Xn],• a(Z(I)) = Z(aI) for each homogeneous ideal I ⊂ k[X0, X1, . . . , Xn],• h(I(Z)) = I(hZ) for each affine variety Z ⊂ kn,• a(I(Z)) = I(aZ) for each projective variety Z ⊂ P

n(k).

23.2 H-bases

In connection with Remark 23.1.3, Macaulay introduced

Definition 23.2.1 (Macaulay).

• For each f = ∑di=1 fi ∈ k[X1, . . . , Xn] where fi are the homogeneous

components of f , and fd = 0 so that deg( f ) = d, write H( f ) := fd .• A subset F := f1, . . . , fs) of the ideal I ⊂ k[X1, . . . , Xn] is called an H -

basis if HF := H( f1), . . . , H( fs) is a basis of the homogeneous idealH(I) generated by HI := H(g) : g ∈ I.

stating immediately

Proposition 23.2.2 (Macaulay). Let I ⊂ k[X1, . . . , Xn] be an ideal and let( f1, . . . , fs) be an H-basis of I. Then

(1) for each f ∈ I, there are polynomials g1, . . . , gs ∈ k[X1, . . . , Xn]such that

f =n∑

i=1

gi fi and, for each i, deg( f ) ≥ deg(gi ) + deg( fi );

(2) ( f1, . . . , fs) is a basis of I;(3) hI = (h f1, . . . ,

h fs).

2 Compare, for example, O. Zarishi and P. Samuel, Commutative Algebra Vol. I, Van Nostrand(1958), p. 190.

23.2 H-bases 115

Proof. We need to prove only (1) since (2) is an obvious consequence and then(3) follows directly from Lemma 23.1.4.

The proof of (1) essentially mimics that implied by the algorithm of Fig-ure 22.1: let f ∈ I; by assumption there are homogeneous polynomials gi suchthat

H( f ) =∑

i

gi H( fi ) and, for each i, deg(gi ) = deg( f ) − deg( fi );

therefore f ′ := f − ∑i gi fi is such that deg( f ′) < deg( f ). The claim then

follows by induction and the required representation can be produced by re-cursive computation.

Historical Remark 23.2.3. While Macaulay’s notion is obviously related to the(Gordan) notion of Grobner bases, and Macaulay proposed (independently?)the same rewriting construction given by Gordan, his definition of H-bases iscompletely unrelated to rewriting. In fact he introduced H-bases in

F. S. Macaulay, The Algebraic Theory of Modular Systems, Section 38

as an ‘immediate consequence’ of Hilbert’s (homogeneous) Basissatz, whichhe stated and proved in the preceding section (Section 37) ‘following substan-tially Konig’s’ proof, a proof not very different from the one we have recordedfor Theorem 20.8.1.

It is worth quoting Macaulay’s introduction to the notion of H-bases:

The following is an immediate consequence of the theorem:3

Any module of polynomials has a basis consisting of a finite number of members.

To prove this it is only necessary to show that a complete linearly independent setof members of any module can be arranged in a definite order in an infinite series. Ifl is the lowest degree of any member we can first take any complete set of membersof degree l, then any complete set of members of degree l + 1 whose terms of degreel + 1 are linearly independent, then a similar set of members of degree l + 2, and so on.In this way a complete linearly independent set of members is obtained in a differentorder.. . .

Consider a complete linearly independent set of members of a given module M ,not an H-module, arranged in a series in the order described above; and make all themembers homogeneous by introducing a new variable x0. We then have a series of ho-mogeneous polynomials belonging to an H-module M0, whose basis consists of a finitenumber of members of the series. The module M0 is called the H-module equivalentto M , and a basis of M obtained from any basis of M0 by putting x0 = 1 is calledan H-basis of M . The distinctive property of an H-basis (F1, F2, . . . , Fk) of M is thatany element F of M can be put in the form A1 F1 + A2 F2 + · · · + Ak Fk where Ai Fi

3 Hilbert’s Basissatz.

116 Macaulay I

(i = 1, 2, . . . , k) is not of greater degree than F . Every module has an H-basis, whichmay necessarily consist of more members than would suffice for a basis in general.. . .

In any basis (F1, F2, . . . , Fk) of an H-module in which no member is irrelevant, i.e.no Fi = 0 mod (F1, . . . , Fi−1, Fi+1, . . . , Fk), the number of members of each degreeis fixed; as can be easily seen by arranging F1, F2, . . . , Fk in order of degree. Hence inany H-basis of a module in which no member is irrelevant the number of members ofeach degree is fixed. On account of this and the other properties of an H-basis mentionedabove an H-basis gives a simpler and clearer representation of a module than a basiswhich is not an H-basis.

It is also interesting to note that while, both in this text and in the extendeduse of these notions by the Grobner school, the letter ‘H’ apparently stands forhomogeneous, in a previous paper:

F. S. Macaulay, On the Resolution of a Given Modular System into Primary SystemsIncluding Some Properties of Hilbert Numbers, Math. Ann. 74 (1913), 66–121,

where he had already introduced the notion in a more compact but essentiallysimilar way, the ‘H’ of ‘H-module’ and ‘H-basis’ explicitly stands for ‘Hilbert-module’ as ‘a module having a basis whose members are all homogeneouspolynomials, not necessarily of the same degree’ in contrast to the notion of‘K-module’ which stands for ‘Kronecker-module’ and is ‘a module in general,and as a rule has not any basis all members of which are homogeneous’; and to‘simple N-module (simple Noether-module)’, that is ‘a module which containsthe origin and no other point’ (see Historical Remark 30.4.2).

There is, of course, an obvious connection between Grobner bases and H-bases:

Lemma 23.2.4. Let < be a degree-compatible term ordering and let G be aGrobner basis of I w.r.t. <. Then:

• for each f ∈ k[X1, . . . , Xn], T( f ) = T(H( f )),• G is an H-basis of I, and• HG = H(g), g ∈ G) is a Grobner basis of H(I).

Proof. By assumption, each f ∈ I has a strong Grobner representation

f =µ∑

i=1

ci ti gi , with ci ∈ k \ 0, ti ∈ T , gi ∈ G,

and

T( f ) = t1T(g1) > · · · > ti T(gi ) > · · · .

23.2 H-bases 117

Since < is degree-compatible there is ν such that deg( f ) = deg(ti fi ), ifi ≤ ν, while deg( f ) > deg(ti fi ), if i > ν.

As a consequence we have H( f ) = ∑νi=1 ci ti H(gi ), which is a strong

Grobner representation.

Corollary 23.2.5. Let < be a degree-compatible term ordering and let G bean H-basis of I. Then the following conditions are equivalent:

• G is a Grobner basis of I w.r.t. <;• HG := H(g), g ∈ G) is a Grobner basis of H(I) w.r.t. <.

Proof. G is a Grobner basis of I w.r.t. <, iff for each f ∈ I there is g ∈ G suchthat T(g) = T(H(g)) divides T( f ) = T(H( f )) iff HG is a Grobner basisof H(I) w.r.t. <.

This lemma has a nice converse.

Definition 23.2.6. For any term ordering < on k[X1, . . . , Xn] the homoge-nization of < is the following term ordering <h on k[X0, X1, . . . , Xn]:

t1 <h t2 ⇐⇒ deg(t1) < deg(t2) or deg(t1) = deg(t2) and at1 < at2.

Proposition 23.2.7 (Lazard). Let

( f1, . . . , fs) ⊂ k[X1, . . . , Xn]

be a basis of I and let (g1, . . . , gr ) be a Grobner basis of I := (h f1, . . . ,h fs)

w.r.t. <h.Then (ag1, . . . ,

agr ) is a Grobner basis of I w.r.t. <.

Proof. If f ∈ I, then there is m for which g := Xm0

h f ∈ I. So

T(g) = Xm0 T(h f ) = Xm+e

0 T( f ).

By assumption there are a term t and a basis element gi such that

Xm+e0 T( f ) = T(g) = tT(gi ), and T( f ) = aT(g) = ataT(gi ) = atT(agi).

Corollary 23.2.8. Let I ⊂ k[X0, X1, . . . , Xn] be a homogeneous ideal satis-fying I : X0 = I, ( f1, . . . , fs) a homogeneous basis of I satisfying fi = ha fi

for each i , < a term ordering on k[X1, . . . , Xn].

118 Macaulay I

Then there are homogeneous polynomials

pi j ∈ k[X1, . . . , Xn], deg(pi j ) + deg( f j ) = deg(pil) + deg( fl), ∀ i, j, l

such that

(1) (H(ag1), . . . , H(agr )), H(agi ) = ∑j pi j H(a f j ), is a reduced

Grobner basis of H(aI) w.r.t. <;(2) (ag1, . . . ,

agr ), agi = ∑j pi j

a f j , is a Grobner basis of aI w.r.t. <;

(3) (g1, . . . , gr ), gi = hagi = ∑j pi j f j , is a Grobner basis of I w.r.t. <h.

Proof. Let us consider the homogeneous ideal J ⊂ k[X1, . . . , Xn] generatedby (H(a f1), . . . , H(a fs)); then clearly J ⊂ H(aI).

Moreover, for any homogeneous f ∈ I, by assumption

g := ha f ∈ I : X∞0 = I

so that g = ∑i pi fi and

H(ag) = g(0, X1, . . . , Xn)

=∑

i

pi (0, X1, . . . , Xn) fi (0, X1, . . . , Xn)

=∑

i

pi (0, X1, . . . , Xn)H(a fi )

∈ J

so that J = H(aI).Let (h1, . . . , hr ) ⊂ k[X1, . . . , Xn] be a reduced Grobner basis of J and let

pi j ∈ k[X1, . . . , Xn], deg(pi j ) + deg( f j ) = deg(hi ),

be the homogeneous polynomials such that hi = ∑j pi j H(a f j ) and define

gi :=∑

j

pi j f j , for each i.

Then, for each i ,

agi = ∑j pi j

a f j ,H(agi ) = ∑

j pi j H(a f j ) = hi ,hagi = ∑

j pi jha f j = ∑

j pi j f j = gi .

Then

(1) by construction, (H(ag1), . . . , H(agr )) is a Grobner basis of H(aI)w.r.t. <,

(2) (ag1, . . . ,agr ), agi = ∑

j pi ja f j is a Grobner basis of aI w.r.t. < by

Corollary 23.2.5,

23.3 Macaulay’s Lemma 119

(3) for each f ∈ I, there is e ∈ N so that f = Xe0

ha f ; a f ∈ aI and there arei and a term t so that T<(a f ) = tT<(agi ) = tT<(gi ) and T<h ( f ) =Xe

0tT<(gi ).

23.3 Macaulay’s Lemma

Oddly,4 while Macaulay explicitly applied the same construction as Gordan,he did not connect the underlying term reduction with the maximal-form-reduction implied and used (but in fact never stated) by the notion of H-bases,being just interested in producing, via a weaker version of Lemma 22.2.12, amonomial ideal J which has the same Hilbert function of a given ideal I:

Corresponding to any given H-ideal M [. . . ] we can deduce two corresponding p.p.-ideals P, P ′, each of which has the same D series D0, D1, D2, . . . as M .5

The first, P , is the ideal whose members of any degree l consist of the first 6 Dl [terms]in (x1, . . . , xn)l .. . .The second [monomial] ideal, P ′, is obtained thus: write the Dl members of the H-idealM of degree l so that their terms are in ascending order, and modify them linearly bymeans of one another so that no two members begin with the same term. The [terms]with which they begin are then the Dl [terms] of P ′ of degree l.F. S. Macaulay, Some Properties of Enumeration in the Theory of Modular Systems,Proc. London Math. Soc. 26 (1927), 533–4

In other words 7 Macaulay proved

4 Understanding the rest of this chapter requires the preliminary reading of Sections 20.6 and 20.7.5 That is two monomial ideals having the same Hilbert function as M .

Here Macaulay is considering a homogeneous ideal M ⊂ k[x1, . . . , xn ] and denotes, for eachl ∈ N, Dl := (l+n−1

n−1) − hH(l; M).

In this quotation, when he speaks of ‘members’, Macaulay means a linearly independent k-basis.

6 First with respect to

a definite order (which we shall call ascending order) according to the rule that xp11 x

p22 . . . x pn

n

comes before xq11 x

q22 . . . xqn

n if the first of the indices p1, p2, . . . , pn which differs from thecorresponding index in q1, q2, . . . , qn is greater than it.F. S. Macaulay, Some Properties of Enumeration in the Theory of Modular Systems, Proc. Lon-don Math. Soc. 26 (1927), 533

In other words he considers on T the degree-compatible term ordering under which any twoterms in Td are compared according to

Xa11 . . . Xar

n < Xb11 . . . Xbr

n ⇐⇒ there exists j : a j > b j and ai = bi for all i < j,

that is the degrevlex ordering induced by X1 < · · · < Xn .7 Actually, while his argument is obviously general, Macaulay stated his result for a single order-

ing and, oddly, this is not the degrevlex ordering induced by X1 < · · · < Xn which he was

120 Macaulay I

Lemma 23.3.1 (Macaulay). Let I ⊂ P := k[X1, . . . , Xn] and let < be a termordering. Then we have 8 P/I ∼= k[N(I)] ∼= P/T(I).

Proof. As in the proof of Lemma 22.2.12, the statement is a direct conse-quence of the algorithm of Figure 22.2.

Corollary 23.3.2. With the notation above, we have, for each l ∈ N

H(l; I) = H(l; T(I)) = #t ∈ N(I), deg(t) ≤ l.

In stating this result, Macaulay was not interested, as Buchberger, in a mem-bership test: his aim was to solve the following 9

Problem 23.3.3. To define a function Q(n, T ) : N2 → N which, for each n,

describes the bound (l + n

n − 1

)− hH(l + 1; I) ≥ Q(n, l)

explicitly considering, but the degree-lexicographical ordering induced by Xn < · · · < X1.In fact, unlike Buchberger, but similarly to Gordan, Macaulay associates to each (homoge-

neous) polynomial the term by which it ‘begins’, that is its minimal monomial, and uses it in his(Buchberger’s) term-reduction.

8 Where T(I) and N(I) are defined, in terms of <, as in Lemma 22.1.5.9 I am sticking to the original statement, notation and ordering used by Macaulay and by the

excellent report of his result given in

E. Sperner, Uber eine kombinatorischen Satz von Macaulay und seine Anwendungen auf dieTheorie der Polynomideale, Abh. Math. Semn. Hamburg 7 (1930), 149–163.

The reader must be aware that present developments of this theory turned Macaulay’s usageupside-down.

In particular:

• in Macaulay’s formula the homogeneous ideals are contained in k[X1, . . . , Xn ], instead ofk[X0, X1, . . . , Xn ] – while, of course, the theory is generalized to ‘Kronecker-modules’ I ⊂k[X1, . . . , Xn ] by the introduction of the homogenizing variable X0 and by reading the resultfrom that of the homogeneous ideals hI ⊂ k[X0, X1, . . . , Xn ];

• Macaulay considered on T , instead of the (degree) lexicographical ordering induced by X1 >

· · · > Xn , the degrevlex ordering induced by X1 < · · · < Xn or, as Sperner (op.cit. p. 150)put it:

Weiter ordnen wir die Potenzprodukte l-ten Graden lexikographisch. Das heißt,

xα11 · x

α22 · · · xαn

n komme vor xβ11 · x

β22 · · · xβn

n , wenn gilt

α1 = β1, α2 = β2, . . . , αi−1 = βi−1, αi > βi ;• according to Macaulay, L consists of the first – ersten in Sperner op. cit. p. 150 – terms of

each degree, instead of the last ones as in this new age version, so that the defined set is in anycase the same;

• Macaulay considered the polynomials represented as linear combinations of increasing terms,so that the ‘leading term’ is the minimal term, unlike in Buchberger theory where the maximalterm is considered: the effect is that the reduction sketched by Macaulay in the quoted pas-sage can be described as an application of Buchberger’s algorithms using the lexicographicalordering induced by X1 > X2 > · · · > Xn .

23.3 Macaulay’s Lemma 121

satisfied for each l by the Hilbert function hH(T ; I) of any homogeneous idealI ⊂ k[X1, . . . , Xn].

Macaulay’s solution of Problem 23.3.3 is split into two steps and requires usto prove that

(1) to each homogeneous ideal I ⊂ k[X1, . . . , Xn], it is possible to asso-ciate a monomial ideal J such that hH(T ; I) = hH(T ; J); this step isperformed in the statement we have quoted and gives Corollary 23.3.2.

(2) for each monomial ideal J ⊂ k[X1, . . . , Xn], denoting, for each l ∈ N,L(l) ⊂ Tl the set consisting of the first

(l+n−1n−1

) − hH(l; J) monomialsof degree l according to the degree reverse lexicographical ordering10

induced by X1 < · · · < Xn , the set L = ∪l∈NL(l) ⊂ T is a monomialideal and satisfies by construction, for each l,

hH(l; J) = hH(l; L) =(

l + n − 1

n − 1

)− #L(l)

so that

• hH(T ; J) = hH(T ; L),

• D(l) := Xiτ, 1 ≤ i ≤ n, τ ∈ L(l) ⊂ L(l + 1) ⊂ Tl+1 and• hH(l + 1; J) = hH(l + 1; L) = (l+n

n−1

) − #L(l + 1) ≤ (l+nn−1

) − #D(l).

Therefore, if we set Q(n, l) := #D(l) we have(l + n

n − 1

)− hH(l + 1; I) =

(l + n

n − 1

)− hH(l + 1; J)

=(

l + n

n − 1

)− hH(l + 1; L) =

≥ Q(n, l)

thus solving Problem 23.3.3.In this context, the role of Macaulay’s Lemma is just to guarantee the proof

of (1), allowing us to set J := T(I), but his role in the context of this book ismore relevant: it reduces the computation of the Hilbert function of a (homo-geneous or not) ideal to the easier case of monomial ideal for which combina-torial techniques are available.

In fact the result is stronger: knowledge of the Hilbert function of an idealallows us to deduce directly numerical invariants of it describing the propertiesof the corresponding variety (such as the dimension).

10 We recall that the reverse lexicographical ordering induced by X1 < · · · < Xn is the termordering on T defined by

Xa11 . . . Xar

n < Xb11 . . . Xbn

n ⇐⇒ there exists j : a j > b j and ai = bi for i < j;

122 Macaulay I

The same solution of Problem 23.3.3 illustrates the general scheme: a quitedifficult problem, like Macaulay’s bound, can be reduced to a combinatorialproblem.

In fact:

• if I ∈ k[X0, . . . , Xn] is homogeneous, then for any term ordering < we have

hH(l; I) = hH(l; T(I)) = #t ∈ N(I), deg(t) = l;• if I ∈ k[X1, . . . , Xn] is an (affine) ideal, then, for any degree-compatible

term ordering < we have

H(l; I) = H(l; T(I)) = #t ∈ N(I), deg(t) ≤ l =l∑

j=0

hH( j; T(I));

• we also have

H(T ; I) = H(T ; T(I)) = H(T ; T(H(I))) =T∑

l=0

hH(l; H(I))

and H(T ; I) = hH(T ; hI).

We are therefore able to reduce the computation of the Hilbert function of anideal I to that of the monomial ideal T(I).

23.4 Resolution and Hilbert Function for Monomial Ideals

The computation of the Hilbert function for any ideal being reduced in thisway to the monomial ideal case we will now discuss this combinatorialproblem: following Hilbert’s argument, we reduce the problem of computingHilbert function for a monomial ideal, to that of presenting a free resolutionof it.

If we are given a basis t1, . . . , ts of a monomial ideal 11

M ⊂ k[X1, . . . , Xn] =: P,

we will write, for 0 ≤ k < s:

• Ik := (i0, . . . , ik) : 1 ≤ i0 < i1 < · · · < ik ≤ s, which we will assume tobe ordered lexicographically;

• rk := #Ik = ( sk+1

);

• e(i0, . . . , ik) : (i0, . . . , ik) ∈ Ik for the canonical basis of the P-modulePrk ;

11 In practical situations, we will be given a Grobner basis g1, . . . , gs of an ideal I and we willhave ti := T(gi ), for each i, and M := T(I).

23.4 Resolution and Hilbert Function 123

• for each i := (i0, . . . , ik) ∈ Ik :

• T(i) := T(i0, . . . , ik) := lcm(ti0 , . . . , tik ),

• d(i) := deg(T(i0, . . . , ik)),

• for each j, 0 ≤ j ≤ k,

i j := (i0, . . . , i j−1, i j+1, . . . , ik) ∈ Ik−1, τ(i; j) := T(i)/T(i j) = T(i0, . . . , ik)/T(i0, . . . , i j−1, i j+1, . . . , ik), e(i; j) := e(i j) := e(i0, . . . , i j−1, i j+1, . . . , ik),

• for each j, l, 0 ≤ l < j ≤ k,

τ(i; l, j) := T(i0, . . . , ik)/T(i0, . . . , il−1, il+1, . . . , i j−1, i j+1, . . . , ik), e(i; l, j) := e(i0, . . . , il−1, il+1, . . . , i j−1, i j+1, . . . , ik).

We will also set

• δ0 the map δ0 : Pr0 → P defined by δ0(e(i)) = ti ,• δk , 0 < k < s, the map δk : Prk → Prk−1 defined by

δk(e(i0, . . . , ik)) =k∑

j=0

(−1) j+1τ(i0, . . . , ik; j)e(i0, . . . , i j−1, i j+1, . . . , ik).

Then, under this notation we have

Lemma 23.4.1 (Taylor). For a monomial ideal M = (t1, . . . , ts) ⊂ P, usingthe notation above, the sequence

0 → Prs−1δs−1−→ Prs−2 · · ·Prk+1

δk+1−→ Prkδk−→ Prk−1 · · ·Pr1

δ1−→ Pr0δ0−→ M

is a free-resolution (the Taylor resolution) of M.

Proof. The required verification that δk−1δk = 0, 1 ≤ k < s, is boring butstraightforward: we have just to note that for each i := (i0, . . . , ik) ∈ Ik :

δk−1δk(e(i)) =k∑

j=0

(−1) j+1τ(i; j)δk−1(e(i; j))

=k∑

j=0

(−1) j+1j−1∑l=0

(−1)l+1τ(i; l, j)e(i; l, j)

+k∑

j=0

(−1) j+1k∑

l= j+1

(−1)lτ(i; j, l)e(i; j, l)

=k∑

j=0

j−1∑l=0

((−1) j+l + (−1) j+l+1

)τ(i; l, j)e(i; l, j)

= 0.

124 Macaulay I

Im(δ0) = M is obvious and ker(δs−1) = 0 is a consequence of the fact thatrs−1 = 1.

Example 23.4.2. Let us reconsider the example we have developed throughoutSection 22.3, that is the ideal I generated by G = g1, g2, g3, g4 ⊂ k[X, Y ]where

g1 := Y 5 − Y 3, g2 := X2Y 2 − X2, g3 := X5 − X, g4 := XY 2 − X

which is a (non-reduced) Grobner basis, with respect to the lexicographicalorder < induced by X < Y . Therefore we can consider the (redundant) basis

T(G) = T(gi ), 1 ≤ i ≤ 4 = Y 5, X2Y 2, X5, XY 2of the monomial ideal T(G) = T(I) whose monomial structure is picturedin Remark 22.3.13 and which is again reproposed here in a slightly differentdescription:

......

......

......

......

• • • • • • • • · · ·• • • • • • • • · · ·A E F • • H • • · · · • • • • • • • · · · • • • • • • • · · · B C • • G • • · · · • • • · · · D • • · · ·

where

represents the terms t ∈ N(G),• represents the terms t ∈ T(G),A represents the term Y 5 = T(1),B represents the term XY 2 = T(4),C represents the term X2Y 2 = T(2) = T(2, 4),D represents the term X5 = T(3),E represents the term XY 5 = T(1, 4),F represents the term X2Y 5 = T(1, 2) = T(1, 2, 4),G represents the term X5Y 2 = T(2, 3) = T(3, 4) = T(2, 3, 4),H represents the term

X5Y 5 = T(1, 3) = T(1, 2, 3) = T(1, 3, 4) = T(1, 2, 3, 4).


The corresponding resolution is

0 → P δ3−→ P4 δ2−→ P6 δ1−→ P4 δ0−→ M (23.1)

where

δ1(e(1, 2)) = X2e(1) − Y 3e(2),

δ1(e(1, 3)) = X5e(1) − Y 5e(3),

δ1(e(1, 4)) = Xe(1) − Y 3e(4),

δ1(e(2, 3)) = X3e(2) − Y 2e(3),

δ1(e(2, 4)) = e(2) − Xe(4),

δ1(e(3, 4)) = Y 2e(3) − X4e(4);

δ2(e(1, 2, 3)) = −X3e(1, 2) + e(1, 3) − Y 3e(2, 3),

δ2(e(1, 2, 4)) = −e(1, 2) + Xe(1, 4) − Y 3e(2, 4),

δ2(e(1, 3, 4)) = −e(1, 3) + X4e(1, 4) − Y 3e(3, 4),

δ2(e(2, 3, 4)) = −e(2, 3) + X3e(2, 4) − e(3, 4),

δ3(e(1, 2, 3, 4)) = e(1, 2, 3) − X3e(1, 2, 4) + e(1, 3, 4) − Y 3e(2, 3, 4).

Corollary 23.4.3. For a monomial ideal M = (t1, . . . , ts) ⊂ P, using thenotation above, we have

HM(T ) =(

T + n − 1

n − 1

)+

s−1∑k=0

(−1)k+1∑i∈Ik

(T + n − d(i) − 1

n − 1

).

Example 23.4.4. Hilbert’s argument proving Corollary 20.7.1 is easily illus-trated by picturing, for each k, how many terms te(i0, . . . , ik) ∈ Prk satisfytT (i0, . . . , ik) = τ for each term τ ∈ T :

k = 0:

......

......

......

......

1 2 3 3 3 4 4 4 · · ·1 2 3 3 3 4 4 4 · · ·A E F 3 3 H 4 4 · · · 1 2 2 2 3 3 3 · · · 1 2 2 2 3 3 3 · · · B C 2 2 G 3 3 · · · 1 1 1 · · · D 1 1 · · ·

126 Macaulay I

k = 1:

......

......

......

......

0 1 3 3 3 6 6 6 · · ·0 1 3 3 3 6 6 6 · · ·A E F 3 3 H 6 6 · · · 0 1 1 1 3 3 3 · · · 0 1 1 1 3 3 3 · · · B C 1 1 G 3 3 · · · 0 0 0 · · · D 0 0 · · ·

k = 2:

......

......

......

......

0 0 1 1 1 4 4 4 · · ·0 0 1 1 1 4 4 4 · · ·A E F 1 1 H 4 4 · · · 0 0 0 0 1 1 1 · · · 0 0 0 0 1 1 1 · · · B C 0 0 G 1 1 · · · 0 0 0 · · · D 0 0 · · ·

k = 3:

......

......

......

......

0 0 0 0 0 1 1 1 · · ·0 0 0 0 0 1 1 1 · · ·A E F 0 0 H 1 1 · · · 0 0 0 0 0 0 0 · · · 0 0 0 0 0 0 0 · · · B C 0 0 G 0 0 · · · 0 0 0 · · · D 0 0 · · ·

In the following table we summarize the result: in each line we write howmany monomials of the module Prk exist at each term in the region labelledby one of the letters; the last row is the alternative sum of the values in each


column; of course, the result is always 1:

k = A B C D E F G H0 1 1 2 1 2 3 3 41 0 0 1 0 1 3 3 62 0 0 0 0 0 1 1 43 0 0 0 0 0 0 0 1

1 1 1 1 1 1 1 1

The resolution (23.2) is far from being minimal. For instance, it is sufficientto apply δ1 to the relation

δ2(e(1, 2, 3)) = −X3e(1, 2) + e(1, 3) − Y 3e(2, 3),

to deduce, since δ1δ2 = 0, that

δ1(e(1, 3)) = X3δ1(e(1, 2)) + Y 3δ1(e(2, 3)).

In this case, it is then possible to simplify the resolution: if we have, fori := (i0, . . . , ik) a relation

δk(e(i)) =k∑

j=0

(−1) j+1τ(i; j)e(i; j),

where

τ(i0, . . . , ik; J ) = 1

or, equivalently,

T(i0, . . . , ik) = T(i0, . . . , i J−1, i J+1, . . . , ik),

and J is the lowest value for which this happens,12 – in which case we willsay that that (i0, . . . , i J−1, i J+1, . . . , ik) is a consequence of i or i defines(i0, . . . , i J−1, i J+1, . . . , ik) – we can

• remove e(i) from Ik ,• replace with 0 each instance of e(i) in the definitions of δk+1(e), for each

e ∈ Ik+1,• remove e(i0, . . . , i J−1, i J+1, . . . , ik) from Ik−1,• replace with

k∑j=0j =J

(−1) j−J+1τ(i; j)e(i; j)

each instance of e(i, J ) in the definition of δk(e), for each e ∈ Ik .

12 So that (i0, . . . , i J−1, i J+1, . . . , ik ) is lexicographically higher than any other element(i0, . . . , i j−1, i j+1, . . . , ik ) for which τ(i; j) = 1.

128 Macaulay I

The same operation can be performed, more generally, if – after some suchsimplifications – we have a relation

δk(e(i0, . . . , ik)) =∑i∈Ik

citie(i), ti ∈ T ,

in which case we choose the lexicographically highest element j such thatts = 1, cs = 0, and we remove both e(i0, . . . , ik) and e(j), replacing them,respectively, with 0 and − ∑

i∈Iki=j

c−1j citie(i).

Algorithm 23.4.5. Once we have a resolution in order to make it minimal, weperform the following computations: while there is some relation

δk(e(i0, . . . , ik)) =∑i∈Ik

citie(i),

in which tj = 1 and cj = 0 holds for some j, we choose (among such relations)that relation for which k is maximal, and e(i0, . . . , ik) is lexicographicallyhighest in Ik+1, and we remove both e(i0, . . . , ik) and the lexicographicallyhighest such j, replacing them, respectively, with 0 and − ∑

i∈Iki=j

c−1j citie(i).

The final result is a minimal resolution; if this algorithm is applied to theTaylor resolution, the output is called the Taylor minimal resolution.

Example 23.4.6. Continuing our example,

• e(1, 2, 3, 4) defines e(1, 2, 3) = X3e(1, 2, 4) − e(1, 3, 4) + Y 3e(2, 3, 4),• e(2, 3, 4) defines e(3, 4) = −e(2, 3) + X3e(2, 4),• e(1, 3, 4) defines

e(1, 3) = X4e(1, 4) − Y 3e(3, 4) = X4e(1, 4) + Y 3e(2, 3) − X3Y 3e(2, 4),

• e(1, 2, 4) defines e(1, 2) = Xe(1, 4) − Y 3e(2, 4),• e(2, 4) defines e(2) = Xe(4);thus we obtain the minimal resolution

0 → P2 δ1−→ P3 δ0−→ M (23.2)

whereδ1(e(1, 4)) = Xe(1) − Y 3e(4),

δ1(e(2, 3)) = −Y 2e(3) + X4e(4).

Algorithm 23.4.7 (Easy hand-resolution algorithm). I want to discuss here analgorithm which allows us easily to compute by hand the resolution of a mono-mial ideal M := (t1, . . . , ts); such a resolution, while not minimal, is usuallymuch shorter than that of Taylor.


If we have, for each k, 0 ≤ k < s

(1) a partition Ik = R(s)k C(s)

k D(s)k ,

(2) a T -degree-compatible ordering 13 ≺ on C(s)k , that is an ordering such

that

T(i) < T(j) ⇒ i ≺ j,

(3) a bijection Φ(s)k : C(s)

k → D(s)k+1 such that, for j ∈ C(s)

k , i := Φ(s)k (j) ∈

D(s)k+1,

• there is J such that j = i J ,• T(i) = T(j), so that τ(i; J ) = 1,

• for any j = J such that τ(i; j) = 1, then i j ≺ j, 14

so that

δk(e(j)) =k∑

j=0j =J

(−1) j−J+1τ(i; j)δk(e(i; j)),

and, if we define

• sk := #R(s)k ;

• Psk the P-module whose canonical basis is

e(i0, . . . , ik) : (i0, . . . , ik) ∈ R(s)k ;

• Ψk : Prk → Psk the morphism such that, for each j ∈ Ik ,

• Ψk(e(j)) := e(j) if j ∈ R(s)k ,

• Ψk(e(j)) := 0 if j ∈ D(s)k ,

• Ψk(e(j)) := ∑kj=0j =J

(−1) j−J+1τ(i; j)Ψk(e(i; j)), 15 if j ∈ C(s)k and

i := Φ(s)k (j) ∈ D(s)

k+1 and J are such that j = i J ;

• γk : Psk → Psk−1 the morphism such that, for each i ∈ R(s)k ,

γk(e(i)) =k∑

j=0

(−1) j+1τ(i; j)Ψk−1e(i; j),

13 In the example we will solve ties on tuples lexicographically.14 So that, for any j = J , i j ≺ j since

τ(i; J ) = 1 ⇒ T(i j) | T(i) = T(j) ⇒ T(i j) < T(j) ⇒ i j ≺ j.

15 Note that Ψk (e(j)) is inductively well-defined since C(s)k is well-ordered by ≺.

130 Macaulay I

then

0 → Psk−1γk−1−→ · · ·Psk+1

γk+1−→ Pskγk−→ Psk−1 · · · γ1−→ Ps0

γ0−→ M (23.3)

is a (not necessarily minimal) resolution of M.Let us now inductively 16 assume that we have already produced the required

data for (t1, . . . , tσ−1) so that in order to extend the same data for (t1, . . . , tσ ),we essentially need to deal with the elements i := (i0, . . . , ik) ∈ Ik such thatik = σ, producing, for k, 1 ≤ k < σ , the partitions

(i0, . . . , ik) ∈ Ik, ik = σ = Q(σ )k E (σ )

k C (σ )k D(σ )

k R(σ )k ,

R(σ−1)k−1 = S(σ )

k−1 B(σ )k−1,

on the basis of which we define

R(σ )k−1 := R(σ )

k−1 ∪ S(σ )k−1,

C(σ )k−1 := C (σ )

k−1 ∪ B(σ )k−1 ∪ C(σ−1)

k−1 ∪ Q(σ )k−1,

D(σ )k := D(σ−1)

k ∪ D(σ )k ∪ E (σ )

k ,

for k, 1 ≤ k < σ , and

R(σ )σ−1 := R(σ )

σ−1, C(σ )σ−1 := C (σ )

σ−1 ∪ C(σ−1)σ−1 ∪ Q(σ )

σ−1.

Then for any

i := (i0, . . . , ik−1, σ ) ∈ Ik and j := i k = (i0, . . . , ik−1) ∈ Ik−1

we set

• i ∈ Q(σ )k and Φ

(σ)k (i) = I := (i0, . . . , il−1, L , il , . . . , ik−1, σ ) ∈ D(σ )

k+1 if

j ∈ C(σ−1)k−1 and Φ

(σ−1)k−1 (j) = (i0, . . . , il−1, L , il , . . . , ik−1);

• i ∈ E (σ )k if j ∈ D(σ−1)

k−1 ;• the difficult case is when j ∈ R(σ−1)

k−1 ; in this case

• if T(i) = T(j) then we include i in D(σ )k and j in B(σ )

k−1 and we set Φ(σ)k (j) =

i;• if T(i) = T(j) and there is a µ < k such that

i µ ∈ D(σ )k−1,

T(i µ) = T(i), so that τ(i; µ) = 1,

then for the maximal such µ, we

16 For σ = 1 we have R(1)0 := 1, while all the other data are ∅.


include i in D(σ )k ;

remove i µ from R(σ )k−1 and include it in C (σ )

k−1, setting Φ(σ)k−1(i µ) = i;

include j in S(σ )k−1;

• if T(i) = T(j) and, for each µ, T(i µ) = T(i) so that τ(i; µ) = 1, thenwe include i in R(σ )

k and j in S(σ )k−1.

Finally, we order

C(σ )k−1 := C (σ )

k−1 ∪ Q(σ )k−1 ∪ B(σ )

k−1 ∪ C(σ−1)k−1

by choosing any T -degree-compatible ordering < on both C (σ )k−1 and B(σ )

k−1 and

extending the T -degree compatible ordering ≺ from C(σ−1)k−1 to C(σ )

k−1 by set-ting 17

i ≺ j ⇐⇒

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

i ∈ C (σ )k−1 and j ∈ C (σ )

k−1 ⇒ i < j

i ∈ Q(σ )k−1, j ∈ C (σ )

k−1 and j ∈ Q(σ )k−2 ⇒ i′ < j′

i ∈ B(σ )k−1, j ∈ C (σ )

k−1 ∪ Q(σ )k−1 and j ∈ B(σ )

k−1 ⇒ i < j

i ∈ C(σ−1)k−1 , j ∈ C(σ−1)

k−1 and i ≺ j,

for any two elements i and j such that T(i) = T(j).The required data can therefore be iteratively computed – thus allowing us

to compute each value sk and each function γk and obtain the resolution shownin Equation 23.3 – by means of

• the algorithm of Figure 23.1 which computes each R(s)k , C (σ )

k−1, D(σ )k , R(σ )

k ,

S(σ )k−1, B(σ )

k−1, and• the algorithm of Figure 23.2 which, for any element i ∈ Ik , deduces in which

partition it is contained and recursively computes Ψk(e(i)).

Example 23.4.8. To illustrate this algorithm we consider the monomial ideal(see Example 23.5.5) I := (t1, . . . , t6) where

t1 := X4, t2 := X3Y 3, t3 := X3Y 2 Z , t4 := X3Y Z2, t5 := Y T 5, t6 := Y 2T .

Since it is easy to realize that the minimal resolution of (t1, t2, t3) is the Taylorresolution, we begin with

R(3)0 := 1, 2, 3,R(3)

1 := (1, 2), (1, 3), (2, 3),R(3)2 := (1, 2, 3),

17 Where, when i = (i0, i1, . . . , ik−1) ∈ Q(σ )k−1, we have ik−1 = σ, i = (i0, i1, . . . , ik−2, σ ) and

we set i′ := i k − 1 = (i0, i1, . . . , ik−2); analogously we set j′ := j k − 1 for j ∈ Q(σ )k−1.

132 Macaulay I

Fig. 23.1. Easy hand-resolution algorithm

(R(s)k , γk) := Resolution(t1, . . . , ts)

whereM is the monomial ideal generated by t1, . . . , ts ∈ Tsk := #R(s)

k , 0 ≤ k < s,

e(i0, . . . , ik) : (i0, . . . , ik) ∈ R(s)k is the canonical basis of the P-module

Psk , 0 ≤ k < s,the sequence (23.3) is a resolution of M.

σ := 1,R(σ )0 := σ , γ0(e(σ ) := tσ

While σ < s doσ := σ + 1,R(σ )

0 := σ , γ0((e(σ )) := tσFor k = 1..σ − 1 do

J (σ )k := (i0, . . . , ik−1, σ ) : (i0, . . . , ik−1) ∈ R(σ−1)

k−1 C(σ )

k−1 := D(σ )k := R(σ )

k := S(σ )k−1 := B(σ )

k−1 := ∅For j := (i0, . . . , ik−1) ∈ R(σ−1)

k−1 doi := (i0, . . . , ik−1, σ )If T(i) = T(j) then

D(σ )k := D(σ )

k ∪ iB(σ )

k−1 := B(σ )k−1 ∪ j

Φ(σ)k−1(j) := i

If exists µ < k such that• T(i µ) = T(i)• i µ ∈ D(σ )

kthen for the maximal such value µ do

D(σ )k := D(σ )

k ∪ iR(σ )

k−1 := R(σ )k−1 \ i µ

C(σ )k−1 := C(σ )

k−1 ∪ i µS(σ )

k−1 := S(σ )k−1 ∪ j

Φ(σ)k−1(i µ) := i

ElseR(σ )

k := R(σ )k ∪ i

S(σ )k−1 := S(σ )

k−1 ∪ jR(σ )

k−1 := R(σ )k−1 ∪ S(σ )

k−1

R(σ )σ−1 := R(σ )

σ−1For k = 0..s − 1 do

Rk := R(s)k

For (i0, . . . , ik) ∈ Rk doi := (i0, . . . , ik)For j = 0..k do

(case, Ψk(e(i; j)))γk(e(i)) := ∑k

j=0(−1) j+1τ(i; j)Ψk(e(i; j))


Fig. 23.2. Easy hand-resolution algorithm (cont.)

(case, Ψk(e(j)), ) := Proj(e(j))where

j := (i0, . . . , ik) ∈ Ikif

j ∈ R(s)k ⇒ case := R, Ψk(e(j)) := e(j)

j ∈ D(s)k ⇒ case := D, Ψk(e(j)) := 0

j ∈ C(s)k ⇒ case := C, Ψk(e(j)) := ∑

i∈R(s)k

citie(i), where

ci ∈ k, ti ∈ T , δk(e(j)) = ∑i∈R(s)

kcitiδk(e(i))

σ := ikIf

j ∈ R(σ )k then case := R, Ψk(e(j)) := e(j)

j ∈ D(σ )k then case := D, Ψk(e(j)) := 0

j ∈ C(σ )k ∪ B(σ )

k thencase := Ci := Φk(j)J such that j = i J

Ψk(e(j)) :=k+1∑j=0j =J

(−1) j−J+1τ(i; j)Ψk(e(i; j))

j ∈ R(σ )k ∪ D(σ )

k ∪ C(σ )k ∪ B(σ )

k thenLet µ < k be the highest value such that (i0, . . . , iµ) ∈ Rµ

Let τ > iµ be the highest value such that (i0, . . . , iµ) ∈ R(τ )k

ifτ < iµ+1 then

case := Ci := Φk(j) = (i0, . . . , iµ, τ, iµ+1, . . . , ik)Ψk(e(j)) = Ψk(e(i; µ + 1))

:=k+1∑j=0

j =µ+1

(−1) j−µτ(i; j)Ψk(e(i; j))

τ = iµ+1 and (i0, . . . , iµ+1) ∈ D(τ )µ+1 then

case := DΨk(e(j)) := 0

τ = iµ+1 and (i0, . . . , iµ+1) ∈ C(τ )µ+1 then

case := CLet ρ such that Φµ+1(i0, . . . , iµ+1) = (i0, . . . , iµ+1, ρ) ∈

Dµ+2i := Φk(j) = (i0, . . . , iµ+1, ρ, iµ+2, . . . , ik)Ψk(e(j)) = Ψk(e(i; µ + 2))

:=k+1∑j=0

j =µ+2

(−1) j−µ+1τ(i; j)Ψk(e(i; j))

134 Macaulay I

and we produce the following results 18

R(4)0 := 4,

R(4)1 := (1, 4), (3, 4),

C (4)1 := (2, 4),

D(4)2 := (2, 3, 4),

R(4)2 := (1, 3, 4),

C (4)2 := (1, 2, 4),

D(4)3 := (1, 2, 3, 4;

R(5)0 := 5,

R(5)1 := (1, 5), (2, 5), (3, 5), (4, 5),

R(5)2 := (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 4, 5), (3, 4, 5),

R(5)3 := (1, 2, 3, 5), (1, 3, 4, 5);

R(6)0 := 6,

R(6)1 := (1, 6), (2, 6), (3, 6), (5, 6),

S(6)1 := (1, 2), (1, 3), (2, 3), (1, 4), (3, 4), (1, 5), (4, 5),

B(6)1 := (2, 5), (3, 5),

C (6)1 := (4, 6),

D(6)2 := (3, 4, 6), (2, 5, 6), (3, 5, 6),

R(6)2 := (1, 2, 6), (1, 3, 6), (2, 3, 6), (1, 5, 6), (4, 5, 6),

S(6)2 := (1, 2, 3), (1, 3, 4), (1, 4, 5),

B(6)2 := (1, 2, 5), (1, 3, 5), (2, 3, 5), (3, 4, 5),

C (6)2 := (1, 4, 6),

D(6)3 := (1, 2, 5, 6), (1, 3, 5, 6), (2, 3, 5, 6), (3, 4, 5, 6), (1, 3, 4, 6),

R(6)3 := (1, 2, 3, 6), (1, 4, 5, 6),

S(6)3 := ∅,

B(6)3 := (1, 2, 3, 5), (1, 3, 4, 5),

C (6)3 := ∅,

D(6)4 := (1, 2, 3, 5, 6), (1, 3, 4, 5, 6);

so that

R(6)0 := (1), (2), (3), (4), (5), (6),

R(6)1 := (1, 2), (1, 3), (2, 3), (1, 4), (3, 4), (1, 5), (4, 5), (1, 6), (2, 6), (3, 6), (5, 6),

18 Where the notation (1, 3, 4) is a shorthand for

(1, 3, 4) ∈ D2, Φ−1(1, 3, 4) = (3, 4) ∈ C1.


R(6)2 := (1, 2, 3), (1, 3, 4), (1, 4, 5), (1, 2, 6), (1, 3, 6), (2, 3, 6), (1, 5, 6), (4, 5, 6),

R(6)3 := (1, 2, 3, 6), (1, 4, 5, 6).

The corresponding resolution is

0 → P2 γ3−→ P8 γ2−→ P11 γ1−→ P6 γ0−→ M (23.4)

where 19

i = e(1, 2) d(i) = 7 γ1(i) = Y 3e(1) − Xe(2),

i = e(1, 3) d(i) = 7 γ1(i) = Y 2 Ze(1) − Xe(3),

i = e(2, 3) d(i) = 7 γ1(i) = Ze(2) − Y e(3),

i = e(1, 4) d(i) = 7 γ1(i) = Y Z2e(1) − Xe(4),

i = e((3, 4) d(i) = 7 γ1(i) = Ze(3) − Y e(4),

i = e(1, 5) d(i) = 10 γ1(i) = Y T 5e(1) − X4e(5),

i = e(4, 5) d(i) = 11 γ1(i) = T 5e(4) − X3 Z2e(5);i = e(1, 6) d(i) = 7 γ1(i) = Y 2T e(1) − X4e(6),

i = e(2, 6) d(i) = 7 γ1(i) = T e(2) − X3Y e(6),

i = e(3, 6) d(i) = 7 γ1(i) = T e(3) − X3 Ze(6),

i = e(5, 6) d(i) = 7 γ1(i) = Y e(5) − T 4e(6),

i = e(1, 2, 3) d(i) = 8 γ2(i) = −Ze(1, 2) + Y e(1, 3) − Xe(2, 3),

i = e(1, 3, 4) d(i) = 8 γ2(i) = −Ze(1, 3) + Y e(1, 4) − Xe(3, 4),

i = e(1, 4, 5) d(i) = 12 γ2(i) = −T 5e(1, 4) + Z2e(1, 5) − Xe(4, 5),

i = e(1, 2, 6) d(i) = 8 γ2(i) = −T e(1, 2) + Y e(1, 6) − Xe(2, 6),

i = e(1, 3, 6) d(i) = 8 γ2(i) = −T e(1, 3) + Ze(1, 6) − Xe(3, 6),

i = e(2, 3, 6) d(i) = 8 γ2(i) = −T e(2, 3) + Ze(2, 6) − Y e(3, 6),

i = e(1, 5, 6) d(i) = 11 γ2(i) = −Y e(1, 5) + T 4e(1, 6) − X4e(5, 6),

i = e(4, 5, 6) d(i) = 12 γ2(i) = −Y e(4, 5) + T 4Ψ (e(4, 6)) − X3 Z2e(5, 6)

= −Y e(4, 5) + T 5e(3, 4)

+T 4 Ze(3, 6) − X3 Z2e(5, 6),

i = e(1, 2, 3, 6) d(i) = 9 γ3(i) = −Ze(1, 2) + Y e(1, 3) − Xe(2, 3),

i = e(1, 4, 5, 6) d(i) = 13 γ3(i) = Y e(1, 4, 5) − T 4Ψ (e(1, 4, 6))

+Z2e(1, 5, 6) − Xe(4, 5, 6)

= Y e(1, 4, 5) − T 5e(1, 3, 4) + T 4e(1, 3, 6)

+Z2e(1, 5, 6) − Xe(4, 5, 6).

19 Note that we are using the formulas

Ψ (e(4, 6)) = −T e(3, 4) + Ze(3, 6),

Ψ (e(1, 4, 6)) = −T e(1, 3, 4) + e(1, 3, 6) + XΨ (e(3, 4, 6)),

Ψ (e(3, 4, 6)) = 0.

136 Macaulay I

This computation allows us to apply Corollary 23.4.3 in order to deduce theHilbert function of M which is

HM(T ) =(

T + 3

3

)−

(T

3

)−

(T − 1

3

)− 4

(T − 3

3

)+ 9

(T − 4

3

)

− 5

(T − 5

3

)+

(T − 6

3

)+

(T − 7

3

)− 2

(T − 9

3

)+

(T − 10

3

)

= 7T + 7.

23.5 Hilbert Function Computation: the ‘Divide-and-Conquer’Algorithms

While this preliminary introduction, which essentially follows Hilbert’s ap-proach and takes advantage of Buchberger’s algorithm in order to effectivelyapply Macaulay’s suggestion of reducing the problem to the monomial case,gives an effective algorithm to compute Hilbert function, we are still very farfrom getting a sufficiently acceptable solution. The algorithm deduced fromCorollary 23.4.3 can only be considered as the starting point of ten years ofresearch towards an efficient solution, which culminated with what is consid-ered the ultimate proposal for Hilbert function computation, the ‘Divide-and-Conquer’ Algorithms.

Lemma 23.5.1. Let I = (t1, . . . , ts) ⊂ P be a monomial ideal and let τ ∈ T .Writing I′ := (t1, . . . , ts, τ ), and

I′′ := (I : τ) =(

t1gcd(t1, τ )

, . . . ,ts

gcd(ts, τ )

),

we have

H(I, T ) = H(I′, T ) + T deg(τ )H(I′′, T ). (23.5)

Proof. The disjoint union of

t ∈ N(I′) : deg(t) ≤ d and tτ ∈ N(I) : t ∈ T , deg(t) ≤ d − deg(τ )is t ∈ N(I) : deg(t) ≤ d. Moreover,

t ∈ T : deg(t) ≤ d −deg(τ ), tτ ∈ N(I) = t ∈ N(I′′) : deg(t) ≤ d −deg(τ )whence the result.

23.5 Hilbert Function Computation 137

Example 23.5.2. For instance in Example 23.4.2 if we set

I := (T(1), T(2), T(3)) = (Y 5, X2Y 2, X5), τ := T(4) = XY 2

we obtain I′′ := (Y 3, X) and we have

N(I′) = N(I) ∪ τ, Y τ, Y 2τ , N(I′′) = 1, Y, Y 2.

Corollary 23.5.3. We have

(1) H(I′, T ) = (1 − T c)(1 − T )−n, for I′ = (Xcm) ⊂ P;

(2) H(I′, T ) = (1 − T )−n ∏mi=1(1 − T ci ) for I′ = (Xc1

1 , . . . , Xcmm ) ⊂

P, m ≤ n.

(3) H(I′, T ) = (1 − T )−n(∏m

i=1(1 − T ci ) − T α∏m

i=1(T bi − T ci ))

forI′ = (Xc1

1 , . . . , Xcmm , ω) ⊂ P and

ω = Xb11 · · · Xbm

m Xam+1m+1 · · · Xan

n , ci > bi > 0, ai > 0, m ≤ n, α

=n∑

i=m+1

ai .

Proof. We obtain the results from Equation (23.5) and Corollary 20.7.4 bysetting

(1) I := (1), τ := Xcm so that I′′ := (1);

(2) I := (Xc11 , . . . , Xcm−1

m−1 ), τ := Xcmm so that I′′ := I.

(3) I := (Xc11 , . . . , Xcm

m ), τ := ω so that I′′ := (Xc1−b11 , . . . , Xcm−bm

m )

Algorithm 23.5.4 (‘Divide-and-Conquer’ Algorithms). Different algorithmsproposed in the early nineties produced, by means of Lemma 23.5.1, theHilbert series H(I, T ) as a combination of expressions T αi H(Ii , T ) where eachmonomial ideal Ii has the shape

Ii = (Xc11 , . . . , Xcm

m , ω), ω ∈ T , m ≤ n.

To reach this result, iteratively, one chooses an element Ii and a term τ andreplaces Ii with I′i := Ii + (τ ) and I′′i := (Ii : τ).

The difference in these algorithms is in the choice of the pivot τ to splitIi ; a deep analysis of these algorithms, taking into account also the cost ofdivisibility tests and of series expansion, has been performed in

A. M. Bigatti, Computation of Hilbert–Poincare series J. Pure Appl. Algebra 119(1997), 237–253.

138 Macaulay I

which suggests choosing as pivot a simple-power Xc jj ‘of the indeterminate

occurring most, with exponent being that of this indeterminate in the GCD oftwo randomly chosen generators.’

Example 23.5.5. Let us apply this algorithm to the monomial ideal

I := (X4, X3Y 3, X3Y 2 Z , X3Y Z2, Y 2T, Y T 5).

The computation performed chooses:

• I and τ := Y , returning

I1 := (X4, Y ), I2 := (X4, X3Y 2, X3Y Z , X3 Z2, Y T, T 5)

and

H(I, T ) = (1 − T 4)(1 − T )−3 + T H(I2, T );

• I2 and τ := Y , returning I3 := (X4, Y, X3 Z2, T 5), I4 := (X4, X3Y,

X3 Z , T ) and

H(I, T ) = (1 − T 4)(1 − T )−3 + T (1 − T 4)(1 − T 5)(1 − T )−3

− T 6(1 − T )−3 + T 2H(I4, T );

• I4 and τ := X3, returning I5 := (X3, T ), I6 := (X, Y, Z , T ) and

H(I, T ) = (1 − T 4)(1 − T )−3 + T (1 − T 4)(1 − T 5)(1 − T )−3

− T 6(1 − T )−3 + T 2(1 − T 3)(1 − T )−3 + T 5.

23.6 H-bases and Grobner Bases for Modules

In order to show how one can apply to the computation of ideal resolutionMacaulay’s paradigm of reducing a computational problem for ideals to a com-binatorial one over monomials, we first need to discuss briefly the generaliza-tion of H -bases and Grobner bases to the case of modules.

Let then P := k[X1, . . . , Xn] and let us consider a free-module Pm , whosecanonical basis is denoted by e1, . . . , em.

We have already remarked that in order to impose a graduation on Pm , itis sufficient to impose a degree on each ei , deg(ei ) := di , and then consideran element (g1, . . . , gm) ∈ Pm to be homogeneous of degree R if and only ifeach gi is either 0 or a homogeneous polynomial of degree R − di .

23.6 H-bases and Grobner Bases for Modules 139

More generally, we can see that Pm as a k-vectorspace has the basis

T (m) = tei , t ∈ T , 1 ≤ i ≤ m= Xa1

1 · · · Xann ei , (a1, . . . , an) ∈ N

n, 1 ≤ i ≤ m.Then, imposing on each ei a degree di is equivalent to imposing on each

modulo-term tei the degree deg(tei ) = di + deg(t).Then forms (i.e. homogeneous elements) of degree R in Pm are naturally the

linear combinations of the modulo-terms of degree R, that is those elements(g1, . . . , gm) ∈ Pm such that each gi is a homogeneous polynomial of degreeR − di (if not 0).

In this context, the notion (and the properties) of H-bases obviously gener-alizes:

Definition 23.6.1. If, for each f = ∑di=1 fi ∈ Pm – where fi are forms of

degree i and d = deg( f ) – we write H( f ) := fd , a subset (g1, . . . , gs) of themodule I ⊂ Pm is called an H -basis if H(g1), . . . , H(gs) is a basis of thehomogeneous module H(I) generated by HI := H(g) : g ∈ I.

As the syzygy module of a homogeneous module is homogeneous, weshould expect that something similar would happen also for the syzygies ofa monomial ideal. To obtain that we have just to generalize what is done forhomogeneous modules, and we get a hint from the monomial resolutions wehave already discussed: if, for each k and each i := (i0, . . . , ik) ∈ Ik we asso-ciate to e(i) the value

T − deg(e(i)) := T(i0, . . . , ik),

then in each relation

δk(i) =k∑

j=0

(−1) j+1τ(i; j)e(i; j),

we have

T − deg(i) = T(i0, . . . , ik)

= T(i0, . . . , ik)

T(i0, . . . , i j−1, i j+1, . . . , ik)T(i0, . . . , i j−1, i j+1, . . . , ik)

= τ(i; j)T − deg(e(i; j)),

making each relation ‘homogeneous’.Therefore if we define a ‘term-degree’ on Pm by assigning a term ωi ∈ T

to each ei , T − deg(ei ) := ωi , we can define a function

T − deg : T (m) → T by T − deg(tei ) = tωi ,

140 Macaulay I

call T -forms ( T -homogeneous elements) of T -degree ω any element

(γ1, . . . , γm) ∈ Pm

such that for each i

γi ∈ T , and γiωi = ω unless γi = 0,

and speak of T -homogeneous modules and T -homogeneous components.Then, both in the Taylor resolution and in the Taylor minimal resolu-

tion, each syzygy module is T -homogeneous and each morphism is T -homogeneous of T -degree 20 1.

We note that we can generalize the notion of H-basis in this context, pro-vided that we have imposed a term ordering < on T :

Definition 23.6.2. For each f = ∑t∈T ft ∈ Pm, where ft are T -forms of

T -degree t and

τ = T − deg( f ) = max<

t : ft = 0,

we denote L( f ) := fτ the leading form of f .A subset G := g1, . . . , gs of a module I ⊂ Pm is called a T-basis if

LG := L(g1), . . . ,L(gs)is a basis of the T -homogeneous module L(I) generated by LI = L(g) :g ∈ I.

The generalization of the notion of Grobner basis to a module is as straight-forward as that of H-basis;21 we have to impose a well-ordering ≺ on T (m)

and it seems natural to assume it is compatible with a fixed term ordering < onT , that is it is such that

t1 ≤ t2, τ1 τ2 ⇒ t1τ1 t2τ2

holds for each t1, t2 ∈ T , τ1, τ2 ∈ T (m). Then for any element

f =∑

τ∈T (m)

c( f, τ )τ ∈ P(m)

its maximal term is the term T( f ) := max≺t : c( f, τ ) = 0; its leading coeffi-cient is lc( f ) :=c( f, T( f )) and its maximal monomial is M( f ) := lc( f )T( f ).

As one can expect, the rest of the definitions are verbatim generalizations; forinstance:

20 In the classical case, the degrees are in the additive semigroup N whose identity is 0; here thedegrees are in the multiplicative semigroup T whose identity is 1.

21 More details can be found in Section 24.3.

23.6 H-bases and Grobner Bases for Modules 141

Definition 23.6.3. A subset G ⊂ I will be called a Grobner basis of the moduleI ⊂ Pm if

T(G) = T(I),

that is TG := T(g) : g ∈ G generates the module T(I) = TI := T(g) :g ∈ Iand

Lemma 23.6.4. Let ω1, . . . , ωm ∈ T , let di := deg(ωi ). Impose on Pm thegraduations defined by

deg(ei ) := di , T − deg(ei ) := ωi .

Let < denote a degree-compatible term ordering on T and ≺ a well-orderingon T (m) compatible with <.

Let I ⊂ Pm be a module and G be a basis of it. Then

• if G is a T-basis of I, then G is an H-basis of I and HG = H(g), g ∈ Gis a T-basis of H(I);

• if G is a Grobner basis of I, then G is a T-basis of I and LG = L(g), g ∈G is a Grobner basis of L(I);

• if G is a Grobner basis of I, then G is an H-basis of I and HG =H(g), g ∈ G is a Grobner basis of H(I).

Proof. It is sufficient to repeat verbatim the proof of Lemma 23.2.4

There is only one point which must be stressed and remembered: sincefor a module element f ∈ Pm , we have T( f ) = tei , while the notion ofS-polynomials is the same, it is possible that two module elements do notpossess an S-polynomial. Namely, for each f1, f2 ∈ Pm such that lc( f1) =1 = lc( f2), let us write

T( f1) =: t1ei1 , T( f2) =: t2ei2;then, if ei1 = ei2 =: ε, it is natural to define lcm(T( f1), T( f2)) := lcm(t1, t2)ε; but, if ei1 = ei2 , there is no way of combining the two elements in order tointerreduce their maximal terms. Therefore, in the module case the definiton is

Definition 23.6.5. For each f1, f2 ∈ Pm such that

lc( f1) = 1 = lc( f2), T( f1) = t1ei1 , T( f2) = t2ei2 ,

the S-polynomial of f1 and f2 exists only in case ei1 = ei2 in which case it is

S( f2, f1) := lcm(t1, t2)

t1f1 − lcm(t1, t2)

t2f2.

142 Macaulay I

23.7 Lifting Theorem

It is worth analysing the computation 22 of the H-basis of the ideal alreadypresented in Example 23.1.2:

Example 23.7.1. Let P := k[X1, X2, X3], hP := k[X0, X1, X2, X3], I :=( f1, f2) ∈ P where

f1 := X21, f2 := X2 + X1 X3.

Macaulay searched all elements (g0, g1, g2) ∈ (hP)3 such that

X0g0 = g1h f1 + g2

h f2; (23.6)

and, in order to do so, he

• affinized the equation, setting X0 = 0 and producing the equation

0 = H(g1)H( f1) + H(g2)H( f2) = H(g1)X21 + H(g2)X1 X3;

• solved it – that is he computed the syzygies among H( f1) and H( f2) –obtaining the set

(pX3, −pX1), p ∈ P, homogeneouswhich satisfies

(pX3)H( f1) + (−pX1)H( f2) = (pX3)X21 + (−pX1)X1 X3 = 0;

• substituted each solution in Equation (23.6), where we set

g1 = H(g1)+X0h1 = pX3+X0h1, g2 = H(g2)+X0h2 = −pX1+X0h2,

obtaining

X0g0 = (pX3 + X0h1)h f1 + (−pX1 + X0h2)

h f2

= X0(h1h f1 − X1 X2 p + h2

h f2);• and, putting f3 = h f3 := X1 X2 ∈ I deduced g0 ∈ ( f1, f2, f3) = I.

Again, he solved the equation

X0g0 = g1h f1 + g2

h f2 + g3h f3, (23.7)

by

22 In F. S. Macaulay, The Algebraic Theory of Modular Systems, Cambridge University Press(1916), p. 40.

23.7 Lifting Theorem 143

• considering the equation

0 = H(g1)H( f1) + H(g2)H( f2) + H(g3)H( f3)

= H(g1)X21 + H(g2)X1 X3 + H(g3)X1 X2;

• obtaining the solutions

(p1 X3 + p2 X2, −p1 X1 + p3 X2, −p2 X1 − p3 X3), p1, p2, p3 ∈ P hom.;• and substituting each solution in Equation (23.7) obtaining

X0g0 = (p1 X3 + p2 X2 + h1 X0)h f1

+ (−p1 X1 + p3 X2 + h2 X0)h f2

+ (−p2 X1 − p3 X3 + h3 X0)h f3

= X0(h1h f1 − X1 X2 p1 + X2

2 p3 + h2h f2 + h3

h f3),

getting g0 ∈ ( f1, f2, f3, f4) = I, where f4 = h f4 := X22 ∈ I.

Finally, the equations

X0g0 = g1h f1 + g2

h f2 + g3h f3 + g4

h f4,

0 = H(g1)H( f1) + H(g2)H( f2) + H(g3)H( f3) + H(g4)H( f4)

do not give new members, allowing us to deduce again g0 ∈ ( f1, f2, f3, f4),and that ( f1, f2, f3, f4) is the required H-basis.

Historical Remark 23.7.2. I leave it to the reader to evaluate whether this al-gorithm 23 anticipates of Buchberger’s S-polynomials and the lifting theorembelow.

In any case, I think that it is quite correct to present such algorithm as fol-lows:

Given a module I ⊂ Pr through a generating basis F := f1, . . . , ft , compute thesyzygy module

s := Syz((H( f1), . . . , H( ft )))

= (h1, . . . , ht ) ∈ P t :∑

i

hi H( fi ) = 0

and check whether, for each homogeneous syzygy σ ∈ s, exists

Σ := S(σ ) := (g1, . . . , gs) ∈ P t

such that

• H(Σ) = σ ,• Σ ∈ Syz( f1, . . . , ft , that is• ∑

i gi fi = 0.

23 ‘The method given is a general one’ comments Macaulay at the end of this computation.

144 Macaulay I

If this is the case, then

• F is an H-basis of the module it generates;• S := S(σ ) : σ ∈ s = Syz( f1, . . . , ft ;• s = H(S).

If this is not the case, one obtains elements f ∈ I such that

H( f ) ∈ H(F) = (H( f1), . . . , H( ft ));adding such elements to F one obtains a better basis F ′ of I such that(

H( f ) : f ∈ F)

(H( f ) : f ∈ F ′) = H(F ′) ⊂ H(I).

Therefore, this computation seems to me to be another instance of‘Macaulay’s paradigm’ for solving computation problems on ideals by reduc-ing them to their initial ideals, and a good introduction to the Lifting Theorem,which was independently discovered by Spear and Schreier.

Theorem 23.7.3 (Lifting Theorem). Let

• P := k[X1, . . . , Xn],• I ⊂ Pr be a T -graded module,• G := g1, . . . , gs be a basis of it,• Ps be graded so that T − deg(ei ) = T − deg(gi ), for each element in its

canonical basis e1, . . . , es,• σ1, . . . , σt be a (T -homogeneous) basis of

s := Syz(TG) = Syz(T(g1), . . . , T(gs)

=

(h1, . . . , hs) :∑

i

hi T(gi ) = 0

⊂ Ps ,

• S := Syz(G) = Syz(g1, . . . , gs = (h1, . . . , hs) :∑

i hi gi = 0 ⊂ Ps .


• G is a T-basis of I,• for each i, 1 ≤ i ≤ t, there is Σi := S(σi ) ∈ S such that L(Σi ) = σi ,

and imply that Σ1, . . . , Σt is a T-basis of S.

Proof. Compare Proposition 24.5.4.

This result holds (and was stated by Schreier) in particular if the T-homogeneous basis σ1, . . . , σt of s is the set of all S-polynomials amongthe elements of TG. In this context we can impose on Ps the term ordering≺ defined by t1ei1 ≺ t2ei2 iff

T −deg(t1ei1) < T −deg(t2ei2) or T −deg(t1ei1) = T −deg(t2ei2) and i1 < i2.

23.7 Lifting Theorem 145

Let us now write, for each i , gi := tiηki + ri , where T(gi ) = tiηki > T(ri )

and η1, . . . , ηr is the canonical basis of Pr .Then, for each i, j, i < j, ηki = ηk j , let us write

σ(i, j) := t (i j)j e j − t (i j)

i ei := lcm(ti , t j )

t je j − lcm(ti , t j )

tiei ∈ s

and note that

• σ(i, j), i < j, ηki = ηk j is a homogeneous basis of s;• each S-polynomial S(gi , g j ) is obtained from σ(i, j) by evaluating each ek

as gk .

Moreover, for each σ(i, j), since, by assumption, G is a T-basis,

S(gi , g j ) = t (i j)j g j − t (i j)

i gi

has a Grobner representation∑

k h(i j)k gk ; therefore if we define

Σ(i, j) := t (i j)j e j − t (i j)

i ei −∑

k

h(i j)k ek,

we have

T(Σ(i, j)) = lcm(ti , t j )

t je j , L(Σ(i, j)) = σ(i, j).

In this context Theorem 23.7.3 informs us that if G is a T-basis of I then

Σ1, . . . , Σt := Σ(i, j), S(gi , g j ) there exists is a T-basis of S. But something more can be stated:

Proposition 23.7.4 (Schreier). With the notation and assumptions above, theconditions of Theorem 23.7.3 imply that Σ1, . . . , Σt is a Grobner basis of S

w.r.t. <.

Proof. Let us consider any element Σ := (h1, . . . , hs) ∈ S and let T(h j )

e j := T(Σ). Since∑

k hk gk = 0, there are some i < j such that η := ηki =ηk j and

T(hi )T(gi ) = T(hi )tiηki = T(h j )t jηk j = T(h j )T(g j )

and a term τ ∈ T such that

T(h j )T(g j ) = T(h j )t jη = τ lcm(ti , t j )η = T(hi )tiη = T(hi )T(gi ),

T(h j )T(g j ) = T(h j )t jηk j = τ lcm(ti , t j )ηk j

146 Macaulay I

so that

T(Σ) = T(h j )e j = τlcm(ti , t j )

t je j = τT(Σ(i, j)).

23.8 Computing Resolutions

Theorem 23.7.3 and Proposition 23.7.4 give directly the algorithm which (withsome improvement) is implemented in most computer algebra systems:

Algorithm 23.8.1 (Schreier). Let P := k[X1, . . . , Xn] and M0 ⊂ Pr be themodule generated by a basis F := f1, . . . , ft . Then:

• compute a Grobner basis G0 := g(0)1 , . . . , g(0)

r0 of M0 producing at thesame time:

• the set

U0 :=(i, j) : 1 ≤ i < j ≤ r0, S(g(0)

i , g(0)j ) there exists

so that

t (i j)j T(g(0)

j ) − t (i j)i T(g(0)

i ) : (i, j) ∈ U0

generates Syz(T(g(0)

1 ), . . . , T(g(0)r0 ));

• for each (i, j) ∈ U0, a Grobner representation

t (i j)j g(0)

j − t (i j)i g(0)

i =∑

k

h(i j)k g(0)

k ;

• the Grobner basis (w.r.t. ≺)

G1 :=

g(1)1 , . . . , g(1)

r1

:=

t (i j)j e(0)

j − t (i j)i e(0)

i −∑

k

h(i j)k e(0)

k : (i, j) ∈ U0

of Syz(M0) =: M1;• the morphism δ0 : Pr0 → Pr such that δ0(e

(0)k ) = g(0)

k , 1 ≤ k ≤ r0 sothat Im(δ0) = M0;

• the morphism δ1 : Pr1 → Pr0 such that δ1(e(1)k ) = g(1)

k , 1 ≤ k ≤ r1 sothat Im(δ1) = M1 = ker(M0);

• set := 1 and iteratively apply Buchberger’s algorithm to the Grobner basisG of M thus obtaining the already known fact that G is a Grobner basis,

23.8 Computing Resolutions 147

but producing at the same time the relevant information:

• a set

U :=(i, j) : 1 ≤ i < j ≤ r, S(g()

i , g()j ) there exists

so that

t (i j)j T(g()

j ) − t (i j)i T(g()

i ) : (i, j) ∈ U

generates Syz(T(g()

1 ), . . . , T(g()r

));• for each (i, j) ∈ U, a Grobner representation

t (i j)j g()

j − t (i j)i g()

i =∑

k

h(i j)k g()

k ;

• the Grobner basis (w.r.t. ≺)

G+1 :=

g(+1)1 , . . . , g(+1)

r+1

:=

t (i j)j e()

j − t (i j)i e()

i −∑

k

h(i j)k e()

k : (i, j) ∈ U

of Syz(M) =: M+1;• the morphism δ+1 : Pr+1 → Pr such that δ+1(e

(+1)k ) = g(+1)

k , 1 ≤k ≤ r+1 so that Im(δ+1) = M+1 = ker(M),

until G+1 = ∅.

Example 23.8.2. Let us illustrate this algorithm with the ideal (see Exam-ple 23.4.2) I generated by (G) = (g(0)

1 , g(0)2 , g(0)

3 , g(0)4 ) ⊂ k[X, Y ] where

g(0)1 := Y 5 − Y 3, g(0)

2 := X2Y 2 − X2, g(0)3 := X5 − X, g(0)

4 := XY 2 − X

which is a (redundant) Grobner basis, with respect to the lexicographical order< induced by X < Y .

From

X2g(0)1 − Y 3g(0)

2 = 0,

X3g(0)2 − Y 2g(0)

3 = −g3 + g4,

Xg(0)1 − Y 3g(0)

4 = 0,

g(0)2 − Xg(0)

4 = 0

we obtain the syzygies

g(1)1 := X2e(0)

1 − Y3e(0)2 ,

g(1)2 := X3e(0)

2 − Y2e(0)3 + e(0)

3 − e(0)4 ,

148 Macaulay I

g(1)3 := Xe(0)

1 − Y3e(0)4 ,

g(1)4 := e(0)

2 − Xe(0)4 ,

from which we have a single S-polynomial

Xg(1)3 − Y 3g(1)

4 = g(1)1

so we get the (redundant) resolution

0 → P δ2−→ P4 δ1−→ P4 δ0−→ I (23.8)

where

δ1(e(1)1 ) = X2e(0)

1 − Y3e(0)2 ,

δ1(e(1)2 ) = X3e(0)

2 − Y2e(0)3 + e(0)

3 − e(0)4 ,

δ1(e(1)3 ) = Xe(0)

1 − Y3e(0)4 ,

δ1(e(1)4 ) = e(0)

2 − Xe(0)4 ,

δ2(e(2)) = Xe(1)3 − Y 3e(1)

4 − e(1)1 ,

whose minimization gives, iteratively

0 → P δ2−→ P3 δ1−→ P3 δ0−→ I, (23.9)

δ1(e(1)1 ) = X2e(0)

1 − XY 3e(0)4 ,

δ1(e(1)2 ) = X4e(0)

4 − Y 2e(0)3 + e(0)

3 − e(0)4 ,

δ1(e(1)3 ) = Xe(0)

1 − Y 3e(0)4 ,

δ2(e(2)) = Xe(1)3 − e(1)

1 ,

and

0 → P2 δ1−→ P3 δ0−→ I (23.10)

δ1(e(1)2 ) = X4e(0)

4 − Y 2e(0)3 + e(0)

3 − e(0)4 ,

δ1(e(1)3 ) = Xe(0)

1 − Y 3e(0)4 .

Algorithm 23.8.3 (Moller). At that time there was also an alternative proposal,namely

• compute a Grobner basis G0 := g(0)1 , . . . , g(0)

r0 of M0 producing at thesame time:


• a subset 24

U :=(i, j) : 1 ≤ i < j ≤ r0, S(g(0)

i , g(0)j ) exists

⊂ U0

sufficient for t (i j)

j T(g(0)j ) − t (i j)

i T(g(0)i ) : (i, j) ∈ U

to generate Syz(T(g(0)

1 ), . . . , T(g(0)r0 )),

• for each (i, j) ∈ U a Grobner representation

t (i j)j g(0)

j − t (i j)i g(0)

i =∑

k

h(i j)k g(0)

k ,

• the morphism δ0 : Pr0 → Pr such that δ0(e(0)k ) = g(0)

k , 1 ≤ k ≤ r0, sothat Im(δ0) = M0,

• compute 25 a minimal resolution of the monomial module T(M0)

0 → Prργρ−→ · · ·Prk+1

γk+1−→ Prkγk−→ Prk−1 · · ·Pr1

γ1−→ Pr0γ0−→ T(M0),

• for each j, 1 ≤ j ≤ r1

• compute a Grobner representation 26∑k

t ( j)k g(0)

k =∑

k

h( j)k g(0)

k

of∑

k t ( j)k g(0)

k where ∑k

t ( j)k T(g(0)

k ) = γ1(e(1)j ),

• and set g(1)j := ∑

k(t( j)k − h( j)

k )e(0)k and δ1(e

(1)j ) := g(1)

j , noting that

L(δ1(e(1)j )) = L(g(1)

j ) = γ1(e(1)j ),

24 Removing for instance the ‘useless’ pairs detected by Buchberger’s Second Criterion(Lemma 22.5.3).

Note that the ‘useless’ pairs detected by Buchberger’s First Criterion (Lemma 22.5.1) couldbe unredundant generators of

Syz(T(g(0)1 ), . . . , T(g(0)

r0 ))and they must not be removed.

25 The original proposal was aiming towards a combinatorial computation (for instance an ap-plication of Algorithm 23.4.7), but, in fact, the best way for producing such a resolution is toapply Algorithm 23.8.1, restricting it, with much more efficiency, to the monomial case, andthen minimizing it.

26 Note that such Grobner representations can be freely deduced from the previous computation

of the Grobner representation of the S-polynomials S(g(0)i , g(0)

j ), (i, j) ∈ U .

150 Macaulay I

thus producing

• the T-basis (w.r.t. ≺) G1 :=

g(1)1 , . . . , g(1)

r1

of Syz(M0) =: M1,

• the morphism δ1 : Pr1 → Pr0 such that δ1(e(1)k ) = g(1)

k , 1 ≤ k ≤ r1, sothat

Im(δ1) = M1 = ker(M0) and L(Im(δ1)) = Im(γ1),

• set := 1 and iteratively, for each j, 1 ≤ j ≤ r

• compute a representation 27∑k

t ( j)k g()

k =∑

k

h( j)k g()

k , T − deg(h( j)k g()

k ) < T − deg(γ+1(e(+1)j ))

of∑

k t ( j)k g()

k where∑k

t ( j)k L(g()

k ) = γ+1(e(+1)j ),

• and set g(+1)j := ∑

k(t( j)k − h( j)

k )e()k and δ+1(e

(+1)j ) := g(+1)

j , notingthat

L(δ+1(e(+1)j )) = L(g(+1)

j ) = γ+1(e(+1)j ),

thus producing

• the T-basis (w.r.t. ≺)

G+1 :=

g(+1)1 , . . . , g(+1)

r+1

of Syz(M) =: M+1,

• the morphism δ+1 : Pr+1 → Pr such that δ+1(e(+1)k ) = g(+1)

k , 1 ≤k ≤ r+1, so that

Im(δ+1) = M+1 = ker(M) and L(Im(δ+1)) = Im(γ+1),

until G+1 = ∅.

Example 23.8.4. From the minimal resolution (see Example 23.4.6)

0 → P2 δ1−→ P3 δ0−→ M (23.11)

whereδ1(e(1, 4)) = Xe(1) − Y 3e(4),

δ1(e(2, 3)) = −Y 2e(3) + X4e(4),

27 By iteratively producing, via linear algebra, an expression of each homogeneous component in

terms of γ+1(e(+1)j ), 1 ≤ j ≤ r+1.


one only has to compute the normal forms

Xg(0)1 − Y 3g(0)

4 = 0,

Y 2g(0)3 − X4g(0)

4 = X5 − XY 2 = g(0)3 − g(0)

4

to obtain directly the syzygies

g(1)1 := Xe(0)

1 − Y 3e(0)3

g(1)2 := Y 2e(0)

3 − X4e(0)4 − e(0)

3 + e(0)4

and the minimal resolution

0 → P2 δ1−→ P3 δ0−→ I (23.12)

δ1(e1) = Xe(0)1 − Y 3e(0)

3 ,

δ1(e2) = Y 2e(0)3 − X4e(0)

4 − e(0)3 + e(0)

4 .

The potential advantage of this algoritm, w.r.t. the previous one, is thatthe pre-computation of the minimal resolution of T(M0) guarantees thatGrobner representations are to be computed for a minimal basis U ofSyz(T(g()

1 ), . . . , T(g()r

)) only and not for the redundant set of all S-polynomials S(i, j), thus minimizing those useless normal form computationsof S(i, j) whose only aim is to prove the redundancy of the syzygy related toS(i, j).

On the basis of this algorithm Gebauer and Moller performed a thoroughinvestigation of the minimization of S-polynomials (see Section 25.1) whichled to a dramatic improvement of Buchberger’s algorithm.

On the other side it must be remarked that Algorithm 23.8.1 has a nice the-oretical consequence, which we will present using freely its notation.

We will moreover assume wlog that

• the term ordering used on P is the lexicographical ordering induced by X1 <

· · · < Xn ,• the canonical basis of Pr−1 , where r−1 := r , is denoted e(−1)

1 , . . . , e(−1)r ,

• Pr−1 is T -graded so that T − deg(e(−1)i ) := 1, for each i ,

• each Pr , −1 ≤ ≤ ρ, is ordered using the term ordering such that

t1e()i1

≺ t2e()i2

⇐⇒T − deg(t1e()

i1) < T − deg(t2e()

i2) or

T − deg(t1e()i1

) = T − deg(t2e()i2

) and i1 < i2,

• each basis G is ordered so that T(g()i ) < T(g()

j ) iff i < j.

152 Macaulay I

Lemma 23.8.5 (Janet–Schreier). With these notations and assumptions, foreach , and each j, 1 ≤ j ≤ r, T(g()

i ) = te(−1)k is such that

t ∈ k[X1, . . . , Xn−].

Proof. By induction, the case = 0 being trivial.For each (i, j) ∈ U, let

T(g()i ) := τi Xdi

n−+1e(−1)ki

, and T(g()j ) := τ j X

d jn−+1e(−1)

k j,

where τi , τ j are terms in k[X1, . . . , Xn−].

Since S(g()i , g()

j ) = 0, e(−1)ki

= e(−1)k j

, and, since i < j , di < d j .

Therefore t (i j)i = (lcm(τi , τ j )/τi )X

d j −din−+1 and

t (i j)j = lcm(τi , τ j )

τ j∈ k[X1, . . . , Xn−].

Moreover t (i j)i e(−1)

i < t (i j)j e(−1)

j , since T − deg(t (i j)i e(−1)

i ) = T −deg(t (i j)

j e(−1)j ) and i < j .

Corollary 23.8.6. Let P := k[X1, . . . , Xn] and M0 ⊂ Pr be any module.Then the minimal resolution of M0 has length ρ ≤ n.

23.9 Macaulay’s Nullstellensatz Bound

Assume we are given an ideal Jν ⊂ k[X1, . . . , Xν] generated by a basisGν := f1, . . . , fs consisting of homogeneous polynomials, all having thesame degree D. In this setting, Macaulay analysed Theorem 20.4.1 in order togive a degree bound of the polynomials di ∈ Fν−1 obtained by Kronecker’selimination:

Lemma 23.9.1 (Macaulay). With the same assumptions and notation as inTheorem 20.4.1, if, each fi ∈ Gν is homogeneous and D := deg( fi ), for eachi, then

• each dρ ∈ Fν−1 is homogeneous and deg(dρ) = D2;• each dρ has a representation dρ = ∑

i giρ fi where

deg(giρ) = D2 − D.

Proof. We will use the same notation as in the proof of Theorem 20.4.1 and wewill denote by α j ∈ k[X1, . . . , Xν−1] and β j ∈ k[U2, . . . , Us][X1, . . . , Xν−1]

the coefficients of the polynomials f1 and G, so that f1 = ∑Dj=0 α j X D− j

ν and

G = ∑Dj=0 β j X D− j

ν .

23.9 Macaulay’s Nullstellensatz Bound 153

If we denote by γi j the entries of the 2D × 2D Sylvester matrix of f1 andG, we have the relations

Ri :=2D∑j=1

γi j X2D− jν =

X D−i

ν f1, if 1 ≤ i ≤ D,

X2D−iν G, if D + 1 ≤ i ≤ 2D,

because

Ri =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

D+i∑j=i

α j−i X2D− jν =

D∑j=0

α j X2D−i− jν = f1 X D−i

ν if i ≤ D,

i∑j=i−D

β j−i+D X2D− jν =

D∑j=0

β j X3D−i− jν = G X2D−i

ν if D < i,

so that 28

deg(Ri ) =

2D − i if 1 ≤ i ≤ D,

3D − i if D + 1 ≤ i ≤ 2D

and

deg(γi j ) = deg(Ri ) − (2D − j) =

j − i if 1 ≤ i ≤ D,

D + j − i if D + 1 ≤ i ≤ 2D.

Each term∏2D

i=1 γiπ(i) of the resultant – where π( · ) is any permutation –has the same degree

D∑i=1

(π(i) − i) +2D∑

i=D+1

(D + π(i) − i) = D2 −2D∑i=1

i +2D∑i=1

π(i) = D2.

Therefore in the homogeneous representation

Res( f1, G) = p f1 + qs∑

i=2

Ui fi

we have deg(p) = deg(q) = D2 − D.

Corollary 23.9.2. For each finite homogeneous set

F := f1, . . . , fs ⊂ k[X0, . . . , Xn],

such that deg( fi ) = D for each i , if Z(F) = ∅ then there are homogeneouspolynomials g1, . . . , gs ∈ k[X0, . . . , Xn], such that

• deg(gi ) = D2n − D,

• X D2n

0 = ∑si=1 gi fi .

28 The degree we evaluate is of course the one such that deg(Xi ) = 1 and deg(U j ) = 0.

154 Macaulay I

Proof. One only has to iterate the result of Lemma 23.9.1. Using the samenotation as in Section 20.4, the iterated resultant computations produce a seriesof ideals Iν = Jν ⊂ k[X0, . . . , Xν], n ≥ ν ≥ 0, the last of which J0 ⊂ k[X0]is a power of X0. If we denote by δν the common degree of the elements in thebasis Gν of Jν , by Lemma 23.9.1, we have δν−1 = δ2

ν . Since by assumption

δn = D, we obtain δ0 = D2nand J0 =

(X D2n

0

).

The requirement that a homogeneous ideal be given by means of a ho-mogeneous basis whose elements have the same degree D can be easily ob-tained without changing the roots of the ideal: if we are given an ideal I ⊂k[X1, . . . , Xn] by means of a basis g1, . . . , gs, deg(gi ) = di , D := max(di ),

we have just to consider the basis (X1 − 1)D−di gi , X D−di1 gi , 1 ≤ i ≤ s so

that each polynomial gi gives rise to two polynomials whose leading coeffi-cients (needed for resultant consideration) and whose common zeros are thesame as the leading coefficient and the zeros of gi . As a consequence

Corollary 23.9.3 (Macaulay’s Projective Nullstellensatz Bound). For eachfinite homogeneous set F := f1, . . . , fs ⊂ k[X0, . . . , Xn], if Z(F) =∅ then, writing D := max(deg( fi )), there are homogeneous polynomialsg1, . . . , gs ∈ k[X0, . . . , Xn], such that

• deg(gi ) = D2n − D,

• X D2n

0 = ∑si=1 gi fi .

Corollary 23.9.4 (Macaulay’s Affine Nullstellensatz Bound). For each fi-nite basis F := f1, . . . , fs ⊂ k[X1, . . . , Xn], if Z(F) = ∅ then, writingD := max(deg( fi )), there are polynomials g1, . . . , gs ∈ k[X1, . . . , Xn], suchthat

• deg(gi ) ≤ D2n − D,• 1 = ∑s

i=1 gi fi .

Proof. One only has to consider the finite basis

h f1, . . . ,h fs ⊂ k[X0, . . . , Xn],

in order to deduce the relation

X D2n

0 =s∑

i=1

Gih fi , deg(Gi ) = D2n − D,

and, by dehomogenization, to obtain the claim with gi = aGi .

23.9 Macaulay’s Nullstellensatz Bound 155

Historical Remark 23.9.5. This result by Macaulay can be put in perspectiveby comparing his comment in the preface of his book

The present state of our knowledge of the properties of Modular Systems [i.e. polyno-mial ideals] is chiefly due . . . to J. Konig’s profound exposition and numerous exten-sions of Kronecker’s theory. Konig’s treatise might be regarded as in some measurecomplete if it were admitted that a problem is finished with when its solution has beenreduced to a finite number of feasible operations. If however the operations are too nu-merous or too involved to be carried out in practice the solution is only a theoretical one;and its importance then lies not in itself, but in the theorems with which it is associatedand to which it leads. Such a theoretical solution must be regarded as a preliminary andnot the final stage in the consideration of the problem.F. S. Macaulay, The Algebraic Theory of Modular Systems, p. v.

with the comment (in Section 17) which follows his exposition (in Sections13–16) of ‘Konig’s exposition of Kronecker’s method of solving equations bymeans of the resultant.’

Geometrically the resultant enables us to resolve the whole spread represented by anygiven set of algebraic equations into definite irreducible spreads (Section 21). It hasbeen supposed the complete resolvent[29] also supplies a definite answer to certainother questions. The following examples disprove this to some extent.Example i. Find the resolvent of n homogeneous equations F1 = F2 = · · · = Fn = 0[∈ k[X1, . . . , Xn]] of the same degree l and having no proper solution.Since there are no solutions of rank < n the complete resolvent is [D1]. The first de-rived set of polynomials [Fn−1] are homogeneous and of degree l2, the 2nd set [Fn−2]are homogeneous and of degree l4, and the (n − 1)th set [F1] are homogeneous and

of degree l2n−1. This last set involve only one variable [X1], and therefore have the

common factor [Xl2n−1

1 ], which is therefore the required complete resolvent.[30]

We should arrive at a similar result if we changed xi to xi −ai (i = 1, 2, . . . , n) before-hand, thus making the polynomials non-homogeneous. The complete resolvent would

then be [(X1 − a1)l2n−1]. The resultant[31] would be [(X1 − a1)ln

]. The differencein the two results is explained by the fact that the resultant is obtained by a processapplying uniformly to all the variables, and the resolvent by a process applied to thevariables in succession.F. S. Macaulay, The Algebraic Theory of Modular Systems, pp. 21–2.

I personally consider this first ‘disproving’ example proposed by Macaulayas a direct pointer to the complexity problem, the more so since his furtherexamples point mainly to the correctness of the method:

29 That is with the notation of Section 20.4,∏n

ν=1 Dν .

30 That is Di = 1 for i > 1 and D1 = Xl2n−1

1 .31 Which we will discuss in the next volume.

156 Macaulay I

The complete resolvent may indicate imbedded modules which do not exist as in Ex. iior it may give no indication of them when they do exist as in Ex. iii.F. S. Macaulay, The Algebraic Theory of Modular Systems, p. 24.

But I must inform the reader that in his 1913 paper Macaulay had alreadypresented the same example in order to dispel a theoretical claim by Konig: heused the example to show that the multiplicity of the factor related to a primarycomponent in the resolvent

. . . is as a rule greater than the corresponding [multiplicity in] the resultant (in suchcases as allow of a comparison being made) and so cannot be regarded or defined as ameasure of the multiplicity of the primary module, as Konig appears to suggest.F. S. Macaulay, On the Resolution of a Given Modular System into Primary SystemsIncluding Some Properties of Hilbert Numbers, Math. Ann. 74 (1913), Section 33,p. 84.

23.10 *Bounds for the Degree in the Nullstellensatz

In a similar mood as Macaulay’s, let us record

Theorem 23.10.1 (Hermann). We have:

(1) Let F := f1, . . . , fs ⊂ k[X1, . . . , Xn]t be a finite basis generat-ing the module M and write D := max(deg( fi )); then each element(g1, . . . , gs) ∈ k[X1, . . . , Xn]s in a minimal basis of Syz(M) satisfiesthe degree bound

deg(gi ) ≤n∑

i=1

(Dt)2i−1.

(2) Let F := l1, . . . , lt ⊂ k[X1, . . . , Xn]s be a finite basis 32 generatingthe module M and write D := max(deg( fi )); then for each g ∈ Mthere are polynomials h1, . . . , ht ∈ k[X1, . . . , Xn], such that

• deg(hi ) ≤ deg(g) + 2∑n

i=1(Dt)2i−1,

• g = ∑ti=1 hi li .

32 We keep the same notation as Grete Hermann; therefore in the first statement we considers elements in a module of rank t ; in the second statement we consider instead a basis of telements in a module of rank s.

In both cases, we are essentially considering the same matrix⎛⎜⎝

f11 · · · f1s...

. . ....

ft1 · · · fts

⎞⎟⎠

23.10 *Bounds for the Degree in the Nullstellensatz 157

Proof.

(1) Writing, for each i ,

fi := ( f1i , . . . , fti ), f j i ∈ k[X1, . . . , Xn], deg( f j i ) ≤ D,

the problem is essentially a linear algebra problem, that is finding allthe solutions (g1, . . . , gs) ∈ k[X1, . . . , Xn]s of the system of linearequations ⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

l1 := f11g1 + · · · + f1s gs = 0,

· · ·l j := f j1g1 + · · · + f js gs = 0,

· · ·lt := ft1g1 + · · · + fts gs = 0,

and is solved in that way; therefore let us denote by p the rank of thematrix ( f j i ) and, for each sequence i1, . . . , i p, 1 ≤ ik ≤ s,

∆(i1, . . . , i p) =

∣∣∣∣∣∣∣f1i1 · · · f1i p

.... . .

...

f pi1 · · · f pi p

∣∣∣∣∣∣∣and let us note that the degree of each determinant is

µ ≤ Dt.

The argument is by induction on n, being trivial for n = 0. We canwlog assume that

• the equations are linearly independent, so that, t = p and, up to arenumbering,

• ∆(1, . . . , p) = 0 and 33

• ∆ := ∆(1, . . . , p) = cXµn +∑µ−1

j=0 h j (X1, . . . , Xn−1)X jn , c = 0.

33 Up to a generic change of coordinates

Xi →

Xi + ci Xn if i < n,

cn Xn if i = n,ci ∈ k \ 0,

whose inverse

Xi →

Xi − ci c−1n Xn if i < n,

c−1n Xn if i = n

cannot increase the degree of the polynomials to which it is applied, so that the degree boundis independent by the system of coordinates.

158 Macaulay I

Cramer’s Formula gives some solution to the system of the equations,namely, for k, t + 1 ≤ k ≤ s,

g(k)i =

⎧⎨⎩

∆(1, . . . , i − 1, k, i + 1, . . . , t) if 1 ≤ i ≤ t,∆ if i = k,

0 if i = k, t < k ≤ s,

and the property of ∆(1, . . . , t) grants that any other solution(g1, . . . , gs) can be reduced 34 via (g(k)

1 , . . . , g(k)s ) : k > t to a so-

lution (g′1, . . . , g′

s) such that degn(g′i ) < µ for i > t.

Moreover, denoting Fi j the subdeterminant obtained from ∆ by cross-ing out the i th row and the j th column, we have

0 =t∑

i=1Fi1li = ∆g′

1 +s∑

k=t+1∆(k, 2, . . . , t)g′

k,

· · ·0 =

t∑i=1

Fi j li = ∆g′j +

s∑k=t+1

∆(1, . . . , j − 1, k, j + 1, . . . , t)g′k,

· · ·0 =

t∑i=1

Fit li = ∆g′t +

s∑k=t+1

∆(1, . . . , t − 1, k)g′k;

therefore, since

degn(∆) = µ, degn(∆(1, . . . , j − 1, k, j + 1, . . . , t)) ≤ µ,

and

degn(g′k) < µ, for k > t

we can deduce that we have

degn(g′k) < µ also for k ≤ t.

As a consequence, each solution (g1, . . . , gs) can be reduced, via

(g(k)1 , . . . , g(k)

s ) : k > t,to a solution (g′

1, . . . , g′s) such that degn(g′

i ) < µ for each i and ouraim is reduced to finding a basis for such solutions.

34 If, for k > t ,

gk = q∆ + rk , degn(rk ) < degn(∆) = µ

we rewrite each gi with gi − qg(k)i .


If we write each g′k as

g′k(X1, . . . , Xn) :=

µ∑i=1

ξik(X1, . . . , Xn−1)Xµ−in

and substitute it in each l j , obtaining

0 = l j = f j1

µ∑i=1

ξi1 Xµ−in + · · · + f js

µ∑i=1

ξis Xµ−in ,

and we equate to 0 each coefficient of a power of Xn we obtain µt ≤Dt2 linear equations 35 in the µs unknowns ξi j , 1 ≤ j ≤ s, 1 ≤ i ≤ µ,whose coefficients are polynomials

φi j ∈ k[X1, . . . , Xn−1], deg(φi j ) ≤ D;

their solutions, by induction, have degree

deg(ξi j ) ≤n−1∑i=1

(D(Dt2)

)2i−1

=n∑

i=2

(Dt)2i−1;

therefore

deg(gi ) ≤ µ + deg(ξi j ) = Dt +n∑

i=2

(Dt)2i−1 =n∑

i=1

(Dt)2i−1.

(2) We now denote by z1, . . . , zs the canonical basis of k[X1, . . . , Xn]s ,we write, for j, 1 ≤ j ≤ t,

l j =s∑

i=1

f j i zi , f j i ∈ k[X1, . . . , Xn], deg( f j i ) ≤ D,

and we use the same notation as in the proof above. However, in thissetting, while we can still assume that

∆ := ∆(1, . . . , t) = cXµn +

µ−1∑j=0

h j (X1, . . . , Xn−1)X jn , c = 0,

we are no longer allowed to assume that the equations are linearly in-dependent, so we just have p ≤ t .

35 And τ ≤ µt ≤ Dt2 linearly independent ones.

160 Macaulay I

We write 36

m1 =p∑

j=1F1 j l j = ∆z1 +

s∑k=p+1

∆(k, 2, . . . , p)zk,

· · ·mi =

p∑j=1

Fi j l j = ∆zi +s∑

k=p+1∆(1, . . . , i − 1, k, i + 1, . . . , p)zk,

· · ·m p =

p∑j=1

Fpj l j = ∆zp +s∑

k=p+1∆(1, . . . , p − 1, k)zk .

For any element g := ∑si=1 gi zi ∈ k[X1, . . . , Xn]s by division we

obtain, for each i , 1 ≤ i ≤ p,

gi = Gi + ∆γi ,

satisfying 37

degn(Gi ) < degn(∆) ≤ Dt = µ, and deg(γi ) ≤ deg(gi ) ≤ deg(g).

If we therefore set for k, p < k ≤ s,

Gk := gk −p∑

i=1

∆(1, . . . , i − 1, k, i + 1, . . . , p)γi ,

we obtain

g −p∑

i=1

γi mi =p∑

i=1

Gi zi +p∑

i=1

∆γi zi +s∑

k=p+1

gkzk −p∑

i=1

∆γi zi

−p∑

i=1

γi

s∑k=p+1

∆(1, . . . , i − 1, k, i + 1, . . . , p)zk

=p∑

i=1

Gi zi

36 Where Fi j denotes the subdeterminant obtained from ∆ by crossing out the i th row and the j thcolumn.

37 The claim deg(γi ) ≤ deg(gi ) requires a proof: note that

χ := Lp(∆γi ) = c Lp(γi )

is the coefficient of XD+degn (γi )n in gi ; therefore

deg(gi ) ≥ D + degn(γi ) + deg(χ) = deg(γi ).


+s∑

k=p+1

(gk −

p∑i=1

γi∆(1, . . . , i − 1, k, i + 1, . . . , p)

)zk

=s∑

i=1

Gi zi .

Also g ∈ M ⇒ G := ∑si=1 Gi zi ∈ M and deg

( p∑i=1

γi mi

)≤

deg(g) + Dt , so that the claim is proved for g if we are able to prove itfor G.Let us therefore assume that we have a representation

G =t∑

j=1

h j l j

and let us prove that, for each j ,

deg(h j ) ≤ deg(g) + 2n∑

i=1

(Dt)2i−1.

Since, unlike in the previous argument, we can no longer assume p = t ,we have therefore to discuss separately the two different cases p < j ≤t and j ≤ p.For p < j ≤ t : since

∆l j = −p∑

i=1

∆(1, . . . , i − 1, j, i + 1, . . . , p)li

we can assume, via division by ∆, that

degn(h j ) < degn(∆) ≤ Dt, for p < j.

For j ≤ p: we have Gi = ∑tj=1 h j fi j and

p∑i=1

Gi Fi j = ∆h j +t∑

k=p+1

hk

p∑i=1

fik Fi j

where

deg(∑p

i=1 fik Fi j) ≤ Dt,

deg(Fi j ) ≤ D(t − 1),

degn(hk) ≤ degn(∆), p < k ≤ tdegn(Gi ) ≤ degn(∆), p < k ≤ t

162 Macaulay I

whence

degn(h j ) ≤ Dt, for j ≤ p.

Therefore if n = 1 the proof is completed. If instead n > 1 we have

G =t∑

j=1

Dt∑k=0

γ jk Xknl j , γ jk ∈ k[X1, . . . , Xn−1],

and, inductively,

deg(γ jk) ≤ deg(G)+2n−1∑i=1

(D(Dt2))2i−1 ≤ deg(g)+Dt+2n∑

i=2

(Dt)2i−1

and

deg(h j ) ≤ Dt + max(deg(γ jk))

≤ deg(g) + 2Dt + 2n∑

i=2

(Dt)2i−1

≤ deg(g) + 2n∑

i=1

(Dt)2i−1.

Corollary 23.10.2 (Hermann Bound). For each finite basis

F := f1, . . . , fs ⊂ k[X1, . . . , Xn],

generating an ideal I, we have, writing D := max(deg( fi ))

(1) each element (g1, . . . , gs) ∈ k[X1, . . . , Xn]s in a minimal basis ofSyz(I) satisfies the degree bound deg(gi ) ≤ ∑n

i=1 D2i−1;

(2) for each f ∈ I there are polynomials g1, . . . , gs ∈ k[X1, . . . , Xn], suchthat

• deg(gi ) ≤ deg( f ) + 2∑n

i=1(Ds)2i−1,

• f = ∑si=1 gi fi ;

(3) I = (1) iff there are polynomials g1, . . . , gs ∈ k[X1, . . . , Xn], suchthat

• deg(gi ) ≤ 2∑n

i=1(Ds)2i−1,

• 1 = ∑si=1 gi fi .


Example 23.10.3. Let us assume the following scenario: we are given n poly-nomials, of degree at most d , in n variables 38 f1, . . . , fn ∈ k[X1, . . . , Xn],deg( fi ) = d for each i , and we want to compute the syzygies amongthem.

Let us moreover assume that ( f1, . . . , fn) is a regular sequence, meaningthat all the syzygies are generated by the trivial ones fi f j – f j fi = 0, i < j.

If we are unaware of H-bases and Grobner bases, and only aware of theHermann Bound, what we need to do is look for all solutions of the equation

s∑i=1

gi fi = 0, deg(gi ) ≤n∑

i=1

D2i−1.

With this aim in mind, let us assume d = n = 3, so that

deg(g) ≤ 3 + 32 + 34 = 93;since the set of the polynomials in k[X1, . . . , Xn] of degree bounded by d hask-dimension

(d+nn

), we have to solve a system of equations having 3

(93+33

) =428 640 unknowns and

(96+33

) = 156 849 equations, giving us all of the

3(90+3

3

) − (87+33

) = 271 818 solutions.Alternatively using the notion of H-bases we have to solve, for each δ ≤∑ni=1 D2i−1

, the homogeneous equation

s∑i=1

gi H( fi ) = 0, gi homogeneous and deg(gi ) = δ;

each equation has 3(δ+2

2

)unknowns and

(δ+5

2

)equations and gives all the

3(δ−1

2

) − (δ−4

2

)solutions; the total number of equations, unknown and solu-

tions is the same as before but the problem is split into smaller and thereforeeasier problems.

The computation is to be performed by increasing degree δ and for eachsolution (g1, . . . , gs), deg(gi ) = δ found, one should then

• verify whether it belongs in the module generated by the solutions previ-ously obtained, and, if this is not the case,

• compute a representation∑s

i=1 gi fi = ∑si=1 hi fi , deg(hi ) < deg(gi ).

38 This scenario has been familiar since the last century and is connected with the KroneckerModel and theory.

In a very informal analysis of practical performance – as opposed to theoretical complexity –it is quite natural to assume n = d as well and, in this context, it has an obvious significantmeaning, a nonsensical expression such as ‘this implementation is able to solve the problem upto n = 7.5’, which is, in fact, the actual standard for the best Buchberger algorithm implemen-tations.

164 Macaulay I

The first equation (δ = 3) requires us to solve 28 equation 39 in 30 unknownsand gives the 3 solutions H( f j )H( fi )− H( fi )H( f j ); to lift each such syzygyto f j fi − fi f j one has to solve a system of equations having 3

(2+33

) = 30

unknowns and(5+3

3

) = 56 equations.At this point, if we are computing by hand, we will immediately realize that

other independent syzygies cannot exist; if we are instead using a computer,we will have to wait until our system has verified that all the other 271.815solutions are consequences of the first 3 ones.

Alternatively if we used Grobner basis techniques, we would have to com-pute 3 S-polynomials, and even if our software is unaware of Buchberger’sFirst Criterion – which allows it to give us the solution immediately – it justneeds at worst to perform for each S-polynomial

(5+33

) = 56 steps of reduc-

tion, each costing(3+3

3

) = 20 arithmetical operations.

Apparently, the example above suggests that H-bases are good and thatGrobner bases are even better, which is true.

But, on the other hand, if we have n+1 polynomials of degree at most d in nvariables, f1, . . . , fn+1 ∈ k[X1, . . . , Xn], defining the empty variety, we willin any case not realize it unless we find a relation 1 = ∑

i gi fi where (whend = n = 3) each gi has degree 93 and so 142 880 terms.

Somewhere we have to pay for that solution, while it is true that in thiscase both the H-basis and the Grobner basis are 1. The point is that beforewe reach that trivial solution, we will have a sequence of partial solutions ofincreasing degree.

Remark 23.10.4. What amazes me more in that example is not the efficiencyof Grobner bases, but that of H-bases.

It is sufficiently amazing to cause me to wonder whether the same trick canbe repeated: after all, for any ideal I ⊂ k[X1, . . . , Xn] – or equivalently anyhomogeneous ideal J ⊂ k[X0, . . . , Xn] such that I = aJ – the ideal H(I) ⊂k[X1, . . . , Xn] is homogeneous with one variable fewer: what happens if wethen compute aH(I) setting Y = 1 for a suitable linear combination Y of thevariables? And how far can we go in this way?

Not surprisingly, Macaulay posed the same question and solved it. Not sur-prisingly again, his motivation was much less trivial than efficiency in mem-bership tests or syzygy computations.

We will discuss this in Chapter 36 (see Remark 36.3.8).

39 Of which only 27 are linearly independent; in fact the coefficient of X21 X2

2 X23 in that expression

is 0.


The doubly exponential bound given by Hermann for the Weak Nullstellen-satz is in contrast with the single exponential bound deduced 40 by Macaulayusing the resultant. Recent results, 41 which use techniques outside the scopeof this book, prove that the Weak Nullstellensatz is really single exponential:

Fact 23.10.5 (Kollar; Fitchas–Galligo). Let

F := f1, . . . , fs ⊂ k[X1, . . . , Xn],

generating an ideal I; denote di := deg( fi ) for each i and D := max(deg( fi )),and assume that d1 ≥ d2 ≥ · · · ≥ ds > 2.

Then I = (1) iff there are polynomials g1, . . . , gs ∈ k[X1, . . . , Xn], suchthat

• 1 = ∑si=1 gi fi ,

• deg(gi ) + di ≤⎧⎨⎩

d1 · · · · · ds if s ≤ n,

d1 · · · · · dn−1 · ds if s > n > 1,

d1 + ds − 1 if s > n = 1.

Corollary 23.10.6. For each finite basis

F := f1, . . . , fs ⊂ k[X1, . . . , Xn],

generating an ideal I, we have, writing D := max(deg( fi ))

(1) I = (1) iff there are polynomials g1, . . . , gs ∈ k[X1, . . . , Xn], suchthat

• deg(gi fi ) ≤ max(3n, Dn),

• 1 = ∑si=1 gi fi ,

(2) for each f ∈ I, f ∈ √I iff there are polynomials g1, . . . , gs ∈

k[X1, . . . , Xn], such that

• deg(gi fi ) ≤ (deg( f ) + 1) max(3n, Dn),• f e = ∑s

i=1 gi fi ,• e ≤ max(3n, Dn).

Proof. In order to obtain the second result, it is sufficient to use Rabinowitch’sTrick, applying Fact 23.10.5 to f1, . . . , fs, 1 − f T ⊂ k[X1, . . . , Xn, T ].

40 Albeit in a specific case: essentially a primary at the origin generated by a regular sequence.41 Compare J. Kollar, Sharp Effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), 963–975; N.

Fitchas and A. Galligo, Nullstellensatz effectif et conjecture de Serre (theoreme de Quillen–Suslin) pour le Calcul Formel, Math. Nachr. 149 (1990), 231–253.

166 Macaulay I

If, using the same notation as in Theorem 23.10.1, we write, as Hermann did,

m(D, t, n) := max(deg(gi )),

it is clear that Hermann’s Theorem 23.10.1 follows directly by her proof of therecursive relation

m(D, t, n) = m(D, Dt2, n − 1).

A more subtle recursive relation ‘obtained by eliminating two variables asHermann eliminates one variable’ was deduced by Lazard:42

Fact 23.10.7 (Lazard). We have

• m(D, t, n) ≤ Dt+D−2+m(D, t ′, n−2) where t ′ := D2t (t2 + 4t + 3/2)−(Dt2 + Dt/2);

• m(D, t, 1) ≤ Dt;• m(D, t, 2) ≤ Dt + D − minD, 2.Corollary 23.10.8 (Lazard). For each finite basis

F := f1, . . . , fs ⊂ k[X1, . . . , Xn],

generating an ideal I, writing D := max(deg( fi )), we have

(1) each element (g1, . . . , gs) ∈ k[X1, . . . , Xn]s in a minimal basis of

Syz(I) satisfies the degree bound deg(gi ) ≤ (Dt)3n2 ;

(2) for each f ∈ I there are polynomials g1, . . . , gs ∈ k[X1, . . . , Xn], suchthat

• deg(gi ) ≤(

(D)3n2

)n

= (D)3(n2 +log3(n))

,

• f = ∑si=1 gi fi .

Proof.

(1) If D = 0, the result follows from the fact that Cramer rules apply.If D = t = 1 by linear change of coordinates and linear operations onthe rows, the system can be expressed as⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

l1 := x1g1 + · · · + xh gh + c1gh+1 = 0,

· · ·l j := x1g1 + · · · + xh gh + c j gh+ j = 0,

· · ·ls−h := x1g1 + · · · + xh gh + cs−h gs = 0,

42 Compare D. Lazard, Resolution des systemes d’equations algebriques, Theor. Comp. Sciences15 (1981), 71–110; D. Lazard, A Note on Upper Bounds for Ideal-theoretical Problems, J.Symb. Comp. 13 (1992), 231–233.


where c j ∈ k for each j ; for such equations the module of the syzygiesis generated by the trivial ones.So we are left to prove the result for the cases Dt ≥ 2. Then we have

• m(D, t, n) ≤ (Dt)3(n−1)

2, for n ≤ 2.

• if t ≥ 5 and m(D, t ′, n − 2) ≤ (Dt ′)α, for some α, then

m(D, t, n) ≤ Dt + D − 2 + m(D, t ′, n − 2)

≤ Dt + D − 2 + (Dt ′)α

≤ (Dt + D − 2 + Dt ′)α

≤ (Dt)3α;• if t < 5 and m(D, t ′, n − 2) ≤ (Dt ′)α, for some α, then

m(D, t, n) ≤ Dt + D − 2 + m(D, t ′, n − 2)

≤ Dt + D − 2 + (Dt ′)α

≤ (Dt + D − 2 + Dt ′)α

≤ (Dt)5α;• if Dt ≥ 2 then t ′ ≥ 5 and t ′ ≥ t .

As a consequence, for n > 2, by recursion on n, we get

m(D, t ′, n − 2) ≤ (Dt ′)3(n−3)

2 for each t ′,

whence

if t ≥ 5, we obtain m(D, t, n) ≤ (Dt)3(n−3)

2 3 = (Dt)3(n−1)

2

if t < 5, we obtain, since 52 < 33,

m(D, t, n) ≤ (Dt)3(n−3)

2 5 ≤ (Dt)3(n−3)

2 332 = (Dt)3

n2.

(2) If (h10, h11, . . . , h1s), . . . , (hu0, hu1, . . . , hus) is a basis of the syzy-

gies among f, f1, . . . , fs we have deg(hi j ) ≤ D := (D)3n2.

Therefore by Corollary 23.10.6 we obtain elements ai such that

deg(ai ) ≤ Dn − D and 1 =∑

i

ai hi0

so that

f =∑

i

ai hi0 f =s∑

j=1

f j

∑i

−ai hi j and deg(ai hi j ) ≤ Dn .

168 Macaulay I

Example 23.10.9. All these bounds are sharp:

Mayr–Meyer examples (Section 38.4) produce instances of bases

• Fdn := f1, . . . , fs ⊂ k[X1, . . . , X10n+4], deg( fi ) ≤ d + 2, gen-erating an ideal Idn for which Syz(Idn) ≥ d2n−1

,• Gdn := g1, . . . , gs ⊂ k[X1, . . . , X10n+2], deg(gi ) ≤ d + 2 gen-

erating an ideal Jdn for which m(d + 2, 1, 10n + 2) ≥ d2n−1.

This is an example by Moller and myself, produced for different reason,which proves that the bounds of Corollary 23.10.6 are sharp: considerthe ideal in I ⊂ k[X1, . . . , Xn] generated by

X Dn , Xn − X D

n−1, . . . , Xi − X Di−1, . . . , X2 − X D

1 ;then

• X Dn

1 ≡ X Dn−1

2 ≡ · · · ≡ X Dn−i+1

i ≡ · · · ≡ X Dn ≡ 0 mod I;

• X1 ∈ √I;

• since I is a homogeneous ideal w.r.t. the weight w(Xi ) := Di−1 wehave

X Dn

1 =n−1∑i=1

gi

(Xi+1 − X D

i−1+i

)+ gn X D

n

with Dn ≥ w(gi ) + D ≥ deg(gi ) + D;• there is no relation

Xe1 =

n−1∑i=1

gi

(Xi+1 − X D

i

)+ gn X D

n with e < Dn,

since under the projection π : k[X1, . . . , Xn] → k[T ] defined byπ(Xi ) = T Di−1

we have

T e = π(Xe1) =

n−1∑i=1

π(gi )π(Xi+1 − X Di−1+i ) + π(gn)π(X D

n )

= π(gn)T Dn ;• Xe

1 ∈ I ⇒ e ≥ Dn .

A variation of this example gives the ideal generated by

X Dn , Xn − X D

n−1, . . . , Xi − X Di−1, . . . , X3 − X D

2 , X2 X D−11 − 1,

for which

1 = g1

(X2 X D−1

1 − 1)

+n−1∑i=2

gi

(Xi+1 − X D

i

)+ gn X D

n


where

g1 = 1 − (X2 X D−11 )Dn−1

X2 X D−11 − 1

,

gi = −X Dn−Dn−1

1

X Dn−i

i+1 − (X Di )Dn−i

Xi+1 − X Di

, 2 ≤ i ≤ n − 1,

gn = X Dn−Dn−1

1

giving a strong lower bound Dn − Dn−1 for Corollary 23.10.6.

24

Grobner I

Buchberger completed his thesis in 1965 and published his results in 1970.The next year, Grobner quoted them in his notes of a course held by him inTurin and Milan in April–May 1971.1 There, in a section devoted to the deter-mination of the primary components in the Lasker–Noether decomposition ofan ideal, he concluded with the following remark:

OSSERVAZIONE: Riguardo ai calcoli che occorre eseguire per risolvere i problemidella teoria degli ideali negli anelli di polinomi, giova notare che, in linea di principio,tutti i calcoli si possono ridurre alla risoluzione di sistemi di equazioni lineari. Infattibasta risolvere il problema dato nei singoli spazi vettoriali P(t). . . In questo procedi-mento e lecito fermarsi ad un certo grado (finito) T che corresponde al grado massimoattinto dai polinomi che formano la base dell’ideale cercato.Un criterio per determinare tale numero T e stato indagato da B. BUCHBERGER(Aequationes mathematicae, Vol. 4, Fasc. 3, 1970, S. 377–388)REMARK: With regard to the calculations needed to solve the problems in the theoryof ideals of polynomial rings, it is helpful to remark that, in principle, all computationscan be reduced to the resolution of systems of linear equations. In fact it is sufficientto solve the given problem in the single vector spaces P(t) [the set of all polynomialsof degree bounded by t] In this procedure it is sufficient to terminate at a fixed (finite)degree T corresponding to the maximal degree reached by the polynomials which area basis of the required ideal.A criterion to determine such number T has been investigated by B. BUCHBERGER(Aequationes mathematicae, Vol. 4, Fasc. 3, 1970, S. 377–388)

This is a remark which seems to be in the same mood as in the introductionby Macaulay of his H-bases (see Historical Remark 23.2.3).

In his paper Buchberger introduces his algorithm in order to solve the fol-lowing problem.

1 W. Grobner, Teoria degli ideali e geometria algebrica. Rendiconti Sem. Mat. Fis. Milano 46(1971), 171–242.

170

Grobner I 171

Problem 24.0.1. Given an ideal I ⊂ P := k[X1, . . . , Xn] and considering thequotient algebra A := P/I, to calculate the multiplication table of A w.r.t. ak-basis.

The solution (see Lemma 22.2.12) is to consider as k-basis the terms inN(I) =: t1, . . . , ts and to represent the product of ti and t j by

ti · t j := Can(ti · t j , I, <).

The problem was dealt with by Grobner himself in

W. Grobner, Uber die Eliminationstheorie. Monatsch. der Math. 54 (1950), 71–78

where an algorithm was given, about which he commented (p. 78)

Ich habe diese Methode seit etwa 17 Jahren in der verschiedensten, auch kompliziertenFallen werwendet und erprobt und glaube auf Graund meiner Erfaharungen sagen zukonnen, daß sie tatsachlich in allen Fallen ein brauchbares und wertvolles Werkzeugzur Losung von diesen und ahnlichen idealtheoretischen Aufgaben darstellt.I have used and tested this method for 17 years in different and complicated cases andI believe on the basis of my experience that I can say that it represents in all cases auseful and worthwhile tool for solving these and similar ideal-theoretic problems.

Example 24.0.2. Grobner illustrated his method on the ideal

I := (x2 + x1x2 + x22 , x2 − x1x2 + x2

2 , x21 − x3

1).

He began by setting x1 → u1, x2 → u2; then he puts x21 → u2

1 = u3 ‘da nachden bisher vorliegeneden Beziehungen keine lineare Abhangigkeit zwischenu1, u2 und u2

1 aufscheint’.2 In a similar way he put x1x2 → u1u2 = u4 obtain-ing at this time

u1 u2 u3 u4

u1 u3 u4

u2

u3

u4

For x22 → u2

2 ‘erhalten wir mit Benutzung der beiden ersten Basispoly-nome’3

u22 = −u2 + u4 = −u2 − u4 ⇒ u4 = 0, u2

2 = −u2;‘wir mussen also auch in der vorausgehenden Zeile u4 streichen und u1u2 = 0

2 Because until now in the given relations no linear dependency among u1, u2 and u21 emerges.

3 This translates as ‘we obtain from the first two polynomials in the basis’.

172 Grobner I

setzen’.4 obtaining

u1 u2 u3

u1 u3 0u2 0 −u2

u3

The next computation

x31 → u3

1 = u1u21 = u1u3 = u3

used the third polynomial, and gave

u1 u2 u3

u1 u3 0 u3

u2 0 −u2

u3 u3

‘Die weiteren Potenzprodukte liefern keine neuen unabhangigen Großenmehr’5 since

u21u2 = u1(u1u2) = u2u3 = 0 (since u1u2 = 0);

similarly u1u22 = 0,6 and

u32 = u2u2

2 = u2(−u2) = −u22 = u2.

With

u41 = u1u3 = u2

3 = u3

‘die Multiplikationstafel vollstandig und das Verfahren abgesclossen’.7

The solution is

k[X1, X2] \ I = Spank1, u1, u2, u3with multiplication table

u1 u2 u3

u1 u3 0 u3

u2 0 u2 0u3 u3 0 u3

.

4 This translates as ‘we must now also remove u4 from the line above and set u1u2 = 0’.5 This translates as ‘the other terms do not give new independent quantities’.6 I assume its implicit argument is

u1u22 = u1(u1u2) = u1(−u2) = 0.

7 This translates as ‘the multiplication table is complete and the procedure is completed’.

24.1 Rewriting Rules 173

It is very tempting to interpret this ‘method’ as an adaptation of the Todd–Coxeter algorithm for enumerating cosets of finitely generated subgroups offinite index in a finitely presented group.8 In any case both this ‘method’ andthe previous quotation from Buchberger’s paper point directly to the two clas-sical approaches for introducing Grobner Theory:

• the connection with rewriting rules and the Knuth–Bendix Algorithm,• a general interpretation of Grobner bases, H-bases and Hironaka’s standard

bases within the theory of graded and filtered rings.9

The next sections will discuss both these approaches.An informal introduction of the theory of rewriting rules (Section 24.1) will

be followed by a presentation of Grobner theory in that context (Section 24.2).Then, after having discussed in detail (Section 24.3) Buchberger theory for

modules, I will show (Section 24.4) that the common pattern of Grobner basesand Macaulay’s H-bases can be generalized in the context of graded rings,where we can characterize the property of Grobner bases as the ability, al-ready noted by Macaulay, of lifting syzygies (Section 24.5), proving the Lift-ing Theorem; I will then generalize this interpretation within valuation rings(Sections 24.6, 24.7 and 24.8).

I then complete this chapter by giving Erdos’ characterization of term or-derings over polynomial rings (Section 24.9) and Bayer’s analysis of thepolytope structure imposed by an ideal on the space of the term orderings(Section 24.10).

24.1 Rewriting Rules

Let S be any set and let us recall that:

Definition 24.1.1. A relation ∼ on S is called

• reflexive if, for each a ∈ S, a ∼ a;• symmetric if a ∼ b ⇒ b ∼ a;• transitive if a ∼ b, b ∼ c ⇒ a ∼ c;• antisymmetric if, for each a, b ∈ S, a ∼ b, b ∼ a ⇒ a = b;• an equivalence relation if it is reflexive, symmetric and transitive;

8 But the reader should be aware that, while the computations are copied from and with no revisionof the Grobner text, the tables are not present there and have been inserted by me.

9 I would like to remark that the linear algebra approach which is also supported in this book ispresent in the Grobner frame of view, in his remark ‘that, in principle, all computations can bereduced to the solution of systems of linear equations’.

174 Grobner I

• Noetherian if there is no infinite sequence

a1 ∼ a2 ∼ · · · ∼ an ∼ · · · ;

• a quasi-order if it is reflexive and transitive.

If S is a set and ∼ is an equivalence relation then, for each a ∈ S one canconsider the equivalence class R(a) := b ∈ S : a ∼ b and the set of all theequivalence classes

S/ ∼:= R(a) : a ∈ S.As usual, it is good to have a suitable representative for each class R(a);

the classical approach is to impose an appropriate order ≺ on S and, for eacha ∈ S, choose an element Can(a, ∼) such that for each b ∈ S

a ∼ b ⇒ Can(a, ∼) b

and call it the canonical form of a mod ∼.In the ‘classical’ case of congruences modulo a prime in domains like Z and

k[X ], the choice of a suitable order and the computation of canonical formsare easily ruled by the Euclidean algorithm.

Iterative application of the Euclidean algorithm being the central point ofthe Kronecker–Duval Model, canonical forms are granted for the roots of uni-variate polynomials.

The ability to define and compute canonical forms is a crucial tool in orderto deal – keeping in mind the Kronecker–Duval Model – with multivariatepolynomial systems but is, more generally, a central problem within computerscience in the wider class of σ -algebras.

Historical Remark 24.1.2. It is worth remarking that the ‘discovery’ of Buch-berger theory, first by the computer algebra community and immediately afterby the algebraic geometry community, was directly connected with the prob-lem. At that time the first (and oldest) computer algebra systems were deal-ing with representation and manipulation of elementary algebraic objects andthe first non-trivial (i.e. non-solvable by means of the Euclidean or Gaussianapproaches) case to be dealt with was that of a polynomial ring modulo anideal.

When Loos in a discussion with Buchberger quoted this as an open prob-lem, Buchberger’s answer was: ‘I solved that problem in my Ph.D. thesis’. Asa consequence Buchberger was invited to speak at the next computer algebrameeting, Eurosam’79, in Marseilles, June 1979. Thus were Grobner bases pre-sented for the first time to the scientific community.


The classical approach to defining canonical forms essentially has two steps:

(1) since one has to take into consideration the computational aspect, a the-oretical definition of ∼ would in many instances (Euclidean division,Gaussian reduction, canonical representation in the Kronecker–DuvalModel, . . . ) serve no purpose without a practical definition which ex-plicitly allows iterative computation of the required canonical forms;this suggests the assumption that ∼ is defined by a generating subset→⊂∼ such that ∼ is the equivalence closure of →;

(2) for taking explicit advantage of the order ≺ imposed on S, one shouldimpose an orientation on the generating subset → in such a way thata → b ⇒ a b.

The corresponding approach consists of repeatedly rewriting the elementsa ∈ S as much as possible until an ‘irreducible normal form’ N F(a) is ob-tained:

a =: a0 → a1 → · · · → an =: N F(a).

Of course, in order to obtain the canonical form of a mod ∼, one must besure not only that such a normal form is unique, but also, since the canon-ical form should give a suitable representative of R(a), that two congruentelements a ∼ b have the same normal form.

This requires us to characterize the properties which must be satisfied by thegenerating set → in order to be granted the computability of canonical forms.

Before doing that, a preliminary point must be fixed: in order to be able toimpose an orientation on →, we should assume at least that for each a, b ∈ S

a ∼ b ⇒ a b or a ≺ b;this assumption requires us to be more precise in our definition: in fact even inour informal definition of Can(a, ∼) we carefully avoided discussing unique-ness. However, the experience with the Euclidean algorithm shows that unique-ness can be forced only modulo associates. Therefore the ‘suitable ordering’≺ must be at least a quasi-order. Then, given such an ordering, we say that twoelements a, b ∈ S are associate if a b and a b. In this way we force allthe canonical forms of an element to become associated.

We might have to impose another requirement on ≺ for two differentreasons:

• the procedure we have outlined consists of repeatedly rewriting an elementa; in order to guarantee termination we must assume that at least → isNoetherian; if → is oriented by ≺ the same requirement on ≺, while notnecessary, is at least helpful;

176 Grobner I

• many of the proofs will be performed by inductive argument; also for thisreason Noetherianity is necessary at least on →.

In our discussion, we will proceed as carefully as possible, by avoiding ref-erence to ≺ over all the statements and proofs and assuming Noetherianity of→ only when we need it.

Let → be an antisymmetric relation on S and let us denote, respectively by→ and ↔ the reflexive–transitive relation and the equivalence relation bothgenerated by →.

Definition 24.1.3. Let → be an antisymmetric relation on S. Then

• a ∈ S is called irreducible if there is no b ∈ S such that a → b;• b ∈ S is called a normal form of a ∈ S if a → b and b is irreducible.

Remark 24.1.4. Let us consider a set S, an equivalence relation ∼ and a quasi-order ≺ on S such that for each a, b ∈ S

a ∼ b ⇒ a b or a ≺ b;As we have remarked above, → could be considered in some sense to be an

‘oriented restriction’ of the equivalence relation ∼, but its definition must begiven carefully; the most obvious choice is to define → on S by

for each a, b ∈ S, a → b ⇐⇒ a ∼ b, a b;so that ↔ concides with ∼ and → is characterized by

for each a, b ∈ S, a → b ⇐⇒ a ∼ b, a b,

but the definition is purely theoretical since it leaves unsolved the problem ofdeciding, given an element a ∈ S, whether it is irreducible or there existsb ∈ S such that b ≺ a.

In order to arrive at an effective definition of → we must restrict ourselvesto a subset ⊂ ∼ generating ∼ in the sense that, for each a, b ∈ S, thereexist ai ∈ S, 0 ≤ i ≤ n, such that

a =: a0 a1 · · · ai ai+1 · · · an := b.

Then we can define → for each a, b ∈ S, by

a → b ⇐⇒ a b and either a b or b a.

There is another point we must keep in mind and the next trivial examplecan help to clarify that: we do not require that the generating set is finite.


In fact, while we keep in mind the case in which S is a domain and ∼ acongruence relation modulo a prime, we are only interpreting S as a set and ∼as an equivalence relation: in other words, we are intentionally forgetting thedomain structure in this discussion: if it becomes useful, we could reconsiderit again only when stating a procedure to test whether, given a ∈ S, there isb ∈ S such that a → b.

As a consequence the generating set in general is considered infinite and,as the next example will show, this does not affect the procedures.

Example 24.1.5. The example we are considering is S := N, with the naturalordering ≺ and the equivalence relation ∼ defined for each a, b ∈ S by

a ∼ b ⇐⇒ a ≡ b (mod 5).

In this case we can define for each a, b ∈ S, by

a b ⇐⇒ |a − b| = 5

and → would be the infinite set of all pairs

→ := (n, n − 5) : n ∈ N, n ≥ 5.

As stupid as it is, this example stresses the following point: given two ele-ments a, b ∈ N without activating the division algorithm, it is impossible totest whether a ∼ b; it is instead sufficient to test whether b ≥ 5 in order todecide whether it is irreducible, and, in the negative case, to rewrite it as b − 5,thus obtaining, by iteration, the finite sequence

b → b − 5 → b − 10 → · · · → N F(b).

And before protesting that this is trivial, please assume that the only com-puter at your disposal is an abacus . . .

Now that we have discussed the trivial aspects, we must focus on the cen-tral point: to characterize the necessary conditions which guarantee that twocongruent elements have the same normal form.

The effect of orienting ∼ is that if a, b ∈ S are such that a ∼ b, then thereare ai ∈ S, 0 ≤ i ≤ n :

a =: a0 → a1 ← a2 → · · · → ai ← ai+1 → · · · an := b.

We have therefore to focus on a local situation

a ← c → b,

178 Grobner I

where our requirement that congruent elements have the same normal formimplies that N F(a) = N F(b):

c

a b↓ ↓

N F(a) = N F(b)

This remark is sufficient to find the required characterization:

Definition 24.1.6. Let → be an antisymmetric relation on S. Then

• a, b ∈ S are said to have a common successor (in symbols a ↓ b) if thereexists d ∈ S such that a → d ← b;

• c ∈ S is said to have a unique normal form in terms of → if for each irre-ducible a, b ∈ S we have

a ← c → b ⇒ a = b;

• → is said to have canonical forms if

∀a ∈ S, ∃!d := Can(a) ∈ S : ∀b ∈ S, b ↔ a ⇒ b → d;

• → is said to have the Church–Rosser property iff for each a, b ∈ S

a ↔ b ⇒ a ↓ b.

Lemma 24.1.7. Let → be an antisymmetric relation on S . Then the followingconditions are equivalent:

R1 → has canonical forms;R2 each c ∈ S has a unique normal form in terms of →;R3 → satisfies the Church–Rosser property.

Proof.

R1 ⇒ R2 Obviously, if a and b are normal forms of c in terms of →, then

a → Can(c) ← b

and, since all are irreducible, a = Can(c) = b.

R2 ⇒ R3 Let a, b ∈ S : a ↔ b; this implies the existence of elementsa =: a0, a1, . . . , am := b which, for each i satisfy either ai ← ai−1

or ai−1 → ai .

Our proof will be by induction in terms of m.

• If m = 1, then, let N F(a) and N F(b) be the normal forms respec-tively of a and b; then either


• a ← b in which case we have N F(a) ← a ← b → N F(b)

and both N F(b) and N F(a) are normal forms of b so thatN F(a) = N F(b) is the common successor of a and b; or

• a → b, in which case the same argument proves that N F(b) andN F(a) are equal (and the required common successor), beingboth normal forms of a.

• If m > 1, by induction we know that there is a common successord of a and am−1. As a consequence

• if b → am−1 then we have a → d ← am−1 ← b and a ↓ b;• while if b ← am−1, let N F(d) and N F(b) be normal forms of

d and b respectively so that

N F(b) ← b ← am−1 → d → N F(d)

and N F(b) and N F(d), being both normal forms of am−1 areequal, and so the required common successor of a and b.

R3 ⇒ R1 Let d be a normal form of a in terms of → and let b ∈ S be suchthat b ↔ a → d; then b ↓ d and there is e ∈ S such that

a ↔ b↓ ↓

d → e

and, since d is irreducible, d = e ← b.

The next step is to ‘localize’ the Church–Rosser property in order to devisean effective test.

Definition 24.1.8. Let → be an antisymmetric relation on S. Then it is called

• confluent if for each a, b, c ∈ S

a ← c → b ⇒ a ↓ b;

• locally confluent if for each a, b, c ∈ S

a ← c → b ⇒ a ↓ b.

Lemma 24.1.9. Let → be an antisymmetric relation on S. Then the followingconditions are equivalent:

R2 each c ∈ S has a unique normal form in terms of →;R3 → satisfies the Church–Rosser property;R4 → is confluent.

180 Grobner I

Proof. R4 being a particular case of R3 we only need to prove that R4 ⇒ R2.Assume a, b ∈ S are different normal forms of c; this implies that a = b anda ← c → b. Then a ↓ b and there exists d ∈ S such that a → d ← b.Since both a and b are irreducible, this gives the required contradiction a =d = b.

Theorem 24.1.10 (Newman). Let → be a Noetherian relation on S. Then thefollowing conditions are equivalent:

R4 → is confluent.R5 → is locally confluent.

Proof. R5 being a particular case of R4 we only need to prove that R5 ⇒ R4.The argument is by induction: if there exists a triple a, b, c ∈ S such that

a ← c → b and there exists no d ∈ S : a → d ← b

among all possible such triples a, b, c, since → is Noetherian there is one inwhich c is minimal w.r.t. → in the sense that for each a′, b′, c′ ∈ S we have

c → c′, a′ ← c′ → b′ ⇒ there exists d ∈ S : a′ → d ← b′.

For such a ‘minimal’ triple a, b, c we easily find a contradiction.In fact c = b ⇒ a → a ← b getting a contradiction; similarly c = a

gives the contradiction a → b ← b.

Therefore we can deduce the existence of a′ and b′ in S such that

a ← a′ ← c → b′ → b.

By assumption R5 we know the existence of d ∈ S such that a′ → d ←

b′. Moreover

(a′) c → a′, a ← a′ → d ⇒ there exists e ∈ S : a → e ← d;(b′) c → b′, e ← b′ → b ⇒ there exists f ∈ S : e → f ← b;

allowing us to deduce from the scheme

c → b′ → b↓ ↓

a′ → d ↓

↓ ↓

a → e → f

the existence of f ∈ S such that a → f ← b and a ↓ b.

As we will see in the next section, Newman’s formulation R5 of the Church–Rosser property can be reformulated within Grobner theory, giving condition


G7. A further weakening of the Church–Rosser property was therefore pro-posed by Buchberger as a generalization of G8 within rewriting rule theory.

It requires us to take in to consideration the quasi-order ≺ which we usedimplicitly to orient → .

Definition 24.1.11. A Noetherian quasi-ordering ≺ on S will be called com-patible with → if a ← b ⇒ a ≺ b.

Definition 24.1.12 (Buchberger–Winkler). Let ≺ be a Noetherian quasi-ordering on S compatible with →. For a, b, c ∈ S, a and b are said tobe c-connected if there exist a =: c0, c1, . . . , cm =: b such that for eachi, ci ≺ c, ci−1 ↓ ci .

Proposition 24.1.13 (Buchberger–Winkler). Let → be a Noetherian rela-tion on S and ≺ a Noetherian quasi-ordering on S compatible with →. Thenthe following conditions are equivalent:

R5 → is locally confluent.R6 For each a, b, c ∈ S : a ← c → b ⇒ a and b are c-connected.

Proof. R6 being weaker than R5, let us prove R6 ⇒ R5 by induction; if existsa triple a, b, c ∈ S such that

a ← c → b and there exists no d ∈ S : a → d ← b

among all possible such triples a, b, c since → is Noetherian there is one inwhich c is minimal w.r.t. → in the sense that for each a′, b′, c′ ∈ S we have

c → c′, a′ ← c′ → b′ ⇒ there exists d ∈ S : a′ → d ← b′.

By R6 for all possible such triples a, b, c, a and b are at least c-connected;therefore we can choose a minimal element γ c and a pair a, b such that

• a ← c → b,

• a and b are γ -connected,• there is no d ∈ S such that a → d ← b.

Therefore any pair a′, b′ such that

• a′ ← c → b′,• a′ and b′ are γ ′-connected,• γ ′ ≺ γ

is such that a′ ↓ b′.

182 Grobner I

By our assumption we can deduce that

• for each i there exists di : ci−1 → di ← ci since ci−1 ↓ ci ;• for each i , di−1 ↓ di since di−1 ← ci → di and ci ≺ γ c;• for each i , di ≺ γ ′ := max≺ci ≺ γ ,

so that d1 and dm are γ ′-connected and, by inductive assumption d1 ↓ dm ;therefore exists e ∈ S such that

a = c0 → d1 → e ← dm ← cm = b,

giving the required contradiction.

All this analysis can be summarized in

Theorem 24.1.14. Let → be an antisymmetric relation on S . Then the fol-lowing conditions are equivalent

R1 → has canonical forms.R2 Each c ∈ S has a unique normal form in terms of →.R3 → satisfies the Church–Rosser property.R4 → is confluent.

If → is Noetherian, then the following condition is also equivalent:

R5 → is locally confluent.

If moreover ≺ is a Noetherian quasi-ordering on S compatible with →, thefollowing condition is also equivalent:

R6 For each a, b, c ∈ S : a ← c → b ⇒ a and b are c-connected.

Algorithm 24.1.15 (Knuth–Bendix). The conclusion of this analysis is theKnuth–Bendix completion procedure which given a finite, antisymmetric,Noetherian relation → on S tries to produce a larger relation

∼→ such that

the congruences ↔ and∼↔

coincide.The algorithm, which succeeds in the case of termination but could never

stop,

• produces all critical pairs (a, b) for which exists c ∈ S such that a ← c →b;

• tests for each critical pair (a, b) whether a ↓ b by computing normal formsa′ and b′ respectively for a and b and checking whether a′ = b′;

24.2 Grobner Bases and Rewriting Rules 183

• and adds, if a′ = b′, the ordered 10 set (a′, b′) to∼→ and extends the set of

the critical pairs.

It should be noted that, while Buchberger’s algorithm is usually presented asan instance of the Knuth–Bendix completion procedure, both results are com-pletely independent and, somehow, Knuth–Bendix could be essentially consid-ered to be a deep review and a wide generalization of many classical rewritingtechniques (not only Euclid and Gauss, but also group theoretical algorithmslike Todd–Coxeter) with which Buchberger’s algorithm shares the same frameof mind.

24.2 Grobner Bases and Rewriting Rules

In order to interpret Grobner bases within the framework of rewriting rules, wemust first define

• a set S;• a congruence relation ∼ on S;• a Noetherian relation → which generates ∼ in the sense that ∼ is the con-

gruence closure of →;• a Noetherian quasi-ordering ≺ which is compatible with →, that is

a ← b ⇒ a ≺ b for each a, b ∈ S.

Obviously, since we are discussing ideals

I := ( f1, . . . , fs) ⊂ P =: k[X1, . . . , Xn]

we will set

• S := k[X1, . . . , Xn] and• p1 ∼ p2 ⇐⇒ p1 ≡ p2 (mod I);

10 Here there is a problem which can be easily solved for specific sets S possessing an algebraic

structure which imposes a Noetherian quasi-ordering ≺ on S; in this case we can enlarge∼→

with a′ ∼→ b′ if a′ b′ and conversely, thus granting that∼→ is still Noetherian.

But in a general case this is the crux:

• Which one among a′ ∼→ b′ and b′ ∼→ a′ still preserves Noetherianity of∼→?

• And, more crucially, even if both choices are compatible, which one should be chosen by usin order not to stop us from extending it in further computations?

• If no choice is compatible at some stage, is this a consequence of a previous arbitrary unluckychoice?

Rewriting-rule theory has dealt with such difficult problems for twenty years.

184 Grobner I

also since the definition of → is induced by ≺ we must focus immediately onthe definition of Noetherian quasi-ordering imposed on P .

The definition is that obviously suggested by the linear algebra structure ofP , which is generated by the linear basis T : once an ordering < is imposed onT , 11 each element f =∑t∈T c( f, t)t ∈ P can be seen as an (infinite) vector

(c( f, t) : t ∈ T )

and two elements can just be compared componentwise. Therefore we define≺ iteratively for any pair p1, p2 ∈ P by

• if p1 = 0 = p2 then p1 p2;• if p1 = 0 = p2, – so that T(p1) = 0 = T(p2),

• if T(p1) > T(p2) then p1 p2, while• if T(p1) = T(p2), p1 p2 ⇐⇒ q1 q2, where we write qi := pi −

M(pi ).

In order to restrict ∼ to a generating set of → it is sufficient to note that

p1 ∼ p2 ⇐⇒ p1 ≡ p2 (mod I)

⇐⇒ ∃hi ∈ P, 1 ≤ i ≤ s : p1 − p2 =s∑

i=1

hi fi

⇐⇒ ∃c j ∈ k \ 0, t j ∈ T , i j , 1 ≤ i j ≤ s : p1 − p2 =u∑

j=1

c j t j fi j .

Therefore, setting F := f1, . . . , fs, it is sufficent to define p1 ↔ p2 by

p1 ↔ p2 ⇐⇒ there exist c ∈ k \ 0, t ∈ T , f ∈ F : p1 = p2 + ct f.

The orientation ↔ by means of ≺ leads to the following definition:12

Definition 24.2.1 (Buchberger). For each g, h ∈ P

h → g ⇐⇒ ∃ t ∈ T , f ∈ F : c(h, tT( f )) = 0, g = h − c(h, tT( f ))

lc( f )t f.

Newman’s Lemma (Theorem 24.1.10) gives the condition (R5) which al-lows us to verify whether reduction to irreducible elements via ↔ allows us tocompute canonical forms modulo I:

11 And we will assume that < is a term ordering, that is a well-ordering (since this will force ≺ tobe Noetherian) satisfying

t1 < t2 ⇒ t t1 < t t2

for each t, t1, t2 ∈ T .12 Where we omit the implicit dependence of → on the data (F, <, . . .).


Problem 24.2.2. For each h ∈ P, t1, t2 ∈ T , f (1), f (2) ∈ F such that

c(h, t1T( f (1))) = 0 = c(h, t2T( f (2))),

writing

gi := h − c(h, ti T( f (i)))

lc( f (i))ti f (i),

do g1 and g2 have a common successor?

Buchberger’s reduction of condition R5 to condition R6 allows us to reduceProblem 24.2.2 to a finite set of cases to be tested; as can be expected, the testsare exactly the same as the definition of S-polynomials. In order to prove that,we need some lemmata:

Lemma 24.2.3 (Buchberger). The following hold:

(1) for each h, g ∈ P, c ∈ k \ 0, t ∈ T , h → g ⇒ cth → ctg;(2) for each h, g, p ∈ P : h → g there exists q ∈ P : h + p → q ←

g + p;(3) 0 is irreducible.

Proof.

(1) It is sufficient to prove that h → g ⇒ cth → ctg, which is trivial.(2) Also in this case we just need to prove that for each h, g, p ∈ P : h →

g there exists q ∈ P : h + p → q ← g + p. Since h → g there aret ∈ T , f ∈ F such that a := c(h, tT( f )) = 0, and

g = h − a

lc( f )t f.

Write m := tT( f ) and b := c(p, m) and remark that c(g, m) = 0.There are different cases:

b = 0 : in this case clearly h + p → g + p;b = 0 = a + b : in this case c(g + p, m) = b = −a = 0 and

h + p = (g + p) − c(g + p, m)

lc( f )t f

so that g + p → h + p;b = 0 = a + b : in this case let us write

q := h + p − a + b

lc( f )t f = g + p − b

lc( f )t f

so that c(q, m) = 0, h + p q ≺ g + p and h + p → q ←g + p.

186 Grobner I

Theorem 24.2.4 (Buchberger). The following conditions are equivalent

(1) → has canonical forms;(2) for each fi , f j ∈ F, the normal form of the S-polynomial of fi and f j

is 0.

Proof. With the same notation as Problem 24.2.2, we need to prove that foreach h ∈ P, t1, t2 ∈ T , and f (1), f (2) ∈ F , g1 and g2 have a commonsuccessor.

Assuming wlog c( f (i), T( f (i))) = lc( f (i)) = 1, and setting

mi := ti T( f (i)) and ri := f (i) − mi ,

we have two cases to consider; in the first one the proof will be based onLemma 24.2.3(2), while the second one will be a consequence of the assump-tion on S-poynomials:

m1 = m2 : We can wlog assume m1 > m2 and we will decompose

h =∑t∈T

c(h, t)t

as

h := H(h) + c(h, m1)m1 + B(h) + c(h, m2)m2 + L(h)

where

H(h) :=∑t∈T

t>m1

c(h, t)t,

B(h) :=∑t∈T

m1>t>m2

c(h, t)t,

L(h) :=∑t∈T

m2>t

c(h, t)t.

Then we have

g1 = H(h) + B(h) + c(h, m2)m2 + L(h) − c(h, m1)t1r1,

g2 = H(h) + c(h, m1)m1 + B(h) + L(h) − c(h, m2)t2r2,

and we can set

g1,2 := H(h) + B(h) + L(h) − c(h, m1)t1r1 − c(h, m2)t2r2,

so that g2 → g1,2. Also, since h → g1 and

g1 = h − c(h, m1)t1 f (1), g1,2 = g2 − c(h, m1)t1 f (1)


Lemma 24.2.3(2) allows us to conclude that g1 ↓ g1,2 (but not thatg1 → g1,2) so that g1 and g2 are h-connected and the claim followsfrom condition R6.

m1 = m2: In this case, for a suitable term u,

t1T( f (1)) = t2T( f (2)) = u lcm(T( f (1)), T( f (2))) = m1 = m2

and, setting c := c(h, m1), we have

g1 = H(h) + B(h) + L(h) − ct1r1,

g2 = H(h) + B(h) + L(h) − ct2r2,

g1 − g2 = −c(t1r1 − t2r2)

= −c(t1 f (1) − t2 f (2))

= −cuS( f (2), f (1)).

By assumption we know that the normal form of S( f (2), f (1)) is 0which means that there are elements p0, . . . , pi , ps ∈ P such that

S( f (2), f (1)) = p0 → p1 → · · · → pi → · · · → ps = 0

and pi ≺ lcm(T( f (1)), T( f (2))).

Thanks to Lemma 24.2.3(1) we can deduce that

g1 − g2 = −cup0 → −cup1 → · · · → −cupi → · · · → −cups = 0

and −cupi ≺ u lcm(T( f (1)), T( f (2))) = m1 = m2.

It is then sufficient to define p′i := −cupi + g2 and to make reference

to Lemma 24.2.3(2) in order to deduce that for each i : p′i ≺ m1

and p′i−1 ↓ p′

i so that g1 = p′0 and g2 = p′

s are m1-connected, that isg1 ↓ g2.

We are now able to reinterpret the Buchberger algorithm in terms ofrewriting-rules theory as follows: once an ideal I is given by giving a basisF := f1, . . . , fs (wlog fi = 1, for each i), the congruence relation ∼ de-fined by

p1 ∼ p2 ⇐⇒ p1 ≡ p2 (mod I)

can be restricted to the generating set → consisting of the pairs T( fi ) →ri where ri := fi − T( fi ) and all its algebraic consequences, that is (seeDefinition 24.2.1)

ctT( fi ) + g → ctri + g, c ∈ k \ 0, t ∈ T , fi ∈ F, g ∈ P, c(g, tT( f )) = 0.

In order to test whether → has canonical forms, so that the computation ofthe normal form of any element a ∈ P would give the canonical representa-tive Can(a, ∼) of the equivalence class R(a) mod I, one must check whether

188 Grobner I

→ satisfies the Church–Rosser property using, instead of the Newman Lemma,the Buchberger–Winkler result which gives not only Theorem 24.2.4 but alsoBuchberger’s Second Criterion (Lemma 22.5.3).

The computation of normal form is performed by repeated reductions

g → g − ct f , where T(g) = tT( f ), c = c(g, T(g)),

until we obtain either 0 as a normal form,13 or an element h such that

g → h = 0 and T(h) ∈ (T( fi ) : 1 ≤ i ≤ s) .

In this case we know that → does not satisfy the Church–Rosser propertyunless we enlarge it with the new relation T(h) → c(h, T(h))−1h − T(h).

24.3 Grobner Bases for Modules

It is now time to summarize, in the more general case of modules, the resultsproved for Grobner bases of an ideal in Chapter 22.

So let (see Section 23.6) us consider P := k[X1, . . . , Xn], endowed witha term ordering < on T := Xa1

1 . . . Xann : (a1, . . . , an) ∈ N

n, and the free-module Pm – the canonical basis of which will be denoted by e1, . . . , em –which is a k-vectorspace generated by the basis

T (m) := tei , t ∈ T , 1 ≤ i ≤ mon which we impose a well-ordering – denoted, with a slight abuse of notationalso by < – satisfying, for each t1, t2 ∈ T , τ1, τ2 ∈ T (m),

t1 ≤ t2, τ1 ≤ τ2 ⇒ t1τ1 ≤ t2τ2.

Therefore each element

f :=s∑

i=1

gi ei = (g1, . . . , gs) ∈ Pm

has a unique ordered representation as an ordered linear combination of the

13 While within rewriting-rules theory, ‘canonical form’ and ‘normal form’ are essentially syn-onymous, in the early 1980s the small community of researchers and implementers workingon Buchberger theory started to distinguish the reductions g → g − ct f according to whethertT( f ) = T(g), since only those reductions such that tT( f ) = T(g) are sufficient to test theChurch–Rosser property and produce new elements h ∈ I : T(h) ∈ (T( fi ) : 1 ≤ i ≤ s) and thecorresponding useful new relations.

While such elements, in the language of rewriting-rule theory, are not normal forms, it wascommon in that community to call ‘normal form’ the results of such restricted reductions. I stillconsider it helpful to follow this practice.

24.3 Grobner Bases for Modules 189

terms t in T (m) with coefficients in k:

f :=s∑

i=1

c( f, ti )ti : c( f, ti ) ∈ k \ 0, ti ∈ T (m), t1 > · · · > ts .

Then we will denote by,

• T( f ) := t1, the maximal term of f ,• lc( f ) := c1, the leading cofficient of f ,• M( f ) := c1t1, the maximal monomial of f ;

and, for any set F ⊂ Pm , write

• TF := T( f ) : f ∈ F;• T(F) := τT( f ) : τ ∈ T , f ∈ F;• N(F) := T (m) \ T(F);• k[N(F)] := Spank(N(F)).

Definition 24.3.1. Let M ⊂ Pm be a submodule, G ⊂ M, f, h, f1, f2 ∈ Pm .

Then

• G will be called a Grobner basis of M if

T(G) = T(M),

that is TG := T(g) : g ∈ G generates T(M) = TM,• for each f1, f2 ∈ Pm such that

lc( f1) = 1 = lc( f2), T( f1) = t1ei1 , T( f2) = t2ei2 ,

the S-polynomial of f1 and f2 exists only when ei1 = ei2 := ε in which caseit is

S( f1, f2) := δ( f1, f2)

t2f2 − δ( f1, f2)

t1f1,

where δ := δ( f1, f2) := lcm(t1, t2) and δε is called the formal term ofS( f1, f2),

• f has the Grobner representation∑m

i=1 pi gi in terms of G, if

f =m∑

i=1

pi gi , pi ∈ P, gi ∈ G, T(pi )T(gi ) ≤ T( f ), for each i ,

• f has the (strong) Grobner representation∑µ

i=1 ci ti gi in terms of G if

f =µ∑

i=1

ci ti gi , ci ∈ k \ 0, ti ∈ T , gi ∈ G,

with T( f ) = t1T(g1) > · · · > ti T(gi ) > · · ·,

190 Grobner I

• for each f1, f2 ∈ Pm, lc( f1) = 1 = lc( f2), whose S-polynomial existsand has δε as its formal term, we say that S( f1, f2) has a weak Grobnerrepresentation in terms of G if it can be written as S(g, f ) = ∑m

k=1 pk gk,

with pk ∈ P, gk ∈ G and T(pk)T(gk) < δε for each k,• h is called a normal form of f w.r.t. G, if

• f − h ∈ (G) has a strong Grobner representation in terms of G and• h = 0 ⇒ T(h) /∈ T(G).

Lemma 24.3.2. With the notation above, we have

(1) For each f ∈ Pm \ 0, G ⊂ Pm, there is a normal form h :=N F( f, G) of f w.r.t. G.

(2) For each f ∈ Pm \ 0, there is h ∈ k[N(M)] such that f − h ∈ M.(3) For each f ∈ Pm \ 0 and any Grobner basis G of M there is h ∈

k[N(M)] such that f − h ∈ M has a strong Grobner representation interms of G.

Proof.

(1) If the claim is false, among the elements f ∈ Pm\0 which do not havea normal form w.r.t. G let us choose one for which T( f ) is minimal.Since, if T( f ) /∈ T(G), f would be a normal form of itself w.r.t. G,then necessarily, T( f ) ∈ T(F), and there are t1 ∈ T , g1 ∈ G, suchthat T( f ) = t1T(g1). Setting

f1 := f − lc( f ) lc(g1)−1t1g1,

since T( f1) < T( f ) then, by minimality, we know that there are a nor-mal form h := N F( f1, G) of f1, and a strong Grobner representation

f1 − h =µ∑

i=2

ci ti gi

in terms of G.We have got the required contradiction, since

f − h = lc( f )

lc(g1)t1g1 +

µ∑i=2

ci ti gi

is a strong Grobner representation and h is the required normal formof f .

(2) Again let us assume the claim is false and let us consider a counterex-ample f ∈ Pm \ 0 for which T( f ) is minimal.


If T( f ) ∈ N(M), we would get a contradiction since f ′ := f − M( f ),not being zero – otherwise f = M( f ) ∈ k[N(M)] – satisfies T( f ′) <

T( f ); but then there is h′ ∈ k[N(M)] such that

M f ′ − h′ = f − (h′ + M( f )) and h := h′ + M( f ) ∈ k[N(M)].

Therefore we must assume T( f ) ∈ T(M), but also this gives us acontradiction; we only have to choose any element f1 ∈ M such thatT( f1) = T( f ) and define

f ′ := f − lc( f ) lc( f1)−1 f1

so that T( f ′) < T( f ); therefore, there is h ∈ k[N(M)] such that f ′ −h ∈ M and

M f − h = lc( f ) lc( f1)−1 f1 + ( f ′ − h).

(3) In the proof of the previous statement we have just to choose as f1

an element t1g1, t1 ∈ T , g1 ∈ G and to denote by∑µ

i=2 ci ti gi thestrong Grobner representation of f1 in terms of G, whose existenceis known inductively, in order to produce the required contradictorystrong Grobner representation (lc( f )/ lc(g1))t1g1 +∑µ

i=2 ci ti gi of fin terms of G.

Corollary 24.3.3. Let N be a finite P-module, Φ : Pm → N be any surjectivemorphism and set M := ker(Φ). Then we have

(1) Pm ∼= M ⊕ k[N(M)];(2) N ∼= k[N(M)];(3) for each f ∈ Pm, there is a unique g := Can( f, M) ∈ k[N(M)] such

that f − g ∈ M. Moreover,

(a) Can( f1, M) = Can( f2, M) ⇐⇒ f1 − f2 ∈ M,(b) Can( f, M) = 0 ⇐⇒ f ∈ M;

(4) for each f ∈ Pm, f − Can( f, M) has a strong Grobner representationin terms of any Grobner basis;

(5) there is a unique set G ⊂ M – its reduced Grobner basis – such that

(a) TG is an irredundant basis of T(M),(b) for each g ∈ G, lc(g) = 1,(c) for each g ∈ G, g = T(g) − Can(T(g), M).

Proof. If (3) holds, then (1), (2) and (5) follow trivially and (4) follows fromthe lemma above.

192 Grobner I

It is then sufficient to prove that, for each f ∈ Pm, there exists a unique

g := Can( f, M) ∈ k[N(M)] : f − g ∈ M.

The existence of such a g is known from the lemma above, and we onlyhave to prove its uniqueness: the existence of g1, g2 ∈ k[N(M)] such thatf − gi ∈ M, i = 1, 2, implies that

g1 − g2 = ( f − g2) − ( f − g1) ∈ k[N(M)] ∩ M

so that g1 = g2 since otherwise we would obtain the contradiction

0 = T(g1 − g2) ∈ N(M) ∩ T(M).

The same kind of argument allows us to prove both (a) and (b).

Theorem 24.3.4. Let M ⊂ Pm be a sub-module, and g1, . . . , gs =: G ⊂ M,

with lc(g j ) = 1, T(g j ) := t j ei j , for each j; the following conditions – whereS(k, j) denotes S(gk, g j ) and ω(k, j) its formal term – are equivalent:

G1 G is a Grobner basis of M;G2 tg : g ∈ G, t ∈ T is a Gauss generating set;G3 f ∈ M ⇐⇒ it has a Grobner representation in terms of G;G4 f ∈ M ⇐⇒ it has a strong Grobner representation in terms of G;G5 for each f ∈ Pm \ 0 and any normal form h of f w.r.t. G, we have

f ∈ M ⇐⇒ h = 0;

G6 for each f ∈ Pm\0, f −Can( f, M) has a strong Grobner representationin terms of G;

G7 for each k, j, 1 ≤ k < j ≤ m, the S-polynomial S(k, j) (if it exists) has aweak Grobner representation in terms of G;

G8 for each k, j, 1 ≤ k < j ≤ s : eik = ei j =: ε – so that S(k, j) exists –there are k = k0, k1, . . . , kρ, . . . kr = j, 1 ≤ kρ ≤ s :

• lcm(tkρ , 0 ≤ ρ ≤ r) = lcm(tk, t j ),• eikρ

= ε, for each ρ,

• each S-polynomial S(kρ−1, kρ) has a weak Grobner representationin terms of G.

Proof.

G1 ⇐⇒ G2 Both statements are equivalent to

T(M) = T(tg) : g ∈ G, t ∈ T .G1 ⇒ G5 Let f ∈ Pm \ 0 and h be a normal form of f w.r.t. G. Then

either


• h = 0 and f = f − h ∈ (G) ⊂ M, or• h = 0, T(h) /∈ T(G) = T(M), h ∈ M and f ∈ M.

G5 ⇒ G4 If f has a strong Grobner representation in terms of G, then f ∈(G) ⊂ M.

Conversely, if f ∈ M and h is a normal form of f w.r.t. G, then h = 0and f = f − h has a strong Grobner representation in terms of G.

G1 ⇒ G6 follows from Corollary 24.3.3(4).G6 ⇒ G4 Since for each f ∈ M, Can( f, M) = 0, then f has a strong

Grobner representation in terms of G.G4 ⇒ G3 is trivial.G3 ⇒ G1 Let τ ∈ T(M); then there is f ∈ M such that T( f ) = τ.

Let f =∑mi=1 pi gi be a Grobner representation.

Then, for some i , τ = T( f ) = T(pi )T(gi ), that is τ ∈ T(G).

G3 ⇒ G7 Since each S(k, j) ∈ (G) = M, then it has a Grobner representa-tion

S(k, j) =m∑

i=1

pi gi , where T(pi )T(gi ) ≤ T(S(k, j))

< ω(k, j) for each i.

G7 ⇒ G3 Let us consider a generic element h ∈ M; since G is a basis of Mthere is a representation h =∑s

i=1 pi gi .

If γ1 := maxi T(pi )T(gi ) ≤ T(h), the representation is a Grobnerone, and we are through.Otherwise, writing J := i : T(pi )T(gi ) = γ1, we have

0 =∑j∈J

M(p j )T(g j ) =∑j∈J

lc(p j )T(p j )T(g j ) =∑j∈J

lc(p j )γ1

and∑

j∈J lc(p j ) = 0. In this case, we intend to show thatthere is another representation h = ∑s

i=1 p′i gi for which γ2 :=

maxi T(pi )T(gi ) < γ1. Then the thesis follows from an inductiveargument, since < is a well-ordering and we cannot have an infinitedecreasing sequence

γ1 > γ2 > · · · > γν > · · · > T(h).

Let δ ∈ T , ε ∈ ei , 1 ≤ i ≤ m be such that γ1 = δε, and let uswrite ι := min(J ).Since for each j ∈ J, T( j) | γ1, then ei j = ε and t j | δ; therefore, foreach j ∈ J \ ι, S(ι, j) exists and also τ j exists such that

τ j lcm(t j , tι) = δ = T(p j )t j = T(pι)tι and T(p j ) = τ jlcm(t j , tι)

t j.

194 Grobner I

Therefore∑j∈J

lc(p j )T(p j )g j =∑j∈J

lc(p j )τ jlcm(t j , tι)

t jg j

=∑j∈J

lc(p j )τ j

(lcm(t j , tι)

t jg j − lcm(t j , tι)

tιgι

)

+(∑

j∈J

lc(p j )

)τ j

lcm(t j , tι)

tιgι

=∑j∈J

lc(p j )τ j S(ι, j).

By assumption, each S(ι, j) has a weak Grobner representation

S(ι, j) =s∑

i=1

pi j gi : τ j T(pi j )T(gi ) < τ jω( j, ι) = δε = γ1.

Therefore if, for each j ∈ J , we define q j := p j − M(p j ), sinceT(q j ) < T(p j ) we have

h =s∑

i=1

pi gi

=∑j∈J

lc(p j )T(p j )g j +∑j∈J

q j g j +∑i ∈J

pi gi

=∑j∈J

lc(p j )τ j S(ι, j) +∑j∈J

q j g j +∑i ∈J

pi gi

=s∑

i=1

∑j∈J

lc(p j )τ j pi j gi +∑j∈J

q j g j +∑i ∈J

pi gi

which is the required Grobner representation.G7 ⇒ G8 is trivial.G8 ⇒ G7 By assumption, for each ρ there exists τρ such that

τρ lcm(tkρ−1 , tkρ ) = lcm(tk, t j ) =: τ.

Therefore

S(k, j) = τ

t jg j − τ

tkgk

=r∑

ρ=1

τ

tkρ

gkρ − τ

tkρ−1

gkρ−1

=r∑

ρ=1

τρ S(kρ−1, kρ).

24.4 Grobner Bases in Graded Rings 195

Since we have ω(k, j) = τρω(kρ−1, kρ), for each ρ, we just needto substitute for each S(kρ−1, kρ) its weak Grobner representation inorder to produce the one required for S(k, j).

Remark 24.3.5. The reader must be aware that in the module case, while Buch-berger’s Second Criterion still holds and can be used (actually it is implicitlycontained in the statement of (G8)), Buchberger’s First Criterion does not holdany more, as the reader can easily realize by trying to generalize the proof ofLemma 22.5.1.

24.4 Grobner Bases in Graded Rings

The analogy between Grobner bases and H-bases, in their definitions, in theirproperties related to the Lifting Theorem (Section 23.7), in their applicationto test ideal membership and to provide degree-bounded representation, sug-gested these notions should be interpreted in the context of graded rings.

Definition 24.4.1. If Γ is a (commutative) semigroup, a ring R is called aΓ -graded ring if there is a family of subgroups Rγ : γ ∈ Γ such that

• R =⊕γ∈Γ Rγ ,• Rγ Rδ ⊂ Rγ+δ for any γ, δ ∈ Γ.

An R-module M of a Γ -graded ring R is called a Γ -graded R-module ifthere is a family of subgroups Mγ : γ ∈ Γ such that

• M =⊕γ∈Γ Mγ ,• Rγ Mδ ⊂ Mγ+δ for any γ, δ ∈ Γ.

Each element x ∈ Mγ is called homogeneous of degree γ .Each element x ∈ M can be uniquely represented as a finite sum x :=∑γ∈Γ xγ where xγ ∈ Mγ and γ : xγ = 0 is finite; each such element xγ is

called a homogeneous component of degree γ.

Definition 24.4.2. Let us assume that Γ is totally ordered by the semigroup14

ordering <. Then, for each element x ∈ M, x = 0,15 x :=∑γ∈Γ xγ , denote

v(x) := max<γ : xγ = 0 its degree, andL(x) := xv(x) its leading form.

14 When speaking of graded and valuation rings and modules, we sometimes omit to mention theimplicit assumption that the ordering < over Γ is a semigroup one.

15 As usual 0 needs a special treatment; we set L(0) = 0 and we assume that v(0) ∈ Γ and weset v(0) < γ, for each γ ∈ Γ .

196 Grobner I

For any set G ⊂ M write 16 LG := L(x) : x ∈ G and let L(G) ⊂ M bethe submodule generated by LG and remark that, for any submodule M ⊂M,

L(M) = LM = L(x) : x ∈ M.Lemma 24.4.3. Let Γ be a semigroup, totally ordered by <. Let R be a Γ -graded ring and let M be a Γ -graded R-module.

Then for each r, r1, r2 ∈ R \ 0, x, x1, x2 ∈ M \ 0 we have:

• v(r x) = v(r) + v(x),L(r x) = L(r)L(x);• v(x1 − x2) ≤ max(v(x1), v(x2));• v(x1 − x2) < max(v(x1), v(x2)) ⇐⇒ L(x1) = L(x2);• L(x1 − x2) = L(x1) − L(x2) ⇐⇒ v(x1 − x2) = v(x1) = v(x2);• L(x1 − x2) = L(x1) ⇐⇒ v(x1 − x2) = v(x1) > v(x2).

Definition 24.4.4. Let Γ be a semigroup, totally ordered by <. Let R be aΓ -graded ring and let M be a Γ -graded R-module and M ⊂ M be a sub-module of M. A set G ⊂ M is called a Grobner basis or standard basis 17 of Mif LG = L(g) : g ∈ G generates L(M) = LM.

For each h ∈ M a representation

h =∑

i

hi gi : hi ∈ R, gi ∈ G

is called a standard representation in R in terms of G iff

v(h) ≥ v(hi ) + v(gi ), for each i.

For each h ∈ M any element g ∈ M such that

• h − g has a standard representation in R in terms of G, and• g = 0 ⇒ L(g) ∈ L(M)

is called a normal form of h.

Example 24.4.5. The obvious example is the ring R := P = k[X1, . . . , Xn]which is refinable into a graded ring by means of the semigroup

16 The symbol L is chosen as a mnemonics for both the leading form L(x) and the leitideal L(M).17 The notion of Grobner basis was introduced by Buchberger in the context in which R = P and

Γ = T ordered by any term-ordering; the notion of standard basis was introduced by Hironakain a context in which either R = P or R = k[[X1, . . . , Xn ]] and, in general, v(Xi ) < v(1) foreach i.

In this book, I will restrict the term ‘Grobner basis’ to Buchberger’s theory while I will speakof ‘standard bases’ when discussing Macaulay’s and Hironaka’s theories and the generalizationsof Buchberger’s theory.

24.4 Grobner Bases in Graded Rings 197

N under which Ri is the set of the homogeneous polynomials of degree i ,v( · ) := deg( · ), L( · ) := H( · ), and where the notion of standardbasis coincides with that of H-basis;

Nn ∼= T under which

Rt = ct : c ∈ k ∼= k, v( · ) := T( · ),

L( · ) := lc( · )T( · ) = M( · ),

and where the notion of standard basis coincides with the one ofGrobner basis. In the case of the module Rr , the notion of standardbases coincides with that of T-bases (Definition 23.6.2).

Theorem 24.4.6. Let Γ be a semigroup, totally ordered by <. Let R be a Γ -graded ring, M a Γ -graded R-module, M ⊂ M a sub-module of M, and letG := g1, . . . , gs ⊂ M. Then, if < is well-ordered, the following conditionsare equivalent:

(1) G is a standard basis of M;(2) for each h ∈ M, h ∈ M iff it has a standard representation in R in

terms of G;(3) for each h ∈ M either

• h ∈ M and h has a standard representation in R in terms of G, or• h ∈ M and there is g ∈ M \ 0 : L(g) ∈ L(M) and h − g has a

standard representation in R in terms of G;

(4) for each h ∈ M there is a normal form g ∈ M;

and all imply that G is a basis of M.

Proof.

(1) ⇒ (4) The proof can be performed by induction: let us assume that foreach h′ ∈ M : v(h′) < γ there is g ∈ M such that

• h′ − g has a standard representation in R in terms of G, and• g = 0 ⇒ L(g) ∈ L(M)

and let us consider any element h ∈ M : v(h) = γ . Either

• L(h) ∈ L(M) and then we can set g := h, or• L(h) ∈ L(M) and there are homogeneous elements mi ∈ R such

that

L(h) =∑

i

miL(gi ), v(h) = v(mi ) + v(gi ), for each i.

Then h′ := h −∑i mi gi ∈ M is such that v(h′) < v(h). Thereforethere are g ∈ M such that

g = 0 ⇒ L(g) ∈ L(M)

198 Grobner I

and a standard representation h′ − g =∑i hi gi in R such that

v(hi ) + v(gi ) ≤ v(h′) < v(h).

As a consequence h − g =∑i (mi + hi )gi is the required standardrepresentation.

(4) ⇒ (3): Let g be any normal form of h; either

• g = 0 and h has a standard representation in R in terms of G, or• g = 0, so that L(g) ∈ L(M), g ∈ M and h ∈ M.

(3) ⇒ (2): Trivial.(2) ⇒ (1): We need to prove that each m ∈L(M) has a representation m =∑

i∈I riL(gi ), ri ∈ R homogeneous, v(m) = v(ri ) + v(gi ) for eachi.If m ∈ L(M), then there is h ∈ M such that L(h) = m. Let h =∑

i hi gi be a standard representation in R in terms of G.If we set I := i : v(h) = v(hi ) + v(gi ), then

m = L(h) =∑i∈I

L(hi )L(gi )

is the representation we are seeking.

24.5 Standard Bases and the Lifting Theorem

Let Γ be a semigroup, totally ordered by the semigroup ordering <, R be aΓ -graded ring, M be a Γ -graded R-module, and let G := g1, . . . , gs ⊂ Mbe a basis generating a submodule M ⊂ M .

In this generalized context we again discuss the finite S-polynomial testneeded to check whether G is a standard basis; the goal, of course, is to finda finite set of elements for which the existence of a standard representationin R in terms of G is sufficient to guarantee the existence of such a standardrepresentation for any module element.

The idea is to start with the fact that for any f ∈ M there is a representationf = ∑

i hi gi in terms of G and we consider the condition which must besatisfied by this representation in order to be standard, that is that

v( f ) ≥ v(hi ) + v(gi ), for each i.

Let us therefore set

v := maxi

v(hi ) + v(gi ) and J := i : v(hi ) + v(gi ) = v.

Clearly v ≥ v( f ) and, if v = v( f ), the representation is standard.

24.5 Standard Bases and the Lifting Theorem 199

We have therefore to check what happens if v > v( f ): writing

R(hi ) := hi − L(hi )

we can partition the representation f =∑i hi gi as

f =∑

i

hi gi =∑i∈J

L(hi )gi +∑i∈J

R(hi )gi +∑i ∈J

hi gi ,

where

• ∑i∈J L(hi )L(gi ) = 0,

• v(R(hi )) + v(gi ) < v, for each i ∈ J,

• v(hi ) + v(gi ) < v, for each i ∈ J.

Writing f ′ :=∑i∈J L(hi )gi , and remarking that v( f ′) < v, we can deducethat the ability of finding a ‘better’ representation f ′ := ∑

i h′i gi in the sense

that v(h′i ) + v(gi ) < v, for each i , implies the ability of producing a ‘better’

representation

f =∑

i

h′i gi +

∑i∈J

R(hi )gi +∑i ∈J

hi gi =:∑

i

hi gi

for which

v > maxi

v(hi ) + v(gi ) ≥ v( f ).

If for each γ ∈ Γ and each decreasing sequence

γ1 > γ2 > · · · > γ j > · · ·there is n such that γn ≤ γ ,18 it is sufficient to repeat the same argumenton the new representation f = ∑

i hi gi , in order to eventually find either arepresentation

f =∑

i

hi gi : maxi

v(hi ) + v(gi ) = v( f ),

that is a standard representation in R in terms of G, or a set of elementsm1, . . . , ms ∈ R such that

(1) for each i, mi is homogeneous,(2) there exists γ ∈ Γ such that, for each i, mi = 0 ⇒ v(mi )+v(gi ) =

γ ,(3)

∑miL(gi ) = 0,

(4)∑

mi gi does not have a standard representation in R in terms of G.

Therefore, we have proved that:

18 Which holds if, as we are implicitly assuming, < is well-ordered.

200 Grobner I

Lemma 24.5.1. Let Γ be a semigroup, totally ordered by <, R be a Γ -gradedring, M be a Γ -graded R-module, and let G := g1, . . . , gs ⊂ M be a basisgenerating a sub-module M ⊂ M.

Let Φ be the set of elements (m1, . . . , ms) ∈ Rs such that

(1) for each i, mi is homogeneous,(2) there exists γ ∈ Γ such that, for each i, mi = 0 ⇒ v(mi )+v(gi ) = γ ,(3)

∑i miL(gi ) = 0;

and let

H :=∑

i

mi gi : (m1, . . . , ms) ∈ Φ

.

If Γ is inf-limited (see Definition 24.5.2 below) and G is not a standard basis,then there is h ∈ H which does not have a standard representation in R interms of G,

where we use

Definition 24.5.2. Let Γ be a semigroup, totally ordered by <. Then Γ is saidto be inf-limited if for each γ ∈ Γ and each decreasing sequence

γ1 > γ2 > · · · > γ j > · · ·there is n such that γn ≤ γ .

The set Φ introduced in Lemma 24.5.1 is a set of syzygies among the setL(g1), . . . ,L(gs).

Moreover if we consider the module Rs and the morphism

s : Rs → R defined by s(m1, . . . , ms) :=∑

i

miL(gi ),

then the set of the syzygies is the module ker(s).In this context, we impose naturally a Γ -graded module structure on Rs in

such a way that s is homogeneous, that is it maps homogeneous elements tohomogeneous ones of the same degree, following the pattern used to define degand T −deg on a polynomial module in Section 23.6: denoting by e1, . . . , esthe canonical basis of Rs so that s(ei ) = L(gi ), we impose on Rs the structureof graded module such that, for each i , v(ei ) = ωi by setting ωi := v(gi ).

19

19 In other words we define

Rs =⊕γ∈Γ

(Rs )γ

where

(Rs )γ = M1 ⊕ · · · ⊕ Mi ⊕ · · · ⊕ Ms

24.5 Standard Bases and the Lifting Theorem 201

Once this is done, we immediately note that Φ is the set of the homogeneouscomponents of the syzygy module ker(s) and the constant value γ ∈ Γ suchthat, for each i,

mi = 0 ⇒ v(mi ) + v(gi ) = γ

is in fact γ := v(m1, . . . , ms).

In the same context we can now consider the map

S : Rs → R defined by S(h1, . . . , hs) :=∑

i

hi gi =: h

which of course is not homogenous; the best we can obtain is

v(hi ) + v(gi ) ≥ v(h).

This is in fact the nux of Buchberger theory.Let us therefore consider h ∈ H and σ := (m1, . . . , ms) ∈ Φ such that

h := S(σ ) =∑

i

mi gi .

The test suggested by Lemma 24.5.1 in order to check whether G is a standardbasis requires us to compute a standard representation h =∑i hi gi such that

v(hi ) + v(gi ) ≤ v(h) < γ = v(m1, . . . , ms).

If we define hi := mi − hi , for each i , and Σ := (h1, . . . , hs), we have

• ∑i hi gi =∑i mi gi −∑i hi gi = 0,

• v(Σ) = v(σ ) = γ,

• L(Σ) = σ,

so that Σ is a syzygy among G, that is Σ ∈ ker(S). This suggests stating thetest we are considering as

For each homogeneous syzygy σ ∈ ker(s) is there a syzygy Σ ∈ ker(S)

such that L(Σ) = σ?Finally, instead of testing this property for all homogeneous syzygies σ ∈

ker(s), it is sufficient to test it only for a (homogeneous) basis of ker(s).

Let us denote by U a homogeneous basis of ker(s) and, for each u ∈ U, letus pick an element lift(u) ∈ ker(S) such that L(lift(u)) = u. Then for eachhomogeneous σ ∈ ker(s) there are homogeneous elements

m(u) ∈ R : σ =∑u∈U

m(u)u and v(σ ) = v(m(u)) + v(u) ⇐⇒ m(u) = 0.

and

Mi :=

Rγi if there exists γi : γi + ωi = ω,

0 if there is no γi : γi + ωi = ω.

202 Grobner I

Then Σ :=∑u∈U m(u) lift(u) is such that Σ ∈ ker(S) and

L(Σ) =∑u∈U

m(u)L(lift(u)) =∑u∈U

m(u)u = σ.

Conversely, if (h1, . . . , hs) =: Σ ∈ ker(S) then each homogeneous com-ponent of

∑i hi gi must be 0; therefore σ := L(Σ) ∈ ker(s) and there are

homogeneous elements m(u) ∈ R such that

σ =∑u∈U

m(u)u and v(σ ) = v(m(u)) + v(u) ⇐⇒ m(u) = 0;

therefore L(Σ) =∑u∈U m(u)L(lift(u)), that is lift(u) : u ∈ U is a standardbasis of ker(S).

We can summarize this analysis by introducing

Definition 24.5.3. With the notation above, if u ∈ ker(s) is homogeneous andv ∈ ker(S) is such that u = L(v), we say that u lifts to v, or v is a lifting of u,or simply u has a lifting,

and stating

Proposition 24.5.4. Let Γ be a semigroup, inf-limited by <, R be a Γ -gradedring, M be a Γ -graded R-module, G := g1, . . . , gs ⊂ M be a basis generat-ing a submodule M ⊂ M, and U be a homogeneous basis of the module ker(s)

of the syzygies among L(g1), . . . ,L(gs).Then G is a standard basis iff each u ∈ U has a lifting.In this case lift(u) : u ∈ U is a standard basis of ker(S).

Proposition 24.5.4 is a formalization in the context of graded rings ofMacaulay’s result (Historical Remark 23.7.2). Also it is a generalization ofTheorem 23.7.3, whose statement is obtained by setting R := P = k[X1, . . . ,

Xn], M := Pr , M := I, U := σ1, . . . , σs, and Σi := lift(σi ).

Remark 24.5.5. The relation between Grobner bases and H-bases stated inLemma 23.2.4 and discussed in Section 23.6 applies also in the setting ofgraded rings.

Let P := k[X1, . . . , Xn] and let

T := Xa11 . . . Xan

n , (a1, . . . , an) ∈ Nn.

We impose on P a graded ring structure by assigning a weight vector

w := (w1, . . . , wn) ∈ Rn, wi ≥ 0,

and impose on P the weight function vw satisfying, for each i, vw(Xi ) := wi ,

24.6 Hironaka’s Standard Bases and Valuations 203

so that vw(Xa11 . . . Xan

n ) = ∑i wi ai ; then Pγ is the vectorspace spanned by

t ∈ T : vw(t) = γ .Let Pm be a free-module whose canonical basis is denoted by e1, . . . , em

and whose vectorspace basis is T (m) = tei , t ∈ T , 1 ≤ i ≤ m, and let us im-pose a graded module structure on it by choosing a vector d = (d1, . . . , dm) ∈R

m , imposing on each ei the degree di and setting, for each tei ∈ T (m)

vw,d(tei ) = di + vw(t). We also denote by Lw : P → P, and Lw,d : Pm →Pm the corresponding leading-form maps.

Let us now consider on P any term ordering < and on Pm any term ordering(which we will still denote <) compatible with < and let us define the termorderings ≺ on both P and Pm by

t1 ≺ t2 ⇐⇒

vw(t1) < vw(t2) orvw(t1) = vw(t2) and t1 < t2,

and

t1ei ≺ t2e j ⇐⇒

vw,d(t1ei ) < vw,d(t2e j ) orvw,d(t1ei ) = vw,d(t2e j ) and t1ei < t2e j .

In this setting, we have:

Corollary 24.5.6. With the notation above, let f ∈ Pm and M ⊂ Pm be asubmodule; then

• T≺( f ) = T≺(Lw,d( f )) = T<(Lw,d( f )),• T≺(M) = T≺(Lw,d(M)) = T<(Lw,d(M)),• If G is a Grobner basis of M w.r.t. ≺, then it is a standard basis of M and

Lw,d(g) : g ∈ G is a Grobner basis of Lw,d(M) w.r.t. ≺ and <.

Proof. Repeat verbatim the proof of Lemma 23.2.4.

24.6 Hironaka’s Standard Bases and Valuations

As Macaulay’s bases deal with projective varieties, another notion indirectlyrelated with Grobner theory deals with geometrical investigation of varieties,Hironaka’s notion of standard bases.

If we consider a univariate series f (X) := ∑∞i=0 ci Xi ∈ Q[[X ]] which is

the Taylor expansion of an analytic function f : R → R such that f (0) = 0,then the order of f , v( f ), is the least value γ ∈ N : cγ = 0 and its initial formL( f ) := cγ Xγ is the lowest-order non-zero Taylor approximation of f at theorigin.

204 Grobner I

Of course, if we consider f as the basis of an ideal ( f ) =: I ⊂ Q[[X ]], wehave v( f ) = minv(g) : g ∈ I and L( f ) generates L(g) : g ∈ I; if f is apolynomial both v( f ) and L( f ) define the multiplicity of f at the origin.

If we now consider an ideal I ⊂ C[X1, . . . , Xn] defining the variety Z(I) ⊂C

n containing the origin, and we use the same notation, so that for each seriesf = ∑∞

i=0 fi ∈ C[[X1, . . . , Xn]], where fi is a homogeneous polynomial ofdegree i , we denote v( f ) := minγ : fγ = 0 the order of f , and we callL( f ) := fγ the initial form of f ; then the variety defined by the ideal L(I)generated by LI is the cone of all the tangents at the origin of the varietyZ(I).

This leads 20 to

Definition 24.6.1 (Hironaka). Let I ⊂ k[[X1, . . . , Xn]]; a set B ⊂ I is a stan-dard basis of I if LB := L(g) : g ∈ B generates L(I) = LI := L(g) :g ∈ I.

Hironaka’s notion introduced a new crucial twist within Grobner theory. Infact, while his notion, at least once it is restricted to ideals I ⊂ k[X1, . . . , Xn],seems to be another instance of the notion introduced in Definition 24.4.4, thesemigroup Γ , even only being N, is not well-ordered. In fact, for a polynomialf = ∑

fi , unlike Macaulay’s notion which takes into consideration the ho-mogeneous component of highest degree, Hironaka’s takes as its initial formthe lowest one. In other words, the ordering ≺, which we must impose on N

in order to interpret Hironaka’s notion in the context of graded rings, is theconverse of the natural ordering:

· · · ≺ n + 1 ≺ n ≺ · · · ≺ 2 ≺ 1 ≺ 0.

The immediate consequence is that Γ is not well-ordered, ≺ is no longerNoetherian, inductive arguments cannot be applied so that the proof of Theo-rem 24.4.6 does not hold any more; in fact, the recursive algorithm to compute

20 Classically, this concept was denoted by in( f ), in(I), etc. and, if one was considering a homo-geneous component

h := (h1, . . . , hm ) =∑

i

hi ei ∈ k[[X1, . . . , Xn ]]m ,

with the old notation one would have in(h) = h; now the notation is restricted to in(h) := hi eiwhere hi is the first (or last, according to the definition) component which is not zero.

Among these two alternative definitions of in(h) for a generic module element h ∈k[[X1, . . . , Xn ]]m , the older has the big advantage of guaranteeing easier proofs of crucialproperties and has essentially no computational disadvantage (all you need to do is substitute amonomial division test with the solution of m linear equations), therefore we will stick to theolder definition.

In order to avoid possible confusion, we avoid the notation in and denote the same conceptby L, reminding us of the classical notions of ‘leading form’ and ‘Leitideal’.


standard representation in terms of a standard basis does not terminate andthe existence of standard representation in terms of a standard basis is notestablished.

Example 24.6.2. The example is trivial: we only have to consider the idealI := (X) ⊂ k[X ] and the polynomial f := X − X2 ∈ I, which, of course, doesnot generate the ideal I while its initial form L( f ) = X generates L(I) = (X).

If we apply the algorithm implicit in the proof of Theorem 24.4.6 in order toproduce a standard representation of X in terms of f := X − X2 we performthe infinite computation

X = 1 f + X2

= (1 + X) f + X3

= (1 + X + X2) f + X4

. . .

=(

d−1∑i=0

Xi

)f + Xd+1

=(

d∑i=0

Xi

)f + Xd+2

. . .

which gives the standard representation

X =( ∞∑

i=0

Xi

)(X − X2) = (1 − X)−1(X − X2)

of X in terms of f in k[[X ]] but not in k[X ].

Remark 24.6.3. While Example 24.6.2 already shows that there are problemsin computing standard representations in a finite number of steps, it could beilluminating to reconsider in this context the discussion in Section 24.5: in thatsituation, we have a syzygy σ = (m1, . . . , ms) ∈ ker(s) such that∑

i

miL(gi ) = 0, and for each i, mi = 0 ⇒ v(mi ) + v(gi ) = γ1

or, equivalently,∑

i mi gi ≡ 0 mod⊕

δ≺γ1Mδ and we are iteratively com-

puting solutions (h j1, . . . , h js) ∈ M such that∑i

h j i gi ≡ 0 mod⊕δ≺γ j

Mδ,

(h j1, . . . , h js) ≡ (m1, . . . , ms) mod⊕δ≺γ1

Mδ

for values γ1 γ2 · · · γ j · · · .

206 Grobner I

It is then sufficient to remember that, in this setting,⊕δ≺γ j

Mδ = (X1, . . . , Xn)γ j

in order to realize the similarity with Hensel’s lifting (Section 18.1), where asolution g1, h1

g1h1 ≡ f mod p

is iteratively lifted to a solution gn, hn such that

gnhn ≡ f mod pn, gn ≡ g1 mod p, hn ≡ h1 mod p,

and to justify the interpretation of standard bases and standard representationin the setting of valuations.

Definition 24.6.4. Let Γ be a (commutative) semigroup, totally ordered by thesemigroup ordering , and R be a ring with 1.

A valuation is a function v : R → Γ such that for each a1, a2 ∈ R \ 0,(1) v(a1a2) = v(a1) + v(a2);(2) v(a1 − a2) max(v(a1), v(a2)).21

Definition 24.6.5. Let Γ be a (commutative) semigroup, totally ordered by thesemigroup ordering ≺, R be a ring with 1 and v : R → Γ a valuation. Thenwrite

• Fγ := a ∈ R : v(a) γ ∪ 0 ⊂ R, for each γ ∈ Γ ;• Vγ := a ∈ R : v(a) ≺ γ ∪ 0 ⊂ R, for each γ ∈ Γ ;• Gγ := Fγ /Vγ , for each γ ∈ Γ ;• G :=⊕γ∈Γ Gγ ;• L : R → G is the map such that, for each a ∈ R, a = 0,L(a) denotes the

residue class of a mod Vv(a) and L(0) = 0;• finally, since for each g ∈⋃γ∈Γ Gγ , there existsa ∈ R : L(a) = g we will

define

L∗ :⋃γ∈Γ

Gγ → R

21 In the classical definition, the required property is

v(a1 − a2) ≥ min(v(a1), v(a2)).

In fact, in the classical setting (see Example 24.6.7) Γ = N with the canonical ordering <, andFn := In , so that Fn ⊃ Fν ⇐⇒ n < ν.

In order to interpret Grobner and standard bases in this setting (where the initial form isthe homogeneous component of lowest degree), we are forced to switch the classical ordering,imposing on Γ = N the ordering ≺ such that n + 1 ≺ n. As a consequence all the orderingformulas must be reversed.


to be any function for which L∗(1) = 1 and L·L∗ is the identity on⋃γ∈Γ Gγ .

Let now E be an R-module and let w : E \ 0 → Γ be a v-compatiblevaluation on E, that is a map such that, for each a ∈ R \ 0, m, m1, m2 ∈E \ 0,• w(am) = v(a) + w(m),• w(m1 − m2) max(v(m1), v(m2)).

Then we can also write

• Fγ (E) := m ∈ E : w(m) γ ∪ 0 ⊂ E, for each γ ∈ Γ ;• Vγ (E) := m ∈ E : w(m) ≺ γ ∪ 0 ⊂ E, for each γ ∈ Γ ;• Gγ (E) := Fγ (E)/Vγ (E), for each γ ∈ Γ ;• G(E) :=⊕γ∈Γ Gγ (E);• L : E → G(E) is the map such that, for each m ∈ E, m = 0,L(m) denotes

the residue class of m mod Vw(m)(E) and L(0) = 0;• since for each g ∈ ⋃

γ∈Γ Gγ , there exists m ∈ E : L(m) = g we willdenote by

L∗ :⋃γ∈Γ

Gγ (E) → E

any function such that L∗(1) = 1 and L · L∗ is the identity on⋃

γ∈Γ Gγ .

Lemma 24.6.6. With the notation above we have, for each a, a1, a2 ∈ R \ 0,m, m1, m2 ∈ E \ 0, and γ, δ ∈ Γ:

(1) Fγ ⊂ R is an additive subgroup of R;(2) δ ≺ γ ⇒ Fδ ⊂ Fγ ;(3) Fγ Fδ ⊂ Fγ+δ;(4) if a = 0, then a ∈ Fv(a) and a ∈ Fδ if δ ≺ v(a);(5) 0 =⋂γ∈Γ Fγ ;(6) G is a Γ -graded ring, the associated graded ring of R;(7) v(a1a2) = v(a1) + v(a2),L(a1a2) = L(a1)L(a2);(8) v(a1 − a2) max(v(a1), v(a2));(9) v(a1 − a2) ≺ max(v(a1), v(a2)) ⇐⇒ L(a1) = L(a2);

(10) L(a1 − a2) = L(a1) − L(a2) ⇐⇒ v(a1 − a2) = v(a1) = v(a2);(11) L(a1 − a2) = L(a1) ⇐⇒ v(a1 − a2) = v(a1) v(a2);(12) L(a) = 0 ⇐⇒ a = 0;(13) v(1) = 0,L(1) = 1;(14) Fγ (E) ⊂ E is an additive subgroup of E;(15) δ ≺ γ ⇒ Fδ(E) ⊂ Fγ (E);(16) Fγ Fδ(E) ⊂ Fγ+δ(E);

208 Grobner I

(17) if m = 0, then m ∈ Fw(m)(E) and m ∈ Fδ(E) if δ ≺ v(m);(18) 0 =⋂γ∈Γ Fγ (E);(19) G(E) is a Γ -graded G-module, the associated graded module of E;(20) w(am) = v(a) + w(m),L(am) = L(a)L(m);(21) w(m1 − m2) max(w(m1), w(m2));(22) w(m1 − m2) ≺ max(w(m1), w(m2)) ⇐⇒ L(m1) = L(m2);(23) L(m1 − m2) = L(m1) − L(m2) ⇐⇒ w(m1 − m2) = w(m1) =

w(m2);(24) L(m1 − m2) = L(m1) ⇐⇒ w(m1 − m2) = w(m1) w(m2);(25) L(m) = 0 ⇐⇒ m = 0.

Example 24.6.7. The classical example is Γ := N ordered so that d d + 1for each d, a ring R, an ideal L ⊂ R such that ∩d Ld = 0; for instance onecan consider

R := k[[X1, . . . , Xn]] and L := m := (X1, . . . , Xn), where for each f ∈ R,v( f ) is the order of f ; or

R := Z, a prime p ∈ N and L := (p), where for each ν ∈ Z, v(ν) is themaximal value such that pv(ν) | ν.

Then we have

Fd = Ld , Vd = Ld+1, Gd = Ld/Ld+1,L(a) := a mod Lv(a)+1.

When:

R = k[[X1, . . . , Xn]], L = m then

Fd = Vd−1 = f ∈ k[[X1, . . . , Xn]] : v( f ) d,Gd ∼= f ∈ k[X1, . . . , Xn], f homogeneous, deg( f ) = d,

so that G ∼= k[X1, . . . , Xn] and L( f ) is the homogeneous componentof f of lowest degree, that is Hironaka’s notion.

R = Z, L := (p) then Fd = Vd−1 = mpd , m ∈ Z, and

Gd ∼=

m ∈ Z : − p

2< m ≤ p

2

,

⊕d≺δ

Gd ∼=

m ∈ Z : − pδ

2< m ≤ pδ

2

,

Z ∼= G ∼= S :=

d∑i=0

ai Xi ∈ Z[X ] : − p

2< ai ≤ p

2, ∀i

⊂ Z[X ],

under the isomorphism evp : S → Z given by evp(h) = h(p) (seeSection 1.6.4.).


If we then consider another ideal I ⊂ R such that I = ⋂δ Lδ + I and the

R-module A := R/I , with the valuation inherited by the one in R, then wehave

Fd(A) = Vd−1(A) = (I + Ld)/I.

Now, since Ld ∩ (I + Ld+1) ⊃ Ld+1, we have

Gd(A) = I + Ld

I + Ld+1

∼= Ld

Ld ∩ (I + Ld+1)

∼= Ld

Ld+1

/ Ld ∩ (I + Ld+1)

Ld+1

=: Gd/Jd

where

Jd := Ld ∩ (I + Ld+1)

Ld+1

= L(r) : r ∈ I, v(r) = d.If we now write

J :=⊕d∈N

Jd ⊂⊕d∈N

Gd = G,

we remark that

• J is a G-module (and so an ideal),• J =⊕dL(r) : r ∈ I, v(r) = d = (L(r) : r ∈ I ) = L(I );• G(A) =⊕d Gd(A) ∼=⊕d Gd/Jd ∼= G/J = G/L(I ),

so that the ability to compute L(I ) for an ideal I ⊂ R allows us to obtain theassociated graded ring of R/I .

Let then Γ be a (commutative) semigroup, totally ordered by the semigroupordering ≺, R be a ring with 1, v : R → Γ a valuation, E be an R-module andw : E → Γ be a v-compatible valuation. Let then Fγ , Vγ , Gγ , G, L : R →G, L∗ :

⋃γ∈Γ Gγ → R, Fγ (E), Vγ (E), Gγ (E), G(E), L : E → G(E),

L∗ :⋃

γ∈Γ Gγ (E) → E be defined as in Definition 24.6.5.In order to generalize Theorem 24.4.6 to the setting of valuation rings, the

discussion of Remark 24.6.3 suggests that we consider only standard repre-sentations mod Vγ (E) for some γ , and restrict ≺ to be inf-limited (Defini-tion 24.5.2).

We therefore introduce

210 Grobner I

Definition 24.6.8. Under the notation above, let B := g1, . . . , gs ⊂ E andh ∈ E.

A representation h = ∑i hi gi : hi ∈ R, gi ∈ B is called a standard repre-

sentation in R in terms of B iff

w(h) v(hi ) + w(gi ).

A representation

h =∑

i

hi gi + h′ : hi ∈ R, gi ∈ B, h′ ∈ E

is called a truncated standard representation at γ ∈ Γ in terms of B iff

w(h) v(hi ) + w(gi ) and h′ = 0 ⇒ w(h′) ≺ γ.

An element h ∈ E is said to have a Cauchy standard representation in termsof B if, for each γ ∈ Γ , it has a truncated standard representation at γ interms of B,

which allows us to give the best version of Theorem 24.4.6 we can state in thiscontext:

Lemma 24.6.9. With the notation above, let B := g1, . . . , gs ⊂ E and h ∈E and let us recursively define the following sequences of elements in E

fn : n ∈ N, pni : n ∈ N, ∀i, 1 ≤ i ≤ s, hn : n ∈ Nas follows

• f0 := h, p0i := 0, h0 := 0,• if f j = 0 or L( f j ) ∈ L(B) then

f j+1 := f j , p j+1 i := p ji , h j+1 := h j ,

• if f j = 0 and L( f j ) ∈ L(B), and m ji ∈ R are elements such that

L( f j ) =∑

i

L(m ji )L(gi ) and w( f j ) = v(m ji ) + w(gi ), for each i,

then

f j+1 := f j −∑

i

m ji gi , p j+1 i := p ji +m ji , h j+1 := h j +∑

i

m ji gi .

Then, for each j

(1) f j = 0 ⇒ f j+1 = 0,(2) f j = 0,L( f j ) ∈ L(B) ⇒ f j+1 = f j ,(3) f j = 0,L( f j ) ∈ L(B) ⇒ w( f j+1) ≺ w( f j ) = w

(∑i m ji gi

),


(4) f j + h j = h,(5) h j ∈ (g1, . . . , gs) ⊂ E,(6) h j =∑i p ji gi is a standard representation in R in terms of B.

Proposition 24.6.10. Under the notation above, let E ⊂ E be a sub-moduleof E and B := g1, . . . , gs ⊂ E.

If Γ is inf-limited, then the following conditions are equivalent:

(1) B is a standard basis of E,(2) for each h ∈ E, h ∈ E iff it has a Cauchy standard representation in

terms of B;(3) for each h ∈ E either

• h has a Cauchy standard representation in R in terms of B, or• there is g ∈ E \ 0 : L(g) ∈ L(E) and h − g has a standard rep-

resentation in R in terms of B.

Proof.

(1) ⇒ (3) Let

fn : n ∈ N, pni : n ∈ N, ∀i, 1 ≤ i ≤ s, hi : n ∈ Nbe defined as in Lemma 24.6.9. Then

• if there exists j ∈ N such that f j = 0 then h = h j has a standardrepresentation in terms of B;

• if there exists j ∈ N such that L( f j ) ∈ L(M) then h − f j = h j hasa standard representation in R in terms of B.

• Finally, if, for each j, f j = 0,L( f j )∈L(M), writing γ j := w( f j ),the sequence

γ1 γ2 · · · γ j · · ·is a decreasing sequence in Γ so that, for each γ ∈ Γ, there isn : γn ≺ γ ; therefore h has a Cauchy standard representation interms of B, since

• h = hn + fn ,• hn has a standard representation in R and• w( fn) = γn ≺ γ .

(3) ⇒ (2): For each element h ∈ E , either

• h has a Cauchy standard representation in terms of B, or• there is g ∈ E \ 0 : L(g) ∈ L(E) and h − g has a standard

representation in R in terms of B; this implies that g ∈ E and soh ∈ E.

212 Grobner I

(2) ⇒ (1): Let m ∈ L(E); then there is h ∈ E such that L(h) = m.

Let h = ∑i hi gi + h′ be a truncated standard representation at w(h)

in R in terms of B.If we set I := i : w(h) = v(hi ) + w(gi ), then

m = L(h) =∑i∈I

L(hi )L(gi ) ∈ L(B),

thus proving B is a standard basis.

Remark 24.6.11. This reformulation of Theorem 24.4.6 is very much weaker:

• first of all, an element h ∈ E does not necessarily have a standard represen-tation in terms of a standard basis of E;

• moreover, we are unable to characterize membership of E in terms of exis-tence of a suitable ‘normal form’ g.

In order to give a statement equivalent to Theorem 24.4.6 we would need adeeper analysis (see Section 24.8) but what we have discussed so far allowsus in any case to generalize Proposition 24.5.4 to standard bases in valuationrings.

Let us begin by noting that, if we impose on both Rs and Gs the valuationw defined by

w(ei ) = ωi := w(gi ) = w(L(gi )), for each i

where e1, . . . , es denotes the canonical basis of both Rs and Gs , one natu-rally obtains that G(Rs) ∼= Gs .

Under this identification, for each h := (h1, . . . , hs) ∈ Rs , we have

w(h) = max(w(hi ) + ωi ),L(h) = (m1, . . . , ms)

where

mi =L(hi ) if w(h) = w(hi ) + ωi ,

0 otherwise

and for each homogeneous element σ := (m1, . . . , ms) ∈ Gs , L∗(σ ) denotesany element h := (h1, . . . , hs) ∈ Rs : L(h) = σ.

Then we have just to consider the morphisms

s : Gs → G : s(m1, . . . , ms) :=∑

i

miL(gi ),

and

S : Rs → R : S(h1, . . . , hs) :=∑

i

hi gi =: h

in order to state


Lemma 24.6.12. With the same notation as Proposition 24.6.10, let U ⊂ Rs

be such that L(u) : u ∈ U is a homogeneous basis of the module ker(s) of thesyzygies among L(g1), . . . ,L(gs), and assume that, for each u ∈ U, S(u)

has a Cauchy standard representation.Let h ∈ E and γ ∈ Γ , γ ≺ w(h).If there is a representation

h =∑

i

hi gi + h′, hi ∈ R, gi ∈ B, h′ ∈ E,

such that

• w(h) ≺ γ1 := maxv(hi ) + w(gi ) : 1 ≤ i ≤ s and• h′ = 0 ⇒ w(h′) ≺ γ ,

then there is a different representation

h =∑

i

hi gi + h′, hi ∈ R, gi ∈ B, h′ ∈ E,

such that

• w(h) γ2 := maxv(hi ) + w(gi ) : 1 ≤ i ≤ s ≺ γ1 and• h′ = 0 ⇒ w(h′) ≺ γ.

Proof. Let h = ∑i hi gi + h′, hi ∈ R, gi ∈ B, h′ ∈ E , be a representation

such that

w(h) ≺ γ1 := maxv(hi ) + w(gi ) : 1 ≤ i ≤ s and h′ = 0 ⇒ w(h′) ≺ γ.

Setting

J := i : v(hi ) + w(gi ) = γ1, 1 ≤ i ≤ s,w(h) ≺ γ1 implies

∑i∈J L(hi )L(gi ) = 0 so that

∑i∈J L(hi )ei ∈ ker(s) and

there are nι ∈ R, uι ∈ U, 1 ≤ ι ≤ r such that

∑i∈J

L(hi )ei =r∑

ι=1

L(nι)L(uι) and v(nι) + w(uι) = γ1.

Then, denoting by hi , 1 ≤ i ≤ s, the elements such that

∑hi ei =

∑i

hi ei −r∑

ι=1

nιuι,

since∑

i∈J L(hi )ei − ∑rι=1 L(nι)L(uι) = 0, we know that, for each i ,

w(hi ) + v(gi ) ≺ γ1.

214 Grobner I

Therefore we have

h − h′ =∑

i

hi gi

= S

(∑i

hi ei

)

= S

(∑hi ei +

r∑ι=1

nιuι

)

= S(∑

hi ei

)+

r∑ι=1

nιS(uι)

=∑

hi gi +r∑

ι=1

nιS(uι).

Since

w(hi ) + v(gi ) ≺ γ1, v(nι) + w(S(uι)) ≺ v(nι) + w(uι) = γ1,

and, by assumption, each S(uι) has a truncated standard representation at γ ,we obtain a truncated standard representation at γ

h =s∑

i=1

hi gi + h′,

such that

v(hi ) + w(gi ) ≺ γ1 and h′ = 0 ⇒ w(h′) ≺ γ.

Proposition 24.6.13. With the same notation as Proposition 24.6.10, assumethat Γ is inf-limited and B generates E.

Let U ⊂ Rs be such that L(u) : u ∈ U is a homogeneous basis of themodule ker(s) of the syzygies among L(g1), . . . ,L(gs).

Then B is a standard basis of E iff for each u ∈ U, S(u) has a Cauchystandard representation in terms of B.

Proof. We intend to prove that each element h ∈ E has a Cauchy standardrepresentation in terms of B. We therefore fix any γ ∈ Γ.

Since obviously h can be represented as

h =∑

i

hi gi + h′, hi ∈ R, gi ∈ B, h′ ∈ E,

such that

w(h) γ1 := maxv(hi ) + w(gi ) : 1 ≤ i ≤ s, h′ = 0 ⇒ w(h′) ≺ γ,


the claim follows by Lemma 24.6.12 and by the assumption that ≺ is inf-limited so that in any decreasing sequence

γ1 γ2 · · · γ j · · ·there is n such that γn w(h), implying that in a finite number of iterationswe will produce the required representation

h =∑

i

hi gi + h′, hi ∈ R, gi ∈ B, h′ ∈ E,

such that

w(h) = maxv(hi ) + w(gi ) : 1 ≤ i ≤ s, h′ = 0 ⇒ w(h′) < γ.

Remark 24.6.14. The relation (see Remark 24.5.5) between Grobner and stan-dard bases can be generalized in this setting to k[[X1, . . . , Xn]].

Let R := k[[X1, . . . , Xn]] and let

T := Xa11 . . . Xan

n , (a1, . . . , an) ∈ Nn.

We impose on R a valuation ring structure by assigning a weight vector 22

w := (w1, . . . , wn) ∈ Rn, wi ≤ 0,

and impose on R the weight function satisfying, for each i, vw(Xi ) := wi ,

so that vw(Xa11 . . . Xan

n ) = ∑i wi ai ; then Rγ is the vectorspace spanned by

t ∈ T : vw(t) = γ .Let Rm be a free-module whose canonical basis is denoted by e1, . . . , em

and its vectorspace basis is T (m) = tei , t ∈ T , 1 ≤ i ≤ m, and let usimpose a v-valuation structure on it by choosing a vector d = (d1, . . . , dm) ∈R

m , imposing on each ei the valuation di and setting, for each tei ∈ T (m),ww,d(tei ) = di + vw(t).

We also denote by Lw : R → G, and Lw,d : Rm → Gm the correspondingleading-form maps.

Let us now consider on R any term ordering < and on Rm any term ordering(which we will still denote by <) compatible with < and let us define the termorderings ≺ on both R and Rm by

t1 ≺ t2 ⇐⇒

vw(t1) < v(t2) orvw(t1) = vw(t2) and t1 < t2,

22 Note that in Hironaka’s theory of standard bases in k[[X1, . . . , Xn ]] one must have Xi < 1,unlike in Grobner theory in k[X1, . . . , Xn ] where Xi > 1.

216 Grobner I

and

t1ei ≺ t2e j ⇐⇒

ww,d(t1ei ) < ww,d(t2e j ) orww,d(t1ei ) = ww,d(t2e j ) and t1ei < t2e j .

In this setting we have

Corollary 24.6.15. With the notation above, let f ∈ Rm and M ⊂ Rm be asubmodule; then

• T≺( f ) = T≺(Lw,d( f )) = T<(Lw,d( f ));• T≺(M) = T≺(Lw,d(M)) = T<(Lw,d(M));• if G is a standard basis of M w.r.t. ≺, then it is a standard basis of M and

Lw,d(g) : g ∈ G is a standard basis of M w.r.t. ≺ and <.

Let us now specialize our setting, considering

Γ := T := Xa11 . . . Xan

n : (a1, . . . , an) ∈ Nn totally ordered by any inf-

limited and Noetherian ordering 23 ≺,R := k[[X1, . . . , Xn]],v : k[[X1, . . . , Xn]] → T the valuation which associates to each series f =∑

t∈T c( f, t)t the value 24

v( f ) := max≺ t ∈ T : c( f, t) = 0so that G = k[X1, . . . , Xn] and L( f ) = c( f, v( f ))v( f ),

an ideal I ⊂ k[[X1, . . . , Xn]], anda standard basis B := g1, . . . , gs ⊂ I of I, so that L(I) = L(B);

let us also wlog assume lc(gi ) = 1 for each i and let us write Ti := L(gi ) foreach i . Moreover let us write:

T1(B) := ∅,

Ti (B) := t ∈ T : tT(i) ∈ (T(1), . . . , T(i − 1))

,

Ni (B) := t ∈ T : tT(i) ∈ (T(1), . . . , T(i − 1))

= T \ Ti (B),

Li (B) := tT(i) : t ∈ Ni (B),N≺(I) := T \ L(I),

23 Alternatively, one can consider any term ordering < such that, for each γ ∈Γ and each increas-ing sequence

γ1 < γ2 < · · · < γ j < · · · ,there is n such that γn ≥ γ , and define ≺ by τ ≺ ω ⇐⇒ τ > ω. In this interpretationv( f ) = min<t ∈ T : c( f, t) = 0.

24 Which exists by definition since ≺ is Noetherian.


so that, mutatis mutandis, Lemma 22.3.2 is satisfied. Then Lemma 22.2.12 canbe generalized in this setting as

Theorem 24.6.16 (Hironaka). For any series h ∈ k[[X1, . . . , Xn]] there are(not necessarily unique) series

p1, . . . , ps, pi =∑

t∈Ni (B)

c(pi , t)t ∈ k[[Ni (B)]] ⊂ k[[X1, . . . , Xn]]

and a unique canonical form

q := Can(h, I, ≺) =∑

t∈N≺(I)

γ (h, t, ≺)t ∈ k[[N≺(I)]] ⊂ k[[X1, . . . , Xn]]

such that

h = q +∑

i

pi gi , v(h − q) v(pi ) + v(gi )

Proof. Uniqueness of q is obvious:

q ′ +∑

i

p′i gi = h = q ′′ +

∑i

p′′i gi ⇒ q ′ − q ′′ ∈ I ∩ k[[N≺(I)]] = 0.

Let us recursively define the following sequences

fn : n ∈ N ⊂ k[[X1, . . . , Xn]],pni : n ∈ N ⊂ k[Ni (B)], for each i, 1 ≤ i ≤ s,qn : n ∈ N ⊂ k[[N≺(I)]]

as follows (see Lemma 24.6.9)

• f0 := h, p0i := 0, q0 := 0;• if f j = 0 then f j+1 := f j , p j+1 i := p ji , q j+1 := q j ;• if f j = 0 and L( f j ) ∈ N≺(I) then

f j+1 := f j − lc( f j )L( f j ), p j+1 i := p ji , q j+1 := q j + lc( f j )L( f j );• if f j = 0 and L( f j ) ∈ L(B), and l, 1 ≤ l ≤ s, t ∈ Nl(B) are the unique

values such that L( f j ) = tT(l) ∈ Ll(B), then:

f j+1 := f j − lc( f j )tgl , p j+1 l := p jl + lc( f j )t,

p j+1 i := p ji for i = l, q j+1 := q j .

Then, for each j

f j = 0 ⇒ f j+1 = 0,

f j = 0 ⇒ v( f j+1) ≺ v( f j ),

q j ∈ k[N≺(I)], p ji ∈ k[Ni (B)], for each i, 1 ≤ i ≤ s,h = f j +∑i p ji gi + q j ,

218 Grobner I

h − f j − q j = ∑i p ji gi is a standard representation in k[[X1, . . . , Xn]] in

terms of B.

If, for some j , f j = 0 we obtain the required standard representation

h − q j =∑

i

p ji gi ∈ I, q j ∈ k[N≺(I)].

Otherwise, since ≺ is inf-limited, the infinite decreasing sequence

v( f0) v( f1) · · · v( f j ) v( f j+1) · · ·is such that for each τ ∈ T there exists j such that v( f j ) ≺ τ . Therefore,limn→∞ fn = 0 and writing

q := limn→∞ qn, pi := lim

n→∞ pni for each i

we have the required standard representation h − q =∑i pi gi ∈ I.

Historical Remark 24.6.17. This result by Hironaka, which is dated 1964, sub-sumes Buchberger’s result on canonical forms, which is obtained by restrictingR to R := k[X1, . . . , Xn] and the inf-limited ordering ≺ relaxed to the termordering case.25 However, while Buchberger’s result allows us to compute astandard (Grobner) basis effectively, in Hironaka’s theory there is no compu-tational approach in order to deduce a standard basis from a given basis; asolution, in the restricted case in which the given basis consists of polynomialsonly, was only proposed in 1981 and explicitly mimicked Buchberger’s algo-rithm; as far as I know, computing a standard basis of an ideal generated by agiven set of series is still an open problem.

24.7 *Standard Bases and Quotient Rings

Let us consider

a (commutative) semigroup Γ totally ordered by the inf-limited semigroupordering ≺,

a ring R with 1,a valuation v : R → Γ ,an ideal I ⊂ R,the quotient rings A := R/I and G/L(I ),and the projections π : R → A, Π : G → G/L(I ).

25 The restriction to R := k[X1, . . . , Xn ] makes it useless to require Noetherianity in order todefine

v( f ) := max≺ t ∈ T : c( f, t) = 0.

24.7 *Standard Bases and Quotient Rings 219

Lemma 24.7.1. The following conditions are equivalent

(1) for each a ∈ A \ 0 the set v(r) : r ∈ R, π(r) = a has a <-minimalvalue which we will denote by v′(a),

(2) I =⋂Γ I + Fγ .

Proof.

(1) ⇒ (2) Since I ⊂ ⋂Γ I + Fγ we have just to prove the converse in-

clusion. Let us therefore assume the existence of an element r ′ ∈⋂Γ I + Fγ such that r ′ /∈ I ; then for each γ ∈ Γ there are r” ∈

I, ρ ∈ Fγ such that r ′ = ρ + r” so that π(ρ) = π(r ′) and we obtainthe contradiction

min<

v(r) : r ∈ R, π(r) = π(r ′) ≤ v(ρ) ≤ γ.

(2) ⇒ (1) Let a ∈ A\0 and let us fix any element ρ ∈ R such that π(ρ) =a. If v(r) : r ∈ R, π(r) = a = π(ρ) has no <-minimal value, thenfor each γ ∈ Γ there is rγ ∈ R such that π(rγ ) = π(ρ), v(rγ ) = γ

so that ρ − rγ ∈ I , rγ ∈ Fγ and ρ ∈ I + Fγ . Therefore ρ ∈ I , a = 0,and we get the required contradiction.

Lemma 24.7.2. With the notation above, and under the assumption that 26

I =⋂Γ I + Fγ the following hold:

• v′ : A → Γ is a valuation;• Π(L(r1)) = Π(L(r2)) =: L′(a) holds for each a ∈ A \ 0 and r1, r2 ∈ R

such that π(r1) = π(r2) = a and v(r1) = v(r2) = v′(a);• a ∈ A \ 0 : v′(a) = γ = π(Fγ (R)) ∼= (Fγ (R) + I )/I for each γ ∈ Γ ;• G(A) ∼= G/L(I );• Π(L(r)) = L(π(r)) holds for each r ∈ R such that L(r) /∈ L(I );• for each r ∈ R, r /∈ I , there is r ′ := NF(r, I ) such that r − r ′ ∈ I,L(r ′) /∈L(I ).

Proof. All the statements are trivial except the last one, which follows fromthe existence of standard bases B of I : if r /∈ I = ⋂

Γ I + Fγ then r doesnot have a Cauchy standard representation in R in terms of B so that (seeProposition 24.6.10) there is r ′ such that r − r ′ ∈ I, and L(r ′) /∈ L(I ).

Proposition 24.7.3. With the same notation as above, and under the assump-tion that I = ⋂

Γ I + Fγ , for any ideal J ⊂ A, writing J ′ := π−1(J ) ⊂ R,

26 This assumption is trivially satisfied if < is a well-ordering.For the meaning of this assumption in the general case see Corollary 24.8.10.

220 Grobner I

the following hold: 27

(1) If B = g1, . . . , gs is a standard basis of J ′, then

π(g) : g ∈ B,L(g) /∈ L(I )is a standard basis of J .

(2) If each r ∈ J ′ has a standard representation in terms of B =g1, . . . , gs, then each a ∈ J has a standard representation in termsof

π(g) : g ∈ B, g /∈ I .(3) If each r ∈ J ′ has a standard representation in terms of B =

g1, . . . , gs, then each a ∈ J has a standard representation in termsof

π(NF(g, I )) : g ∈ B, g /∈ I .(4) If C = f1, . . . , fu is a standard basis of I and D = g1, . . . , gs ⊂

J ′ is a set such that

• for each g ∈ D,

• π(g) = 0,

• v(g) = v′(π(g)), so that• Π(L(g)) = L(π(g)),

• and π(g) : g ∈ D is a standard basis of J ,

then C ∪ D is a standard basis of J ′ in R.(5) If each r ∈ I has a standard representation in terms of C =

f1, . . . , fu, and D = g1, . . . , gs ⊂ J ′ is a set such that

• for each g ∈ D,

• π(g) = 0,• v(g) = v′(π(g)), so that• Π(L(g)) = L(π(g)),

• and each a ∈ J has a standard representation in terms of π(g) :g ∈ D

then each r ∈ J ′ has a standard representation in terms of C ∪ D.

27 As pointed out in Remark 24.6.11, in general the notions of

• being a standard basis of an ideal,• giving a standard representation of a member of an ideal,• returning a normal form of a member of an ideal,

do not coincide.

24.7 *Standard Bases and Quotient Rings 221

Proof.

(1) Let a ∈ J, a = 0 and let r ∈ J ′ be such that π(r) = a and v(r) = v′(a)

so that Π(L(r)) = L(a) and let hi ∈ R be elements such that

L(r) =s∑

i=1

L(hi )L(gi ) and v(r) = v(hi ) + v(gi ).

Since L(gi ) ∈ L(I ) implies Π(L(gi )) = 0, setting L := i : L(gi ) /∈L(I ) we have

L(a) = Π(L(r)) =s∑

i=1

Π(L(hi ))Π(L(gi )) =∑i∈L

Π(L(hi ))Π(L(gi )).

(2) Let a ∈ J, a = 0 and let r ∈ J ′ be such that π(r) = a and v(r) = v′(a)

so that Π(L(r)) = L(a) and let hi ∈ R be elements such that

r =s∑

i=1

hi gi and v(r) ≥ v(hi ) + v(gi ).

Then, since π(gi ) = 0 for each gi ∈ I , setting L := i : gi /∈ I ) wehave

a = π(r) =s∑

i=1

π(hi )π(gi ) =∑i∈L

π(hi )π(gi )

and

v′(a) = v(r) ≥ v(hi ) + v(gi ) ≥ v′(π(hi )) + v′(π(gi )).

(3) π(g) = π(NF(g, I )) for each g.(4) Let r ∈ J ′.

If L(r) ∈ L(I ) then there are h j ∈ R such that

L(r) =u∑

j=1

L(hj)L( f j ) and v(r) ≥ v(h j ) + v( f j ).

Otherwise, r /∈ I and Π(L(r)) = L(π(r)); therefore there are pi ∈ Rsuch that

Π(L(r)) = L(π(r))

=s∑

i=1

Π(L(pi ))L(π(gi ))

=s∑

i=1

Π(L(pi ))Π(L(gi ))

= Π

(s∑

i=1

L(pi )L(gi )

)

222 Grobner I

and v(r) = v(pi ) + v(gi ), for each i , so that

L(r) −s∑

i=1

L(pi ))L(gi ) ∈ L(I )

and there are h j ∈ R such that

L(r) −s∑

i=1

L(pi )L(gi ) =u∑

j=1

L(h j )L( f j )

and v(r) = v(h j ) + v( f j ), for each i .(5) Let r ∈ J ′; then there are pi ∈ R such that

π(r) =∑

i

π(pi )π(gi ) and v(r) ≥ v(pi ) + v(gi ).

Then r ′ := r −∑i pi gi ∈ I and there are hi ∈ R such that

r ′ =∑

i

hi fi and v(r ′) ≥ v(hi ) + v( fi )

so that r =∑i pi gi +∑i hi fi is the required standard representation.

Remark 24.7.4 (Logar). With the same notation as above, also denoting, witha slight abuse of notation, by π each canonical projection π : Rt → At andidentifying as e1, . . . , et the canonical basis of both Rt and At , let( f1, . . . , ft ) be a standard basis of I and let J ⊂ At be the module gen-erated by G: = π(g1), . . . , π(gs) where g1, . . . , gs ⊂ Rt , π−1(J ) =(g1, . . . , gs) + I t .

Writing, in connection with the Lifting Theorem 23.7.13,

G ′ := g1, . . . , gs ∪ fi e j : 1 ≤ i ≤ r, 1 ≤ j ≤ t,and

S′ :=

(h1, . . . , hs, h′11, . . . , h′

r t ) :s∑

i=1

hi gi +r∑

i=1

t∑j=1

fi e j = 0

= Syz(G ′) ⊂ Rs+r t ,

Σ ′1, . . . Σ

′t a basis of S′ and χ : Rs+r t → At the projection defined by

χ(h1, . . . , hs, h′11, . . . , h′

r t ) = (π(h1), . . . , π(hs))

for each (h1, . . . , hs, h′11, . . . , h′

r t ) ∈ Rs+r t , then

χ(Σ ′1), . . . , χ(Σ ′

t )

24.8 *Standard Bases in Valuation Rings 223

is a basis of

Syz(G) =

(h′1, . . . , h′

s) :s∑

i=1

h′iπ(gi ) = 0

⊂ As .

This allows us to adapt the algorithms discussed in Section 23.8 comput-ing resolutions of k[X1, . . . , Xn]-modules in order to obtain resolutions of A-modules of a quotient ring A := k[X1, . . . , Xn]/I .

24.8 *Characterization of Standard Bases in Valuation Rings

The characterization of a standard basis given by Proposition 24.6.10 is notin terms of standard representations but only in terms of Cauchy ones, whichare essentially the truncations of a standard representation modulo Vγ (E), foreach γ ∈ Γ .

This is already sufficient for us to investigate

(1) whether other elements h ∈ E \ E have such a representation;(2) what happens if we take the ‘limit’ of such representation and, in par-

ticular, what kind of representation would we obtain.

The answers to these questions are trivial: the solutions, as the reader probablyguessed, are:

(1) each element in⋂

Γ E + Fγ (E) has a Cauchy standard representation;(2) such a representation

h =∑

h(i)gi , v(h(i))w(gi ) ≤ v(h),

at least when R := G, E := G(E), will be a standard representation interms of ‘series’ elements

h(i) =∞∑j=1

h(i)γ j , h(i)γ j ∈ Gγ j , homogeneous v(h(i)γ j ) = γ j ;

and, under the natural assumption that Γ is inf-limited, the answer inthe general case will be essentially the same once we have set the ap-propriate notation.

Clearly if Γ is well-ordered, the effect of this ‘limit’ operation on a (necessarilyfinite) sequence gives a representation in terms of finite sums of homogeneouscomponents

h(i) =µ∑

j=1

h(i)γ j , h(i)γ j ∈ Gγ j ,

224 Grobner I

in other words, just a standard representation. To obtain the general result wecan therefore assume that Γ is just inf-limited and we will fix a specific infinitedecreasing sequence

λ1 λ2 · · · λn · · ·to which we will repeatedly make reference in this section.

Let us begin by discussing the first question:

Lemma 24.8.1. With the same notation as Proposition 24.6.10, and assumingthat Γ is inf-limited, write Cl(E) :=⋂Γ E + Fγ (E). Then:

(1) Cl(E) is an R-module;(2) if h ∈ E has a Cauchy standard representation in terms of B, then

h ∈ Cl(E);(3) B is a standard basis of E iff each h ∈ Cl(E) \ 0 has a Cauchy stan-

dard representation in terms of B.

Proof.

(1) For h ∈ Cl(E) and r ∈ R, we need to prove that rh ∈ E + Fγ (E) foreach γ ∈ Γ.

So let us fix γ ∈ Γ and let us take λn : λn +v(r) ≺ γ . Since h ∈ Cl(E),there exist f1 ∈ E, f2 ∈ Fλn (E) : h = f1 + f2; therefore

r f1 ∈ E, r f2 ∈ Fγ (E), r f = r f1 + r f2 ∈ E + Fγ (E).

(2) By assumption, for each γ ∈ Γ there is a representation

h =∑

hi gi + h′, v(hi ) + w(gi ) w(h), w(h′) ≺ γ,

so that h = g + h′ with g :=∑ hi gi ∈ E and h′ ∈ Fγ (E).

(3) Let h ∈ Cl(E) \ 0 and γ ∈ Γ . By assumption

h = f1 + f2, f1 ∈ E, f2 ∈ Fγ (E),

and f1 has a truncated standard representation f1 := ∑hi gi + f ′ at

γ , from which we obtain the required truncated standard representationf :=∑ hi gi + ( f ′ + f2) at γ .

The proof of the claim on the structure of standard representations requiresa deeper analysis of the topology imposed on R and E by their filtrations, inorder to allow us to mimic the Cauchy construction of R as the completion ofQ. Therefore we must begin by proving that the filtration sets Fγ : γ ∈ Γ and Fγ (E) : γ ∈ Γ , imposed on the ring R and on the R-module E by thevaluations v and w, are a basis of the neighbourhood of 0, so imposing on


them also a topology under which the R-module operations 28 are continuous.In other words we have to prove that:

Lemma 24.8.2. For each γ ∈ Γ,

(1) there exist γ ′, γ ′′ ∈ Γ such that for each f ∈ Fγ ′(E), g ∈Fγ ′′(E), f + g ∈ Fγ (E) holds;

(2) there exist γ ′, γ ′′ ∈ Γ such that for each f ∈ Fγ ′ , g ∈ Fγ ′′(E), f g ∈Fγ (E) holds.

Proof.

(1) It is sufficient to take γ ′ := γ ′′ := γ ;(2) fix an arbitary γ ′′ ∈ Γ and consider the infinite decreasing sequence

λ′1 λ′

2 · · · λ′n · · · ,

where λ′n := γ ′′ + λn for each n, and define γ ′ := λn where n is any

element such that λ′n γ,

and to recall that:

Lemma 24.8.3. We have⋂

γ∈Γ Fγ (E) = 0.Proof. For each m ∈ E \ 0, exists n : λn ≺ γ := v(m), implying m ∈Fλn (E),

in order to mimic the Cauchy construction by introducing

Definition 24.8.4. A sequence (an), an ∈ E, n ∈ N, is called a Cauchy se-quence in E if

∀γ ∈ Γ, ∃n ∈ N : ap − aq ∈ Fγ (E), ∀p, q > n.

A Cauchy sequence (an) in E is called a null sequence if

∀γ ∈ Γ, ∃n ∈ N : ap ∈ Fγ (E), ∀p > n,

and proving that

Lemma 24.8.5. For each Cauchy sequences (mn) in E, and each γ ∈ Γ , thereis n ∈ N such that either

• w(m p) γ for each p > n, or• w(m p) = w(mq) γ,L(m p) = L(mq), for each p, q > n.

28 There is no need to consider separately the ring R and the R-modules E , since it is sufficient toconsider R as an R module itself, as we do throughout this discussion.

226 Grobner I

Proof. We know that there exists n ∈ N such that w(m p − mq) γ for eachp, q > n; therefore, if there is p > n such that w(m p) γ , then, for eachq > n, w(m p) = w(mq) and L(m p) = L(mq).

Theorem 24.8.6.

(1) For each Cauchy sequence (mn) in E, there are γ ∈ Γ and n ∈ N suchthat w(m p) γ, for each p > n.

(2) The set C(E) of all Cauchy sequences in E is an R-module under theoperations

(mn) + (µn) := (mn + µn), (mn), (µn) ∈ C(E),

a(mn) := (amn) (mn) ∈ C(E), a ∈ R.

(3) The set C(R) of all Cauchy sequences in R is a ring under the operation

(an) · (bn) := (an · bn), (an), (bn) ∈ C(R).

(4) The set C(E) of all Cauchy sequences in E is a C(R)-module under theoperation

(an) · (mn) := (an · mn) (an) ∈ C(R), (mn) ∈ C(E).

(5) The set N(E) of all null sequences in E is a C(R)-module.(6) R := C(R)/N(R) is a ring.(7) Let φ : R → R be the map which associates, to each a ∈ R, the residue

class mod N(R) of the Cauchy sequence (an) where an = a for eachn.

Then φ is an immersion.(8) N(R) · C(E) ⊂ N(E).(9) E := C(E)/N(E) is an R-module.

(10) Let φ : E → E be the map which associates, to each m ∈ E, theresidue class mod N(E) of the Cauchy sequence (mn) where mn = mfor each n.

Then φ is an immersion.

Proof.

(1) Let us fix δ ∈ Γ so that there exists n such that either

• w(m p) δ for each p > n and the claim holds for γ := δ, or• w(m p) = w(mq) := γ for each p, q > n and the claim trivially

holds.


(2) Let us fix (mn), (µn) ∈ C(E), a ∈ R and γ ∈ Γ .We know that there exists n such that, for each p, q > n, w(m p −mq) γ, w(µp − µq) γ implying w

((m p − µp) − (mq − µq)

) γ .We know also that there exists ν : λν γ − v(a) and n such thatw(m p − mq) λν for each p, q > n, so that

w(am p − amq) = v(a) + w(m p − mq) v(a) + λν γ.

(3) Let us fix (an), (bn) ∈ C(R) and γ ∈ Γ . We know that there are:

• δ1 ∈ Γ , n1 ∈ N : v(ap) δ1, for each p > n1;• γ1 ∈ Γ : δ1 + γ1 ≺ γ ;• n2 ∈ N : v(bp − bq) γ1, for each p, q > n2;• δ2 ∈ Γ , n3 ∈ N : v(bp) δ2, for each p > n3;• γ2 ∈ Γ : γ2 + δ2 γ ;• n4 ∈ N : v(ap − aq) γ2, for each p, q > n4;so that, for each p, q > n := max(n1, n2, n3, n4) :

v(apbp − aqbq) = v(ap(bp − bq) − (ap − aq)bq

) max(v(ap) + v(bp − bq), v(ap − aq) + v(bq))

max(δ1 + γ1, γ2 + δ2)

γ.

(4) Let us fix (an) ∈ C(R), (mn) ∈ C(E) and γ ∈ Γ . We know that thereare:

• δ1 ∈ Γ , n1 ∈ N : v(ap) δ1, for each p > n1;• γ1 ∈ Γ : δ1 + γ1 γ ;• n2 ∈ N : v(m p − mq) γ1, for each p, q > n2;• δ2 ∈ Γ , n3 ∈ N : w(m p) δ2, for each p > n3;• γ2 ∈ Γ : γ2 + δ2 γ ;• n4 ∈ N : w(ap − aq) γ2, for each p, q > n4;so that, for each p, q > n := max(n1, n2, n3, n4) :

w(apm p − aqmq) = w(ap(m p − mq) − (ap − aq)mq

) max(v(ap)+w(m p −mq), v(ap −aq)+w(mq))

max(δ1 + γ1, γ2 + δ2)

γ.

(5) We have to prove that for (mn), (µn) ∈ N(E), (an) ∈ C(R),

(mn) + (µn), (an) · (mn) ∈ N(E).

228 Grobner I

Let us fix γ ∈ Γ . Then exists n : w(m p) γ, w(µp) γ , for allp > n, implying v(m p − µp) γ, for all p > n.

Also there are:

• δ ∈ Γ , n1 ∈ N : v(ap) δ, for each p > n1;• γ1 ∈ Γ : δ + γ1 γ ;• n2 ∈ N : w(m p) γ1, for each p > n2,

implying that, for each p > n := max(n1, n2)

w(apm p) = v(ap) + w(m p) δ + γ1 γ.

(6) N(R) is an ideal.(7) We have to prove that for each a ∈ R \ 0, the Cauchy sequence (an)

defined by an = a, for each n, is not in N(R). This is a consequenceof⋂

γ∈Γ Fγ = 0.(8) Let (mn) ∈ C(E), (an) ∈ N(R), and γ ∈ Γ . Then there are:

• δ ∈ Γ , n1 ∈ N : w(m p) δ, for each p > n1;• γ1 ∈ Γ : γ1 + δ γ ;• n2 ∈ N : v(ap) γ1, for each p > n2,

implying that, for each p > n := max(n1, n2)

w(apm p) = v(ap) + w(m p) γ1 + δ γ.

(9) As an obvious consequence of the previous statement.(10) Since

⋂γ∈Γ Fγ (E) = 0.

Lemma 24.8.7. With the notation of Theorem 24.8.6, let m ∈ E , m = 0, andlet (mn), (µn) be two Cauchy sequences in E which converge 29 to m. Thenthere exists N ∈ N such that, for each p, q > N,

w(m p) = w(µq) =: wˆ(m), L(m p) = L(µq) =: Lˆ(m).

Proof. For each λn there exists d(n) ∈ N such that either

• w(m p) λn for each p > d(n), or• w(m p) = w(mq) λn,L(m p) = L(mq), for each p, q > d(n).

Since

w(m p) λn for each n ∈ N, p > d(n) ⇒ (mn) ∈ N(E) ⇒ m = 0

giving a contradiction, therefore there is n ∈ N such that, for each p, q > d(n),

w(m p) = w(mq) =: wˆ(m), L(m p) = L(mq) =: Lˆ(m).

29 In the sense that they belong to the residue class module N(E) represented by m.


Also (mn − µn) converges to 0 so that exists N ∈ N, N > d(n), such that,for each q > N , w(mq − µq) ≺ wˆ(m) = w(mq), whence

w(µq) = w(mq) = wˆ(m), L(µq) = L(mq) = Lˆ(m).

Corollary 24.8.8. Defining

• wˆ : E → Γ , so that for each m ∈ E, wˆ(m) is the value defined inLemma 24.8.7,

• vˆ : R → Γ , to be wˆ for the module R,• Lˆ : E → G(E), so that for each m ∈ E,Lˆ(m) is the value defined in

Lemma 24.8.7,

then:

(1) vˆ is a valuation on R which extends the valuation v in R;(2) wˆ is a vˆ-compatible valuation on E, which extends the valuation w in

E;(3) Lˆ extends L;(4) vˆ, wˆ, G, G( · ),Lˆ satisfy Lemma 24.6.6;(5) in particular G(R) = G(R) = G, G(E) = G(E);(6) and, for each s, G(Rs) ∼= G(Rs) ∼= Gs .

In this context we can reinterpret Lemma 24.8.1 as

Lemma 24.8.9. E ∩ E = Cl(E) =⋂Γ E + Fγ (E).

Proof. If h ∈ E ∩ E, there is a Cauchy sequence (gn) in E such that, for eachγ ∈ Γ, there exists n ∈ N for which we have

w(h − gn) ≺ γ and h ∈ E + Fγ (E).

On the other hand if h ∈ Cl(E) ⊂ E we know that, for each n ∈ N, thereexist gn ∈ E, hn ∈ Fλn (E) satisfying h = gn + hn .

Since gp − gq = hq − h p for each p, q ∈ N, then, for each γ ∈ Γ , thereexists n ∈ N such that, for each p, q > n,

γ λn w(hq − h p) = w(gp − gq), and γ λn w(h p) = w(h − gp),

so that (gn) is a Cauchy sequence, (h − gn) is a null sequence, (gn) convergesto h, whence h ∈ E,

and we obtain

Corollary 24.8.10. With the same notation as in Lemma 24.7.1, for E = Iand E = R the following conditions are equivalent:

230 Grobner I

(1) A has the valuation v′ : A → Γ defined by

v′(a) := min<

v(r) : r ∈ R, π(r) = a for each a ∈ A \ 0;(2) I =⋂Γ I + Fγ = Cl(I );(3) A ∼= R/ I .

Proof. By Lemma 24.7.1 R/ I ∼= R/ Cl(I ). Let us then consider the projectionσ : R R/ Cl(I ). Then, for any r ∈ R,

r ∈ ker(σ ) ⇐⇒ r ∈ I ∩ R ⇐⇒ r ∈ Cl(I ),

so that

A ∼= R/ I ⇐⇒ ker(σ ) = I ⇐⇒ I = Cl(I ).

We are now able to reinterpret Lemma 24.6.9 as

Lemma 24.8.11. Let Γ be a (commutative) semigroup, inf-limited by thesemigroup ordering ≺, R a ring with 1, v : R → Γ a valuation, E an R-module, w : E → Γ a v-compatible valuation, E ⊂ E a sub-module of E andB := g1, . . . , gs ⊂ E.

With the notation introduced in this and in the previous sections, let us con-sider an element h ∈ E and let us recursively define the following sequences:

fn : n ∈ N ⊂ E, rni : n ∈ N ⊂ R, ∀i, 1 ≤ i ≤ s, hn : n ∈ N ⊂ E,

as follows

• f0 := h, r0i := 0, h0 := 0;• if f j = 0 or L( f j ) ∈ L(B) then

f j+1 := f j , r j+1 i := r ji , h j+1 := h j ;

• if f j = 0 and L( f j ) ∈ L(B), and m ji ∈ R are elements such that

L( f j ) =∑

i

L(m ji )L(gi ), and w( fi ) = v(m ji ) + w(gi ), for each i ,

then

f j+1 := f j −∑

i

m ji gi , r j+1 i := r ji +m ji , h j+1 := h j +∑

i

m ji gi .

Then, for each j:

(1) f j = 0 ⇒ f j+1 = 0;(2) f j = 0,L( f j ) ∈ L(B) ⇒ f j+1 = f j ;(3) f j = 0,L( f j ) ∈ L(B) ⇒ w( f j+1) ≺ w( f j ) = w

(∑i m ji gi

);


(4) f j + h j = h;(5) h j ∈ (g1, . . . , gs) ⊂ E;(6) h j =∑i r ji gi is a standard representation in R in terms of B;(7) if h ∈ E then, for each n, fn ∈ E .

Corollary 24.8.12. With assumptions and notations as in Lemma 24.8.11, ifmoreover, for each n, fn+1 = fn = 0 then, writing γn := w( fn) we have

(1) the sequence γ0 γ1 · · · γ · · · is an infinite decreasing sequ-ence,

(2) ( fn) is a Cauchy sequence converging to 0,(3) (hn) is a Cauchy sequence converging in E to h,(4) for each i, (rni ) is a Cauchy sequence in R, whose limits in R we will

denote ri ,(5) h =∑i ri gi .

Proof. Since, by assumption, w( fn+1) ≺ w( fn) for each n, the claim on ( fn)

is obvious and implies that on (hn) since hn = h − fn, for each n.

By construction, for each i, and each p > q,

rpi − rqi =q∑

j=p+1

m ji and v(rpi − rqi ) = γp − v(gi ),

implying the claim on (rni ).

We are therefore now able to give the complete statement of Proposi-tion 24.6.10:

Theorem 24.8.13. Let Γ be a (commutative) semigroup, inf-limited by thesemigroup ordering ≺, R a ring with 1, v : R → Γ a valuation, E an R-module, w : E → Γ a v-compatible valuation, E ⊂ E a submodule of E andB := g1, . . . , gs ⊂ E.

With the notations introduced in this and in the previous sections, then thefollowing conditions are equivalent:

(1) B is a standard basis of E;(2) B is a standard basis of Cl(E);(3) B is a standard basis of E;(4) for each element h ∈ E, h ∈ Cl(E) iff it has a Cauchy standard repre-

sentation in R in terms of B;(5) for each element h ∈ E , h ∈ E iff it has a Cauchy standard represen-

tation in R in terms of B;(6) for each element h ∈ E, h ∈ Cl(E) iff it has a standard representation

in R in terms of B;

232 Grobner I

(7) for each element h ∈ E , h ∈ E iff it has a standard representation in Rin terms of B;

(8) for each element h ∈ E, h ∈ Cl(E) iff there is a Cauchy sequence(hn) ∈ C(E) converging to h and such that for each n ∈ N, hn has astandard representation in R in terms of B;

(9) for each element h ∈ E , h ∈ E iff there is a Cauchy sequence (hn) ∈C(E) converging to h and such that for each n ∈ N, hn has a standardrepresentation in R in terms of B;

(10) for each h ∈ E \ 0 either

• h ∈ Cl(E) and h has a standard representation in R in terms of B,or

• h ∈ Cl(E) and there is g ∈ E \ 0 : L(g) ∈ L(E) and h − g ∈ Ehas a standard representation in R in terms of B;


• h ∈ E and h has a standard representation in R in terms of B, or• h ∈ E and there is g ∈ E \ 0 : Lˆ(g) ∈ Lˆ(E) and h − g ∈ E has

a standard representation in R in terms of B;


• h ∈ Cl(E) and there is a Cauchy sequence (hn) ∈ C(E) convergingto h and such that for each n ∈ N, hn has a standard representationin R in terms of B, or

• h ∈ Cl(E) and there is g ∈ E \ 0 : L(g) ∈ L(E) and h − g ∈ Ehas a standard representation in R in terms of B;


• h ∈ E and there is a Cauchy sequence (hn) ∈ C(E) converging to hand such that for each n ∈ N, hn has a standard representation in Rin terms of B, or

• h ∈ E and there is g ∈ E \ 0 : Lˆ(g) ∈ Lˆ(E) and h − g ∈ E hasa standard representation in R in terms of B;

and all imply that B is a basis of Cl(E) in R.

Proof.

(2) ⇒ (1) and (3) ⇒ (1) are obvious.(4) ⇒ (2) and (5) ⇒ (3): Let m ∈ L(E); then there is h ∈ E such that

L(h) = m. Let λn ≺ w(h) and h = ∑i hi gi + g be a truncated

standard representation in terms of B at λn . Then

max(v(hi ) + w(gi )) w(h) λn w(g),


so that, setting I := i : w(h) = v(hi ) + w(gi ), we have

m = L(h) =∑i∈I

L(hi )L(gi ),

proving that B is a standard basis.(6) ⇒ (4) and (7) ⇒ (5): If h = ∑

i hi gi is a standard representation inR in terms of B, in order to get a truncated standard representationh = ∑

i ri gi in R at γ ∈ Γ , it is sufficient to truncate each hi takingany element ri ∈ R such that v(hi − ri ) ≺ γ − w(gi ).

(8) ⇒ (4) and (9) ⇒ (5): Let h ∈ Cl(E), (hn) ∈ C(E) be a Cauchy se-quence converging to h, and γ ∈ Γ.

Setting n : w(h − hn) ≺ γ , if hn = ∑i hi gi + g is a Cauchy

truncated representation at γ , then h := ∑i hi gi + (g + h − hn) is

the same.(10) ⇒ (6), (11) ⇒ (7), (12) ⇒ (8), and (13) ⇒ (9) are obvious.(1) ⇒ (10) and (1) ⇒ (11): Let h ∈ E \ 0; with the same notation as

in Lemma 24.8.11, there are three cases:

• there is n ∈ N such that f j = 0 for each j > n, so that h =hn = ∑

i rni gi ∈ E is a standard representation in R in termsof B;

• there is n ∈ N such that f j = fn = 0 for each j > n, so thath = fn + hn ; hn = ∑

i rni gi is a standard representation in R interms of B, and L( fn) ∈ L(E), implying that fn ∈ E and

h = fn + hn ∈ E + Fγn (E) ⊃ Cl(E);

• for each n ∈ N, fn+1 = fn = 0, so that

L( fn) ∈ L(E) and h = fn + hn ∈ E + Fγn ;

also, for each γ ∈ Γ , there exists n ∈ N such that E + Fγn ⊂E + Fγ , implying that h ∈ Cl(E).Moreover Corollary 24.8.12 guarantees that, taking the limit of theCauchy standard representations h = fn +∑ j i r ji gi , one obtains

the standard representation h =∑i ri gi in R in terms of B.

(1) ⇒ (12) and (1) ⇒ (13): Let h ∈ E \ 0; again there are threecases:

• there is n ∈ N such that f j = 0 for each j > n, so that h = hn =∑i rni gi ∈ E is a standard representation in R in terms of B;

234 Grobner I

• there is n ∈ N such that f j = fn = 0 for each j > n, so thath = fn + hn ; hn = ∑

i rni gi is a standard representation in R interms of B, and L( fn) ∈ L(E), implying that fn ∈ E and

h = fn + hn ∈ E + Fγn (E) ⊃ Cl(E);• for each n ∈ N, fn+1 = fn = 0, so that by Corollary 24.8.12, (hn)

converges to h and consists of elements in E having the standardrepresentation hn =∑ j i r ji gi in R in terms of B.

24.9 Term Ordering: Classification and Representation

In order to apply Grobner technology, we need to characterize the term order-ings < on

T := Xa11 . . . Xan

n : (a1, . . . , an) ∈ Nn;

by Definition 22.1.2 they are those orderings which are a semigroup ordering,that is

t1 < t2 ⇒ t t1 < t t2, for each t, t1, t2 ∈ T,

and a well-ordering; however, restricting oneself to well-orderings on T , isunnatural and only has the unpleasant effect of removing Hironaka’s theoryfrom consideration.

Therefore the problem to be solved is to characterize all semigroup orderingson T , or, equivalently, all orderings on the semigroup N

n which are compatiblewith addition; clearly any such ordering can be uniquely extended to a Q-vector space ordering on Q

n .Such a required characterization was already available when Buchberger in-

troduced his theory, because in 1955 Erdos characterized all ordered R-vectorspaces. Here we present that part of his result which characterized the finitecase.

Definition 24.9.1. An ordered R-vectorspace is an R-vectorspace V endowedwith an ordering < such that, for each x, y ∈ V, λ ∈ R

• x > 0, y > 0 ⇒ x + y > 0,• x > 0, λ > 0 ⇒ λx > 0,• x > y ⇐⇒ x − y > 0.

For any two elements x, y > 0 of an ordered R-vectorspace, x > 0 iscalled incomparably smaller than y > 0 (denoted by x ! y) iff λx ≤ y for

24.9 Term Orderings 235

each λ ∈ R; x > 0 and y > 0 are said to be equivalent (x ∼ y) if neitherx ! y nor x " y holds.

For any x ∈ V \ 0, |x | denotes the positive element among x and −x.

Erdos’ characterization proves, for each ordered R-vector space V ,dimR(V ) = n, the existence of a basis b1, . . . , bn such that

b1 " b2 " · · · " bn > 0.

Then, for any b :=∑ni=1 ci bi ∈ V we have

b > 0 ⇐⇒ there exists j : c j > 0 and ci = 0 for i < j.

Lemma 24.9.2. Let V be an ordered R-vectorspace. For any two linearly in-dependent, positive and equivalent elements x, y ∈ V , there is a linear combi-nation ax + by, a, b ∈ R, which is incomparably smaller than both.

Proof. Since x and y are

• linearly independent, then y − λx = 0 for each λ ∈ R,• both positive, y < λx, λ ∈ R, implies λ > 0,• equivalent, it is sufficient to produce a linear combination ax + by ! x in

order to prove the claim.

Also, since they are equivalent, the set λ : y < λx ⊂ R is not empty andhas the lower bound 0; therefore it has a greatest lower bound λ ∈ R; as a

consequence, for each µ ∈ R, µ > 0, we have(λ − 1

µ

)x < y <

(λ + 1

µ

)x

so that −x < µ(y − λx) < x .The positive element among y−λx and λx − y is then incomparably smaller

than x (and also y).

Lemma 24.9.3 (Erdos). Let V be an ordered R-vectorspace, let

b1, . . . , bν ⊂ V

be a linearly independent set consisting of positive elements no two of whichare equivalent and such that SpanR(b1, . . . , bν) V .

Then, for any element b ∈ V \ SpanR(b1, . . . , bν), there exists a positiveelement bν+1 ∈ V \ SpanR(b1, . . . , bν) which is not equivalent to any bi andsuch that

SpanR(b1, . . . , bν, b) = SpanR(b1, . . . , bν, bν+1).

Proof. Let us wlog assume that 0 < b1 ! b2 ! · · · ! bν .

If |b| is not equivalent to any bi , it is sufficient to set bν+1 := |b|.

236 Grobner I

Otherwise, let i ≤ ν be the least value for which bi is equivalent to a form

λi bi + λi+1bi+1 + · · · + λνbν + λb > 0, λ = 0.

Then Lemma 24.9.2 gives the existence of a form

bν+1 := µbi + ν (λi bi + λi+1bi+1 + · · · + λνbν + λb)

which satisfies 0 < bν+1 ! bi ! bi+1 ! · · · ! bν .

Corollary 24.9.4 (Erdos). Let V be any ordered R-vectorspace such thatdimR(V ) = n and let β1, . . . , βn be any basis of V . Then:

(1) V has a basis b1, . . . , bn such that b1 " b2 " · · · " bn > 0;(2) for each b :=∑n

i=1 ci bi ∈ V we have

b > 0 ⇐⇒ there exists j : c j > 0 and ci = 0 for i < j;(3) let (alk) be the n-square matrix such that βk := ∑

l blalk for each k;then for each b :=∑n

k=1 ckβk ∈ V we have

b > 0 ⇐⇒ there exists j :n∑

k=1

a jkck >0 andn∑

k=1

aikck = 0 for i < j.

Proof.

(1) The proof is by induction on n: if n = 1 we set b1 := |β1|; if n > 1 weassume that we have already produced a basis b1, . . . , bn−1 such that

• b1 " b2 " · · · " bn−1 > 0 and• SpanR(b1, . . . , bn−1) = SpanR(β1, . . . , βn−1).

Its condition being satisfied, Lemma 24.9.3, applied to b1, . . . , bn−1and βn , allows us to produce a positive element bn which is not equiv-alent to any bi and such that

SpanR(b1, . . . , bn) = SpanR(β1, . . . , βn).

To complete the proof, we only have to re-order the b → βs.(2) Let us wlog assume b = ∑n

i=k ci bi , ck = 0. For each i > k,since bk " bi , we have bk > (k − n)ci c

−1k bi so that (n − k)bk >∑n

i=k+1(k − n)ci c−1k bi whence b =∑n

i=k ci bi > 0.

(3) Since

b =n∑

k=1

ckβk =∑

l

bl

n∑k=1

alkck,

the claim follows by the previous statement.


Recalling (Remark 24.5.5) that a weight function vw : T → R on T andP := k[X1, . . . , Xn] is the assignment of a vector

w := (w1, . . . , wn) ∈ Rn, wi ≥ 0,

so that

vw(Xa11 . . . Xan

n ) =∑

i

wi ai ,

Erdos’ result can be formulated, within Buchberger’s and Hironaka’s theory,as

Corollary 24.9.5 (Erdos). Each semigroup ordering < on T is characterizedby assiging r ≤ n linearly independent vectors

w1, . . . , w j := (w j1, . . . , w jn), . . . , wr ∈ Rn

– or equivalently an r × n matrix (w j i ) ∈ Rrn of maximal rank – so that for

each t1 := Xa11 . . . Xan

n , t2 := Xb11 . . . Xbn

n in T , we have

t1 < t2 ⇐⇒ ∃ j : vw j (t1) < vw j (t2) and vwi (t1) = vwi (t2) for i < j.

Moreover, such an ordering is a well-ordering iff, for each i , Xi > 1, that isiff, for each i , w j i > 0, where j denotes the minimal value for which w j i = 0.

Finally, if M1, M2 are two r × n matrices, they characterize the same order-ing < iff there is an invertible r-square matrix A = (ai j ) such that

M1 = AM2 and ai j =

0 if i < j1 if i = j

.

Example 24.9.6. To illustrate Erdos’ result let us consider P := k[X1, X2, X3]and the ordering < under which

Xa11 Xa2

2 Xa33 < Xb1

1 Xb22 Xb3

3 ⇐⇒⎧⎨⎩

a1 + a2 + a3 < b1 + b2 + b3 ora2 + a3 < b2 + b3 ora3 < b3,

which is characterized by the matrix⎛⎝1 1 1

0 1 10 0 1

⎞⎠ ,

and let us choose as basis of Xa11 Xa2

2 Xa33 : (a1, a2, a3) ∈ Q

3 the basis

238 Grobner I

X2, X3, X1. Therefore we set b1 := X2 > 0 and, applying Lemma 24.9.3 to

• b1 and b := X3 we obtain b2 := b−11 b = X−1

2 X3 > 0;• b1, b2 and b := X1 we obtain

b3 := b1b2b−1 = X2

(X−1

2 X3

)X−1

1 = X−11 X3 > 0;

and, after re-ordering,

b1 := X2 " b2 := X−11 X3 " b3 := X−1

2 X3 > 0.

In this context we recall the following:

Proposition 24.9.7 (Bayer). Given any finite set of terms T ⊂T and any termordering <, then:

• the set

C(T, <) := w ∈ Rn : vw(t) < vw(τ ) ⇐⇒ t < τ for each t, τ ∈ T

is a relatively open convex polyhedral cone;• there is a weight vector w ∈ Z

n such that, for each t, τ ∈ T ,

vw(t) < vw(τ ) ⇐⇒ t < τ.

Proof. Let us consider the set

T := a := (a1, . . . , an) ∈ Nn : Xa1

1 . . . Xann ∈ T ⊂ N

n

which we order so that

(a1, . . . , an) < (b1, . . . , bn) ⇐⇒ Xa11 . . . Xan

n < Xb11 . . . Xbn

n

and define B := b − a : a, b ∈ T, a 0 for each (β1, . . . , βn) ∈ B

is the intersection of open half-spaces.

Among the term orderings we will quote those which have common andpractical use, and are used in applying this theory.

• The lexicographical (lex) ordering induced by X1 < X2 < · · · < Xn isdefined by

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j j,


and is characterized by the matrix⎛⎜⎜⎜⎜⎜⎝

0 0 · · · 0 10 0 · · · 1 0...

.... . .

......

0 1 · · · 0 01 0 · · · 0 0

⎞⎟⎟⎟⎟⎟⎠ ;

it has good elimination properties since it allows us to compute all the elim-ination ideals I ∩ k[X1, . . . , Xi ]:

Fact 24.9.8. If G is the Grobner basis of I ⊂ k[X1, . . . , Xn] w.r.t. lex thenG ∩ k[X1, . . . , Xi ] is the Grobner basis of I ∩ k[X1, . . . , Xi ] w.r.t. lex.

Proof. Compare Corollary 26.2.4.

• The lexicographical ordering depends on the ordering imposed on the vari-ables (see Remark 24.9.13 below): the lexicographical ordering defined by

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j X2 > · · · > Xn and characterized by the matrix⎛⎜⎜⎜⎜⎜⎝

1 0 · · · 0 00 1 · · · 0 0...

.... . .

......

0 0 · · · 1 00 0 · · · 0 1

⎞⎟⎟⎟⎟⎟⎠ .

• In general, if one is interested in computing a Grobner basis of

I ∩ k[Y1, . . . , Yd ]

for an ideal

I ⊂ k[Y1, . . . , Yd , Z1, . . . , Zr ]

with respect to a particular ordering <Y on k[Y1, . . . , Yd ] characterizedby the matrix MY , an efficient solution is to choose an ordering <Z onk[Z1, . . . , Zr ] characterized by the matrix MZ and to compute the Grobnerbasis G of I w.r.t. the ordering < such that

Y c11 · · · Y cd

d Za11 · · · Zar

r < Y e11 · · · Y ed

y Zb11 · · · Zbr

r

⇐⇒

Za11 · · · Zar

r <Z Zb11 · · · Zbr

r or

Za11 · · · Zar

r = Zb11 · · · Zbr

r , Y c11 · · · Y cd

d <Y Y e11 · · · Y ed

y

characterized by the matrix

(0 MZ

MY 0

). Then:

240 Grobner I

Fact 24.9.9. If G is the Grobner basis of I ⊂ k[Y1, . . . , Yd , Z1, . . . , Zr ]w.r.t. < then G ∩ k[Y1, . . . , Yd ] is the Grobner basis of I ∩ k[Y1, . . . , Yd ]w.r.t. <Y .

Proof. Compare Theorem 26.2.2

Any such ordering is called the block ordering inducing

Y1, . . . , Yd < Z1, . . . , Zr defined by <Y and <Z .

• The reverse lexicographical (rev-lex) ordering induced by X1 < X2 < · · · <

Xn is defined by

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j > b j and ai = bi for i < j,

and characterized by the identical matrix⎛⎜⎜⎜⎜⎜⎝

−1 0 · · · 0 00 −1 · · · 0 0...

.... . .

......

0 0 · · · −1 00 0 · · · 0 −1

⎞⎟⎟⎟⎟⎟⎠ ,

which is the ordering introduced by Macaulay in his theorem (Section 23.3)and is not a well-ordering since · · · < Xd+1

i < Xdi < · · · < X1 < 1.

• Macaulay introduced and applied it only on homogeneous components, so,more correctly, Macaulay’s ordering is the deg-rev-lex (degree reverse lex-icographical) ordering induced by X1 < X2 < · · · < Xn where terms arefirst compared by their degree and the ties are solved using rev-lex: it isdefined by

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j > b j and ai = bi for 0 ≤ i < j,

– where we set a0 := −∑i ai , b0 := −∑i bi – and characterized by thematrix 30 ⎛

⎜⎜⎜⎜⎜⎝

1 1 · · · 1 1−1 0 · · · 0 00 −1 · · · 0 0...

.... . .

......

0 0 · · · −1 0

⎞⎟⎟⎟⎟⎟⎠ .

It has the following important property: 31

30 The last row of the matrix characterizing rev-lex is useless for solving ties and can therefore beremoved.

31 For relevant applications of this characterization, compare Corollary 26.3.13.


Corollary 24.9.10. Denoting, for each i ≤ n, πi : T → T ∩k[X1, . . . , Xi ]the projection 32 defined by

X j :=

X j if j > i1 if j ≤ i

then any two terms t1, t2 ∈ T , setting d ji := deg(π j (ti )), satisfy

t1 < t2 ⇐⇒ ∃ j : d j1 < d j2, and di1 = di2 for each i < j.

Historical Remark 24.9.11. It is worth noting that Buchberger himself, inhis seminal paper, used the same ordering as Macaulay, the deg-rev-lex or-dering induced by X1 < X2 < · · · < Xn .

Macaulay mainly used it in homogeneous components and always con-sidered as leading term the minimal one. Buchberger, who was working onthe non-homogeneous case, in order to be assured that his reduction wasterminating, had no other choice but to require that the leading term was ofmaximal degree and so to consider as leading term the maximal one.

• Dually (see Remark 24.9.13 below) one can consider the rev-lex orderinginduced by X1 > X2 > · · · > Xn which is defined by

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j > b j and ai = bi for i > j,

and characterized by the matrix⎛⎜⎜⎜⎜⎜⎝

0 0 · · · 0 −10 0 · · · −1 0...

.... . .

......

0 −1 · · · 0 0−1 0 · · · 0 0

⎞⎟⎟⎟⎟⎟⎠ ,

• and the deg-rev-lex ordering induced by X1 > X2 > · · · > Xn which isdefined as

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j > b j and ai = bi for n + 1 ≥ i > j,

– where we set an+1 := −∑i ai , bn+1 := −∑i bi – and characterized by⎛⎜⎜⎜⎜⎜⎝

1 1 · · · 1 10 0 · · · 0 −10 0 · · · −1 0...

.... . .

......

0 −1 · · · 0 0

⎞⎟⎟⎟⎟⎟⎠ .

32 Obviousy π0 is just the identity.

242 Grobner I

Note that the revlex ordering < induced by X1 > X2 > · · · > Xn and thelex-ordering ≺ induced by X1 ≺ X2 ≺ · · · ≺ Xn are related by

t1 < t2 ⇐⇒ t1 t2 for each t1, t2 ∈ T .

Dually the lex ordering < induced by X1 > X2 > · · · > Xn and the rev-lex-ordering ≺ induced by X1 ≺ X2 ≺ · · · ≺ Xn are related by

t1 < t2 ⇐⇒ t1 t2 for each t1, t2 ∈ T .

• More generally, given an ordering < on T , characterized by the matrix M ,its degree extension is the ordering ≺ defined as

t1 ≺ t2 ⇐⇒ deg(t1) < deg(t2) or deg(t1) = deg(t2), t1 < t2

and characterized by the matrix obtained by bordering M , adding on top arow of 1s: ⎛

⎜⎜⎝1 · · · 1

M

⎞⎟⎟⎠ .

• In this way we obtain also the degree lexicographical (deg-lex) orderinginduced by X1 < X2 < · · · < Xn (also known as the total degree order-ing) which is obtained by ordering the terms according to their degree andsolving ties via the lexicographical ordering; it is defined as

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j j

– where we set an+1 := ∑i ai , bn+1 := ∑

i bi – and characterized by thematrix ⎛

⎜⎜⎜⎜⎜⎝

1 1 · · · 1 10 0 · · · 0 10 0 · · · 1 0...

.... . .

......

0 1 · · · 0 0

⎞⎟⎟⎟⎟⎟⎠ ,

• and the degree lexicographical ordering induced by X1 > X2 > · · · > Xnwhich is defined as

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j < b j and ai = bi for 0 ≤ i < j

– where we set a0 := ∑i ai , b0 := ∑

i bi – and characterized by the


matrix ⎛⎜⎜⎜⎜⎜⎝

1 1 · · · 1 11 0 · · · 0 00 1 · · · 0 0...

.... . .

......

0 0 · · · 1 0

⎞⎟⎟⎟⎟⎟⎠ .

• If we have a weight vector w := (w1, . . . , wn) ∈ Rn \ 0 and a term

ordering < represented by a matrix M , the construction leading to the degreeextension of < can be performed leading to the weight extension ≺ of <

(or the refinement of vw with <) defined as

t ≺ T ⇐⇒ vw(t) < vw(T ) or vw(t) = vw(T ), t < T

and characterized by ⎛⎜⎜⎝

w1 · · · wn

M

⎞⎟⎟⎠ .

Bayer and Stillman proved that the revlex ordering is the ‘most efficient’ refine-ment of a weight function vw: recalling that, for a homogeneous ideal I ⊂ P ,the regularity of M0 := I is the least value reg(I) := m for which each i thsyzygy Mi := Syz(Mi−1) is generated in degree bounded by m + i , in gen-eral we have reg(T<(I)) ≥ reg(I); they proved that, with the same notation asCorollaries 24.5.6 and 24.6.15 and under the wlog assumption 33

w1 ≤ w2 ≤ · · · ≤ wn,

we have:

Fact 24.9.12 (Bayer–Stillman). For any homogeneous ideal

I ⊂ k[X1, . . . , Xn] =: P

and any matrix M ∈ GL(n, k), we have

reg(T≺(M(I))) = reg(T<(Lw(M(I)))) ≥ reg(Lw(M(I))).

If, moreover, < is the revlex ordering induced by X1 < · · · < Xn, then thereis a non-empty Zariski open set U ⊂ GL(n, k) such that

reg(T≺(M(I))) = reg(T<(Lw(M(I)))) = reg(Lw(M(I))), for each M ∈ U.

33 This is only required in order to re-number the variables so that

vw(X1) ≤ vw(X2) ≤ · · · ≤ vw(Xn).

244 Grobner I

Proof. Compare Theorem 38.5.12

Remark 24.9.13. I fear I am not only left-handed but also left-brained. WhatI have always called (and I am calling here) the lexicographical ordering ink[X1, . . . , Xn] is the term ordering defined by

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j j,

which induces over the variables the ordering

1 < X1 < X2 < · · · < Xn;nearly everybody else called lexicographical ordering the ordering defined by

Xa11 . . . Xan

n < Xb11 . . . Xbn

n ⇐⇒ ∃ j : a j < b j and ai = bi for i < j,

which induces over the variables the ordering

1 < Xn < Xn−1 < · · · < X1.

When I asked why they do so, the only explanation obtained, apart from theipse dixit approach, is that it is exactly what happens with the alphabetical lex-icographical order. In fact if we have to compare (1, 0, 0, 0) with (0, 1, 0, 0)

I can agree that the obvious choice is to say that (1, 0, 0, 0) > (0, 1, 0, 0);but this, while it has sense only if we are thinking a la Erdos of the monomi-als as elements in N

n, has much less sense if we consider the monomials aspolynomial elements in k[X1, X2, . . . , Xn] or in k[X, Y, Z , T, W ] where thecommon definition of lex-ordering implies that Xn < · · · < X2 < X1 or,horribile visu, W < T < Y < X ; the difference between the two defini-tions in fact essentially boils down to deciding whether we consider morenormal having 1 < Xn < Xn−1 < · · · < X1 (as everybody thinks) 34 or1 < X1 < X2 < · · · < Xn (as I think).

34 It is worth noting that Macaulay, in his combinatorial research on T ∼= Nn , where he used as

ordering degrevlex, assumed that the variables were ordered as X1 < X2 < · · · < Xn .Also Grobner in his algorithmic solutions of Problem 24.0.1 and Buchberger in his thesis

listed the monomials using degrevlex with X1 < X2 < · · · < Xn .The position of Gjunter is very illuminating: in his deep analysis of the structure of

numerations, that is a specific class of degree compatible term orderings, he states:

On peut choisir d’autres numerations qui different des deux numerations ci-dessus [deglex anddegrevlex with X1 < X2 < · · · < Xn ]. Par ex. . . . on peut convenit que [X

α11 . . . X

α44 ] ait un

n inferieur [of Xβ11 . . . X

β44 ] si

β4 − α4 > 0 ouβ4 − α4 = 0, α1 − β1 > 0 ouβ4 − α4 = 0, α1 − β1 = 0, β2 − α2 > 0.

[ . . . ] Nous appellerons numeration reguliere toute numeration base sur la compairason desdifferences entre les exposants des monomes correspondants.Remarque. Si le degre des monomes est egal a l’unite, chaque exposant etant egal a l’unite oua 0, toutes les numerations regulieres conduisent aux memes resultatsthat is X1 < X2 < · · · < Xn .


From my point of view,35 if one thinks a la Kronecker, variables are in-troduced consecutively in order to define a new – algebraic or transcenden-tal – element in terms of the previous ones, and polynomial ideals are thecollection of all the algebraic relations between these orderly defined algebraic

35 I try in this note to justify my position by describing my frame of mind.If we consider a field k and we successively construct a sequence of ‘arithmetical expres-

sions’ (see Section 8.1) β1, . . . , βd , βd+1, . . . , βd+r , where (up to reordering) we can assumethat

• β1 is transcendental over k,• for i, 1 < i ≤ d , βi is transcendental over k(β1, . . . , βi−1),• α1 := βd+1 is algebraic over k(β1, . . . , βd ) and satisfies the algebraic relation

f1(β1, . . . , βd , α1) = 0, f1 ∈ k[Y1, . . . , Yd ][Z1],

• αi := βd+i is algebraic over k(β1, . . . , βd )[α1, . . . , αi−1] for each i , 1 < i ≤ r , andsatisfies the algebraic relation

fi (β1, . . . , βd , α1, . . . , αi−1, αi ) = 0,

fi ∈ k[Y1, . . . , Yd , Z1, . . . , Zi−1][Zi ],

the Kronecker–Duval Model gives us that (up to factorization/squarefree splitting) we can as-sume that for each i, 1 ≤ i ≤ r , setting di := degi ( fi ), we have

• fi is monic,• deg j ( fi ) < d j , for all j < i,

so that we have a tower of rings

k[Y1, . . . , Yd ] ∼= D0 ⊂ · · · ⊂ Di ⊂ · · · ⊂ Dr = k[β1, . . . , βd , βd+1, . . . , βd+r ]

where, for each i, 1 ≤ i ≤ r ,

Di := k[β1, . . . , βd , α1, . . . , αi ]∼= k[Y1, . . . , Yd , Z1, . . . , Zi ]/( f1, . . . , fi )∼= Di−1[Zi ]/ fi ,

and (in the Kronecker Model) a corresponding tower of fields

k(Y1, . . . , Yd ) ∼= K0 ⊂ · · · ⊂ Ki ⊂ · · · ⊂ Kr = k(β1, . . . , βd , βd+1, . . . , βd+r )

where

Ki := k(β1, . . . , βd , α1, . . . , αi )

∼= K0[Z1, . . . , Zi ]/( f1, . . . , fi )∼= Ki−1[Zi ]/ fi .

In this context, if we set n = d + r and we identify the polynomial rings P := k[X1, . . . , Xn ]and k[Y1, . . . , Yd , Z1, . . . , Zr ] by

Xi :=

Yi if i ≤ dZi−d if i > d

it is natural to assume that X1 < X2 < · · · < Xn and that the k-basis T of P is well-ordered under the lexicographical ordering induced by X1 < X2 < · · · < Xn . If r =n, under this ordering the admissible sequence ( f1, . . . , fr ) is the reduced Grobner basis

246 Grobner I

expressions; it seems at least natural to preserve the order by which they aredefined, ensuring that X1 < X2 < · · · < Xn .

Forgetting to think a la Kronecker can have some unpleasant consequences:there are statements which are awkward to make and difficult to prove using

(Theorem 34.1.2).

In the same setting, the Primitive Element Theorem (Theorem 8.4.5) informs us thatthere are an element γ ∈ Kr and polynomials g0, g1, . . . , gr ∈ K0[Z ] such that

Kr = K0[γ ] ∼= K0[Z ]/g0 and αi = gi (γ ), 1 ≤ i ≤ r.

An ideal

I ⊂ k[X1, . . . , Xn ] = k[Y1, . . . , Yn ]

where Y1, . . . , Yn is a ‘generic’ system of coordinates (see Section 27.8) satisfies

• dim(I) = d ⇐⇒ I ∩ k[Y1, . . . , Yd ] = (0) = I ∩ k[Y1, . . . , Yd+1] (Corollary 27.11.3);• setting d := dim(I), r := n − d there are polynomials

g1, g2, . . . , gr ∈ k(Y1, . . . , Yd )[Yd+1]

such that (Corollary 34.3.4)

Ie := Ik(Y1, . . . , Yd )[Yd+1, . . . , Yn ]

=(

g1(Yd+1), Yd+2 − g2(Yd+1), . . . , Yn − gr (Yd+1)

);

• I is unmixed (Definition 27.13.1) iff (Corollary 27.13.6)

I = Ice := Ik(Y1, . . . , Yd )[Yd+1, . . . , Yn ] ∩ k[Y1, . . . , Yn ];• for any term ordering < such that Y1 < Y2 < · · · < Yn , there are polynomials

h1, h2, . . . , hr ∈ k[Y1, . . . , Yn ] such that (Chapter 37)

T<(hi ) = Yδid+i , for each i

δ1 ≤ · · · ≤ δi ≤ δi+1 ≤ · · · ≤ δr

Therefore if I ⊂ k[X1, . . . , Xn ] is a radical, unmixed ideal, d = dim(I), r = n − d,Y1, . . . , Yn is a ‘generic’ system of coordinates and G is the Grobner basis of I ink[Y1, . . . , Yn ] w.r.t. the lexicographical ordering induced by Y1 < Y2 < · · · < Yn thenthere are q2, . . . , qr ∈ k[Y1, . . . , Yd ] and p0, p2, . . . , pr ∈ k[Y1, . . . , Yd , Yd+1] such that

G ∩ k[Y1, . . . , Yi ] = ∅ ⇐⇒ i ≤ d,

G ∩ k[Y1, . . . , Yd+1] = (p0),qi Yd+i − pi ∈ G, 2 ≤ i ≤ r ,

for each i , 2 ≤ i ≤ r , p0, Yd+2 − p2q−12 , . . . , Yd+i − pi q−1

i ) is the Grobner basisof Ik(Y1, . . . , Yd )[Yd+1, . . . , Yd+i ] w.r.t. the lexicographical ordering induced byYd+1 < · · · < Yd+i .

Let

I ⊂ k[X0, . . . , Xn ] be a homogeneous ideal;X0, . . . , Xn a ‘generic’ system of coordinates;< a term ordering on k[X0, . . . , Xn ];for each i , <i its restrictions to k[Xi , . . . , Xn ];M0(I) := H(a I) ⊂ k[X1, . . . , Xn ];Mi (I) := H(a(Mi−1(g)) ⊂ k[Xi+1, . . . , Xn ] for each i – where the homogenization/

affinization variable is Xi ;

then

(1) depth(I) = λ ⇐⇒ λ is the maximal value for which X0, . . . , Xλ−1 is a regular sequenceof I (Definition 36.1.1 and Lemma 36.2.3);

24.10 *Grobner Bases and the State Polytope 247

the common notion, while the same proof is more elementary with my defini-tion;36 compare for instance

Each 0-dimensional radical ideal has a basis of the form

(g1(X1), X2 − g2(X1), . . . , Xi − gi (X1), . . . , Xn − gn(X1))

which is a Grobner basis under my definition of lex, with the same result statedusing the common definition:

Each 0-dimensional radical ideal has a basis of the form(g1(Xn), Xn−1 − g2(Xn), . . . , Xn−i − gi (X1), . . . , X1 − gn(Xn)

).

To avoid confusion, it is now common to specify of which ordering one isthinking by explicitly stating the corresponding ordering on the variables.

The reader must be aware that all through this book, if there is no specifica-tion, lex, deglex, revlex, degrevlex are the ones induced by X1 < X2 < · · · <

Xn notwithstanding that most papers, books and software use the ones inducedby Xn < · · · < X2 < X1. I am probably really left-brained, but I refuse tofollow this nonsensical mood.

24.10 *Grobner Bases and the State Polytope

Let P := k[X1, . . . , Xn] and let

T := Xa11 . . . Xan

n , (a1, . . . , an) ∈ Nn.

For any weight function

w := (w1, . . . , wn) ∈ Rn \ 0

we consider the valuation vw : P → R induced by vw(Xi ) = wi for each i , andthe corresponding leading-form map Lw : P → P; in the same mood, for anysemigroup ordering (not necessarily a term ordering) < on T we consider thecorresponding valuation v< : P → T and leading-form map T< : P → T .

(2) if (Remark 36.3.8)

• X0, . . . , Xλ−1 is a regular sequence of I• for each i < λ and each t1, t2 ∈ T [i, n] we have

t1 <i−1 t2 ⇐⇒ deg(t1) < deg(t2) or deg(t1) = deg(t2), at1 <ia t2,

it is sufficient to compute the Grobner basis of Mλ−1(I) w.r.t. <λ and to iteratively applyCorollary 23.2.18 in order to deduce the Grobner basis of each Mi (I), 0 ≤ i < λ andfinally of I.

The only term ordering < on T which satisfies

t1 <i−1 t2 ⇐⇒ deg(t1) < deg(t2) or deg(t1) = deg(t2), at1 <ia t2,

for each t1, t2 ∈ T [i, n], and each i ≤ n, is the degrevlex ordering induced by X1 < X2 <

· · · < Xn .36 I have seen theorems stated over the polynomial ring k[Xn , Xn−1, . . . , X1]!

248 Grobner I

If I ⊂ P is an ideal, while, for any T -valuation <, T<(I) is obviously amonomial ideal, Lw(I), for an R-valuation w ∈ R

n , is not necessarily a mono-mial ideal but just a homogeneous ideal. However, if we fix a term ordering <

and we consider its weight extension ≺ by w, Corollaries 24.5.6 and 24.6.15can be reformulated as

Corollary 24.10.1. With the notation above, the following conditions areequivalent:

• G is a Grobner basis of I w.r.t. ≺;• LwG is a Grobner basis of Lw(I) w.r.t. < and ≺.

Proof. Let f ∈ I; G is a Grobner basis of I w.r.t. ≺, iff there is g ∈ G suchthat T≺(g) = T≺(Lw(g)) = T<(Lw(g)) divides T≺( f ) = T≺(Lw( f )) =T<(Lw( f )) iff LwG is a Grobner basis of Lw(I) w.r.t. < and ≺.

Corollary 24.10.2. With the notation above, T≺(I) = T<(Lw(I)).

Lemma 24.10.3. Let w, w′ be two weight vectors.Let < be a term ordering, let ≺ be its weight extension by w and let G be

the reduced Grobner basis of I w.r.t. ≺.Then

Lw′(I) = Lw(I) ⇐⇒ Lw′(g) = Lw(g), for each g ∈ G.

Proof. Let us denote by ≺′ the weight extension of < by w′.If Lw′(g) = Lw(g) for each g ∈ G, then we have

Lw(I) = Lw(G) = Lw′(G) ⊆ Lw′(I);but Lw(I) Lw′(I) would imply

T≺(I) = T<(Lw(I)) T<(Lw′(I)) = T≺′(I)

which contradicts the consequence of Lemma 22.2.12

k[N≺(I)] ∼= P/I ∼= k[N≺′(I)].

Conversely assume Lw′(I) = Lw(I). Let us fix any g ∈ G and set

m := T≺(g) = T<(Lw(g));we have Lw′(g −m) ∈ k[N≺(I)] because g −m ∈ k[N≺(I)]. Therefore, settingh′ := Lw′(g) ∈ Lw(I), we necessarily have

m = T<(h′) = T≺′(g) and r ′ := h′ − m ∈ k[N≺(I)];since we easily have

m = T<(h) = T≺(g) and r := h − m ∈ k[N≺(I)]


also for h := Lw(g) ∈ Lw(I), we obtain

h − h′ = r − r ′ ∈ Lw(I) ∩ k[N≺(I)] = (0)

so that Lw′(g) = Lw(g).

This lemma allows us to apply the argument of Proposition 24.9.7 in orderto deduce

Corollary 24.10.4. Let I ⊂ P be an ideal and let w ∈ Rn. Then:

(1) the set

C(I, w) := v ∈ Rn : Lv(I) = Lw(I)

is a relatively open convex polyhedral cone;37

(2) there is a weight vector δ ∈ Zn such that Lδ(I) = Lw(I);

(3) let w′ ∈ Rn; if w′ ∈ C(I, w) then C(I, w′) is a face of C(I, w).

Proof. Let < be any term ordering and let ≺ be the weight extension of the <

by w and let G := g1, . . . , gt be the reduced Grobner basis of I w.r.t. ≺; let

37 Recall that

• a polyhedron P ⊂ Rn is a finite intersection of closed half-spaces in R

n ;• it is a cone if there exist vectors w1, . . . , wm ∈ R

n such that

P =

m∑i=1

λi wi : λi ∈ R, λi ≥ 0

;

• for any polyhedron P ⊂ Rn and a vector w ∈ R

n , facew(P) denotes the polyhedron

facew(P) := p ∈ P : w · p ≥ w · q for each q ∈ P;• if P ⊂ R

n is a polyhedron and F is a face of P , the normal cone of F at P is

NP (F) := w ∈ Rn : facew(P) = F;

• a fan is a finite collection F of cones such that

(a) if P ∈ F each face of P is a member of F;(b) if P1, P2 ∈ F, then P1 ∩ P2 ∈ F is a face of both P1 and P2;

• if P ⊂ Rn is a polyhedron the collection

N (P) := NP (F) : F is a face of Pis a fan which is called the normal cone of P;

• the Minkowski sum of two polyhedra P1, P2 ⊂ Rn is the polyhedron

P1 + P2 := p1 + p2 : p1 ∈ P1, p2 ∈ P2 ⊂ Rn

and satisfies the formula

facew(P1 + P2) = facew(P1) + facew(P2),

which implies that for each vertex v of P1 + P2 there are unique vertices p1 of P1 and p2 ofP2 such that v = p1 + p2.

250 Grobner I

us also write, for each i ,

hi := Lw(gi ) and mi := Xe11 . . . Xen

n := T≺(gi ) = T<(hi )),

so that

gi = mi +∑t∈T

c(hi − mi , t)t +∑t∈T

c(gi − hi , t)t.

For any v := (v1, . . . , vn) ∈ Rn we have

Lv(gi ) = Lw(gi )

if and only if both

• ∑i eivi >∑

i aivi , for each t = Xa11 . . . Xan

n : c(gi − hi , t) = 0, and• ∑i eivi =∑i aivi , for each t = Xa1

1 . . . Xann : c(hi − mi , t) = 0.

As a consequence C(I, w) is the intersection of open half-spaces and hyper-planes.

If w′ ∈ C(I, w) then Lw(I) = Lw(Lw′(I)); therefore there is a term ordering< such that, denoting by

• ≺′ the weight extension of < by w′,• ≺ the weight extension of < by w,• G := g1, . . . , gt the reduced Grobner basis of I w.r.t. ≺,

we have T≺(I) = T≺′(I). Therefore, using the same notation as above andsetting ki := Lw′(gi ) we have

Lw(ki ) = Lw(Lw′(gi )) = Lw(gi ) = hi ,

gi = mi +∑t∈T

c(hi − mi , t)t +∑t∈T

c(ki − hi , t)t +∑t∈T

c(gi − ki )t,

and, for any v := (v1, . . . , vn) ∈ Rn

Lv(gi ) = Lw′(gi )

if and only if

• ∑i eivi >∑

i aivi , for each t = Xa11 . . . Xan

n : c(gi − ki , t) = 0,• ∑i eivi =∑i aivi , for each t = Xa1

1 . . . Xann : c(ki − hi , t) = 0 and

• ∑i eivi =∑i aivi , for each t = Xa11 . . . Xan

n : c(hi − mi , t) = 0,

which proves that C(I, w′) is a face of C(I, w).

Theorem 24.10.5. Let I ⊂ P be an ideal and let in(I) be the set consisting ofall monomial ideals M ⊂ P such that M = T<(I) for some semigroup ordering< on T .

Then in(I) is finite.


Proof (Logar). Let ( f1, . . . , fs) be a basis of I and assume that in(I) is infinite.Since each fi is a finite combination of terms there are an infinite subset

Σ1 ⊂ in(I) and monomials mi such that mi = T<( fi ) for each i, 1 ≤ i ≤ s,and each term ordering < for which T<(I) ∈ Σ1. Let us choose an ordering<1 such that T<1(I) ∈ Σ1: if (m1, . . . , ms) T<1(I) then there is a non-zeropolynomial fs+1 ∈ I ∩ k[N<1 ].

As before, since fs+1 is a finite combination of terms there are an infinitesubset Σ2 ⊂ Σ1 ⊂ in(I) and a monomial ms+1 such that mi = T<( fi ) foreach i, 1 ≤ i ≤ s + 1, and each term ordering < for which T<(I) ∈ Σ2. Let uschoose an ordering <2 such that L<2(I) ∈ Σ2: if

(m1, . . . , ms) (m1, . . . , ms+1) T<2(I)

then there is again a non-zero polynomial fs+2 ∈ I ∩ k[N<2 ].Repeatedly we can obtain

• non-zero polynomials fs+ j ∈ I ∩ k[N< j ],• subsets Σ j+1 ⊂ Σ j ⊂ in(I),• monomials ms+ j such that mi = T<( fi ) for each i, 1 ≤ i ≤ s + j , and each

term ordering < for which T<(I) ∈ Σ j+1,• orderings < j+1 such that T< j+1(I) ∈ Σ j+1.

Since, for each j , (m1, . . . , ms+ j−1) (m1, . . . , ms+ j ) T< j+1(I), byNoetherianity, after a finite number of steps we obtain

• a basis G = f1, . . . , fr ,• terms mi , 1 ≤ i ≤ r ,• an infinite subset Σr+1 ⊂ in(I),• a term ordering <r+1 such that T<r+1(I) ∈ Σr+1,

such that

• mi = T<( fi ) for each i, 1 ≤ i ≤ r , and each term ordering < for whichT<(I) ∈ Σr+1,

• G is the Grobner basis of I w.r.t. <r+1.

This implies, for each term ordering < for which T<(I) ∈ Σr+1,

T<r+1(I) = T<r+1(G) = (m1, . . . , mr ) T<(I),

thus contradicting the consequence of Lemma 22.2.12

k[N<r+1(I)] ∼= P/I ∼= k[N<(I)].

Let us assume in(I) = M1, . . . , Mm and let us fix for i, 1 ≤ i ≤ m, a termordering <i such that L<i (I) = Mi and denote by Gi the reduced Grobnerbasis of I w.r.t. <i .

252 Grobner I

Corollary 24.10.6 (Bayer). Let I ⊂ P; there is D ∈ N such that deg(g) ≤ Dfor any term ordering <, and any polynomial g, which is a member of thereduced Grobner basis of I w.r.t. <.

Corollary 24.10.7. For any ideal I ⊂ P ,

G(I) := C(I, w) : w ∈ Rn

is a fan, the Grobner fan of I.

Proof. Finiteness being granted by Theorem 24.10.5, we have to prove theaxioms of being a fan; both are consequences of Corollary 24.10.4(3):

(1) Let F be a face of C(I, w) and let w′ be any vector in the relative interiorof F ; then, by Corollary 24.10.4.(3) F = C(I, w′) is a face of C(I, w).

(2) Let w, w′ ∈ Rn and consider

P := C(I, w) ∩ C(I, w′).

We have proved that for each w′′ ∈ P , C(I, w′′) is a face of both C(I, w)

and C(I, w′). Therefore P is a finite union of such common faces, but,since such union can only be convex, necessarily P = C(I, w′′).

Corollary 24.10.8 (Weispfenning). Each ideal I ⊂ P possesses a finite uni-versal Grobner basis G, that is a basis which is a Grobner basis of I with respectto any term ordering <, namely G :=⋃m

i=1 Gi .

Historical Remark 24.10.9. While Corollary 24.10.6 is an obvious conse-quence of Theorem 24.10.5 the original argument is upside-down: Bayer’s re-sult was deduced as a consequence of a delicate construction; Theorem 24.10.5was originally deduced as a trivial consequence of it: there is at most a finitenumber of monomial ideals generated by terms of degree bounded by D; oncethis was stated, Logar proposed his proof as an easy shortcut for Bayer’s result.

Logar’s proof and Weispfenning’s deduction and proof of Theorem 24.10.5are completely independent of each other; Logar’s proof seems to be a simpli-fied version of that of Weispfenning.

Weispfenning, in fact, discussed an algorithm to compute a universalGrobner basis which performed Buchberger’s algorithm branching when-ever a new basis element f = ∑

i ci ti is produced, in such a way that eachterm ti is chosen as leading term of f . The finiteness of this branching treeis then a consequence of Noetherianity and of the fact that a polynomial is afinite combination of terms.


Let us now impose the assumption that I is homogeneous; for each monomialideal Mi ∈ in(I), write, for each δ, 1 ≤ δ ≤ D,

Miδ := Mi ∩ Tδ = Xa11 . . . Xan

n ∈ Mi ,

n∑i=1

ai = δ,

Aiδ := (a1, . . . , an) : Xa11 . . . Xan

n ∈ Miδ ⊂ Nn,

wiδ :=∑

(a1,...,an)∈Aiδ

(a1, . . . , an) ∈ Nn .

Definition 24.10.10 (Bayer–Morrison). With this notation, the state polytopeof I is the Minkowski sum

P(I) =D∑

δ=1

Pδ(I) :=

D∑δ=1

pδ : pδ ∈ Pδ

of each convex hull

Pδ(I) :=

m∑i=1

λiwiδ : λi ∈ R, λi ≥ 0,

m∑i=1

λi = 1

.

Lemma 24.10.11. Let w, v ∈ Rn be such that

Lv(I) =: M j ∈ in(I) and Lw(I) =: Ml ∈ in(I);then, for each δ, 1 ≤ δ ≤ D, we have

(1) facev(Pδ(I)) = w jδ;(2) Lv(I) ∩ Tδ = Lw(I) ∩ Tδ ⇒ w jδ = wlδ.

Proof. Let us write N := Tδ \ M jδ; therefore, for each t ∈ M jδ we have

t − Can(t, I, < j ) = t −∑τ∈N

γ (t, τ, < j )τ ∈ I, Lv(t) = t,

so that

vv(t) > vv(τ ) for each t ∈ M jδ, τ ∈ N : γ (t, τ, < j ) = 0.

This implies that for any set M ⊂ Tδ for which #(M) = #(M jδ), M = M jδ ,one has

v · w jδ =∑

(a1,...,an)∈A jδ

v · (a1, . . . , an) =∑

t∈M jδ

vv(t) >∑t∈M

vv(t),

whence the claims.

Theorem 24.10.12 (Bayer–Morrison). For any homogeneous ideal I ⊂ PG(I) is the normal cone N (P(I)) of the state polytope of I.

254 Grobner I

Proof. Since all the faces of a fan are determined by the maximal faces, it issufficient to consider just any two vectors w, v ∈ R

n such that

Lv(I) =: M j ∈ in(I) and Lw(I) =: Ml ∈ in(I);for two such vectors we must prove that

Lv(I) = Lw(I) ⇐⇒ facev(P(I)) = facew(P(I)).

Since monomial ideals are equal iff they agree in each degree, we have

Lv(I) = Lw(I) ⇐⇒ Lv(I) ∩ Tδ = Lw(I) ∩ Tδ for each δ, 1 ≤ δ ≤ D;also, for each δ, 1 ≤ δ ≤ D,

Lv(I) ∩ Tδ = Lw(I) ∩ Tδ ⇒ w jδ = wlδ.

Therefore Lv(I) = Lw(I) implies

facev(P(I)) =D∑

δ=1

facev(Pδ(I))

=D∑

δ=1

w jδ

=D∑

δ=1

wlδ

=D∑

δ=1

facew(Pδ(I))

= facew(P(I)).

Conversely if Lv(I) = Lw(I), there exists some δ, 1 ≤ δ ≤ D, for which

Lv(I) ∩ Tδ = Lw(I) ∩ Tδ whence facev(Pδ(I)) = w jδ = wlδ = facew(Pδ(I));the uniqueness of the decomposition of the faces of a Minkowski sum thusimplies facev((P(I)) = facew((P(I)).

25

Gebauer and Traverso

25.1 Gebauer–Moller and Useless Pairs

From the first implementations, it became clear that the bottleneck of Buch-berger’s algorithm was the efficiency of the normal form computations. Thisled to Buchberger’s introduction of his criteria, to the informal notion of use-less pairs and to investigating efficient strategies in which to apply Buch-berger’s Criteria in order to detect useless pairs.

With this in mind, Gebauer and Moller’s investigation1 made clear that theefficiency of the normal form computation strongly depended on the orderingby which the S-polynomials were treated and on the implementation of theinstruction

Choose i, j ∈ B.

Example 25.1.1. An elementary example is the case

G := g1, g2, g3, g4 ∈ k[X1, X2, X3, X4]

whereg1 := X3 X4 − 1, g2 := X1 X3,

g3 := X1 X2 − 1, g4 := Xn1 .

If at any time we picked up the last pair i, j which had been inserted in Bour computation would look like

3, 4: S(3, 4) = Xn−11 := g5; G := G ∪ g5;

4, 5: S(4, 5) = 0;3, 5: S(3, 5) = Xn−2

1 := g6; G := G ∪ g6;

1 With their implementations in SAC2, REDUCE (≈ 1985) and SCRATCHPAD II, documentedin R. Gebauer and H. M. Moller A Fast Variant of Buchberger’s Algorithm. Preprint (1985), andR. Gebauer and H. M. Moller On an Installation of Buchberger’s Algorithm. J. Symb. Comp. 6,pp. 275–286 (1988).

255

256 Gebauer and Traverso

5, 6: S(5, 6) = 0;4, 6: S(4, 6) has a weak Grobner representation in terms of G because T(5) |

T(4, 6) and S(4, 5) and S(5, 6) have such a representation;3, 6: S(3, 6) = Xn−3

1 := g7; G := G ∪ g7;. . .

3, i: S(3, i) = Xn−i+31 := gi+1; G := G ∪ gi+1;

i, i + 1: S(i, i + 1) = 0;i − 1, i + 1: S(i − 1, i + 1) has a weak Grobner representation in terms of

G because T(i) | T(i − 1, i + 1) and S(i − 1, i) and S(i, i + 1) havesuch representations;

· · · j, i + 1: S( j, i+1) has a weak Grobner representation in terms of G because

T(i) | T( j, i+1) and S( j, i) and S(i, i+1) have such representations;· · ·

3, i + 1: S(3, i + 1) = Xn−i+21 := gi+2; G := G ∪ gi+2;

· · ·3, n + 3: S(3, n + 3) = 1 := gn+4; G := G ∪ gn+4;allowing us to deduce that the required Grobner basis is (1).

If, alternatively, any time we had picked up a pair i, j such thatdeg(T(i, j)) were minimal, our computation would have been:

1, 2: S(1, 2) = X1 := g5; G := G ∪ g5;2, 5: S(2, 5) = 0;3, 5: S(3, 5) = 1 := g6; G := G ∪ g6;obtaining immediately the required solution.

Buchberger suggested performing the instruction

Choose i, j ∈ B

in Algorithm 22.6.3 by choosing a pair i, j ∈ B such that deg(T(i, j)) isminimal;2 but this improvement of the normal form computation transferredthe bottleneck of Buchberger’s algorithm to the management of the set of thepairs in B, a set which is quadratic in the number of the elements in G andwhich had to be constantly re-ordered by increasing ordering of deg(T(i, j)).

In order to eliminate this bottleneck, Gebauer and Moller proposed to slimB down while performing the instruction

B := B ∪ i, s, 1 ≤ i < s

2 For a further discussion on that, compare Section 25.3.

25.1 Gebauer–Moller and Useless Pairs 257

by checking whether each pair i, s could be proved to be useless,3 in whichcase it would not be included in B.

The example discussed in Example 22.5.6 made clear that for such an ap-proach, aimed at detecting useless pairs before computing normal forms, it wasimpossible to take advantage of Lemma 22.5.3, which needed a deeper anal-ysis (see Corollary 25.1.6), and this gave a different reformulation of condi-tion G8.

This led to a drastic change of approach: while Buchberger criteria allowedto avoid a posteriori ‘useless’ S-pair computations whose possession of weakGrobner representation was granted by the previous computations of someS-pair normal forms (and perhaps the corresponding Grobner basis enlarge-ment) the aim now is to detect a priori a set, as minimal as possible, of ‘useful’S-pairs whose normal form computation is sufficient to either

• prove that the given basis is Grobner, or• extend the given basis G to a larger one G ′ in terms of which each S-pair

among the elements of G has a weak Grobner representation.

We will use freely the same notation and assumption as in Section 24.3 and,in particular, in Theorem 24.3.4.4

Moreover if we are given a finite basis

G := g1, . . . , gs ⊂ M ⊂ Pm

where we write, for each j, T(g j ) =: t j ei j , we will implicitly consider onlysubsets

(i, j, k, . . .), 1 < i < j < k < · · · ≤ s

such that

eii = ei j = eik = · · · =: ε,

and we will write

T(i, j, k, . . .) := lcm(ti , t j , tk, . . .).

3 Informally, in the lingo of the Buchberger implementation community, an S-pair is called ‘use-less’ if its normal form is 0; this definition, of course, can only be informal since it stronglydepends on the environment: a postponed S-pair will necessarily have a zero normal form, ifits computation is performed after all Grobner basis elements have been produced by previous‘useful’ S-pairs.In fact, while the emphasis has always been to discard useless pairs, the aim of good implemen-tations has always been to detect quickly a minimal set of useful S-pairs to which restrict theircomputation.

4 While I give the theory for the general case of modules, some statements hold only in the caseof an ideal, as will be either explicitly stated or implicitly implied by the assumption that m = 1.


Definition 25.1.2. A subset

GM ⊂ i, j, 1 ≤ i < j ≤ s, S(i, j) exists

is called a Gebauer–Moller set for G = g1, . . . , gs if for each i, j, 1 ≤ i <

j ≤ s, there exist

i1, j1, . . . , iρ, jρ, . . . , ir , jr , 1 ≤ iρ < jρ ≤ s,elements t1, . . . , tr ∈ T ,and coefficients c1, . . . , cr ∈ k,

such that

• S(i, j) = ∑ρ cρ tρ S(iρ, jρ);

• T(i, j) = tρT(iρ, jρ), for each ρ;• for each ρ, either

• iρ, jρ ∈ GM or• (in case M is an ideal) T(iρ, jρ) = T(iρ)T( jρ).


G7 For each i, j, 1 ≤ i < j ≤ m, the S-polynomial S(i, j) (if it exists) has aweak Grobner representation in terms of G.

G9 There is a Gebauer–Moller set GM for G such that for each i, j ∈ GM,S(i, j) has a weak Grobner representation in terms of G.

Proof. For each i, j, 1 ≤ i < j ≤ s, for which S(i, j) exists,

• i, j ∈ GM , and S(i, j) has a weak Grobner representation in terms of Gby assumption, or

• (M is an ideal and) T(i) T(j) = T(i, j), and S(i, j) has a weak Grobnerrepresentation in terms of G by Buchberger’s First Criterion, or

• a weak Grobner representation in terms of G of S(i, j) is obtained fromS(i, j) = ∑

ρ cρ tρ S(iρ, jρ) substituting for each S(iρ, jρ) their weakGrobner representation.

Lemma 25.1.4 (Moller). For each i, j, k : 1 ≤ i, j, k ≤ s we have

T(i, j, k)

T(i, k)S(i, k) − T(i, j, k)

T(i, j)S(i, j) + T(i, j, k)

T(k, j)S(k, j) = 0.


Proof. One has

T(i, j, k)

T(i, k)S(i, k) − T(i, j, k)

T(i, j)S(i, j) + T(i, j, k)

T(k, j)S(k, j)

= T(i, j, k)

T(i, k)

(T(i, k)

T(k)gk − T(i, k)

T(i)gi

)

− T(i, j, k)

T(i, j)

(T(i, j)

T( j)g j − T(i, j)

T(i)gi

)

+ T(i, j, k)

T(k, j)

(T(k, j)

T( j)g j − T(k, j)

T(k)gk

)

=(

T(i, j, k)

T(k)gk − T(i, j, k)

T(i)gi

)

−(

T(i, j, k)

T( j)g j − T(i, j, k)

T(i)gi

)

+(

T(i, j, k)

T( j)g j − T(i, j, k)

T(k)gk

)= 0.

Remark 25.1.5 (Gebauer–Moller). If, in the equation of Lemma 25.1.4 relat-ing three S-polynomials, at least one of the coefficients, say T(i, j, k)/T(i, j),is 1, then the corresponding S-polynomial S(i, j) is a combination of the othertwo S-polynomials; therefore it is sufficient to prove that S(i, k) and S( j, k)

have a weak Grobner representation, in order to deduce that the same alsoholds for S(i, j).

However, the example discussed in Example 22.5.6 shows that very oftenall the three coefficients are constant and in order to avoid aporetic loops onemust consider which of the possible S-polynomials should be considered to be‘useless’.

The solution is implicitly contained in Theorem 23.7.3 (see also Proposi-tion 24.5.4): one needs only to choose a set which is a basis of the syzygymodule Syz(T(g1), . . . , T(gs)).

In order to pick such a basis, it is sufficient to impose on the set

S(s) := (i, j), 1 ≤ i < j ≤ s, S(i, j) exists any ordering ≺, which is compatible with the term ordering < on T (m), that is

T(i1, j1) < T(i2, j2) ⇒ (i1, j1) ≺ (i2, j2),

and choose as ‘useless’ the biggest element among the possible choices.


We will therefore impose on S(s) the ordering ≺ defined by

(i1, j1) ≺ (i2, j2) ⇐⇒⎧⎨⎩

T(i1, j1) < T(i2, j2) orT(i1, j1) = T(i2, j2), j1 < j2 orT(i1, j1) = T(i2, j2), j1 = j2, i1 < i2.

(25.1)Let us assume that

i, k ≺ i, j, j, k ≺ i, j andT(i, j, k)

T(i, j)= 1;

therefore we have

T(i, j, k) = T(i, j), T(k) | T(i, j), T(i, k) | T(i, j), T( j, k) | T(i, j).

There are now three possible cases according to the position of k:

B: i < j < k,M′: i < k < j ,F′: k j then T(i, k) = T(i, j); similarly

( j, k) ≺ (i, j), k > j ⇒ T( j, k) = T(i, j);M′: (k, j) ≺ (i, j), i < k ⇒ T(k, j) = T(i, j);F′: k < i < j .

This simple remark yields

Corollary 25.1.6 (Buchberger’s Second Criterion (strong)). For i, j, 1 ≤i < j ≤ s, if there is k, 1 ≤ k ≤ s, such that

B: i < j < k, T(k) | T(i, j), T(i, k) = T(i, j) = T( j, k) orM: k < j , T(k, j) | T(i, j) = T(k, j) orF: k < i < j , T(k, j) = T(i, j),

then

S(i, j) = T(i, j, k)

T(i, k)S(i, k) + T(i, j, k)

T(k, j)S(k, j).

If, moreover, S(i, k) and S(k, j) have a weak Grobner representation interms of G, the same holds for S(i, j).


Proof. The case F′ can be split into two subcases:

• k < i < j , T(k, j) = T(i, j),• k < i < j , T(k, j) | T(i, j) = T(k, j).

The first case is F, while M is obtained by merging the second case and M′.Therefore the set of all triples (i, j, k) such that

i, k ≺ i, j, j, k ≺ i, j andT(i, j, k)

T(i, j)= 1

can be partitioned into the three cases B, M, F.And for each triple (i, j, k) in this set Lemma 25.1.4 proves the relation

S(i, j) = T(i, j, k)

T(i, k)S(i, k) + T(i, j, k)

T(k, j)S(k, j)

from which the statement on weak Grobner representations follows directly.

Definition 25.1.7 (Gebauer–Moller). An S-polynomial

S(i, j), 1 ≤ i < j ≤ s,

is called redundant if either

(1) there exists k > j such that T(i, j, k) = T(i, j), T(i, k) = T(i, j) =T( j, k), or

(2) there exists k < j : T( j, k) | T(i, j) = T( j, k).

Lemma 25.1.8.

R := i, j, 1 ≤ i < j ≤ s : S(i, j) is not redundantis a Gebauer–Moller set.

Proof. In order to prove the claim by induction, it is sufficient to show that, foreach i, j, 1 ≤ i < j ≤ s, such that S(i, j) is redundant, there are

i1, j1, . . . , iρ, jρ, . . . , ir , jr , 1 ≤ iρ < jρ ≤ s,elements t1, . . . , tr ∈ T , andcoefficients c1, . . . cr ∈ k

such that

• S(i, j) = ∑ρ cρ tρ S(iρ, jρ),

• T(i, j) = tρT(iρ, jρ), for each ρ,

• iρ, jρ ≺ i, j.


In order to show this, we only need to consider the representation

S(i, j) = T(i, j, k)

T(i, k)S(i, k) + T(i, j, k)

T(k, j)S(k, j)

and prove that

i, k ≺ i, j k, j;this holds (according to the two cases of the definition) because

(1) T(i, k) | T(i, j, k) = T(i, j) = T(i, k) implies i, k ≺ i, j and thesame argument proves j, k ≺ i, j;

(2) the same argument as that above proves j, k ≺ i, j, while i, k ≺i, j because T(i, k) ≤ T(i, j) and k < j.

Lemma 25.1.9. Let G := g1, . . . , gs and let

GM∗ ⊂ i, j, 1 ≤ i < j < sbe a Gebauer–Moller set for G∗ = g1, . . . , gs−1.

Let

T := T( j, s) : 1 ≤ j < sand let T′ ⊂ T be the set of the elements τ ∈ T such that either

• there exists τ ′ ∈ T : τ ′ | τ = τ ′ or• there (in case M is an ideal) exists iτ : 1 ≤ iτ < s, T(iτ )T(s) = T(iτ , s) =

τ .

For each τ ∈ T \ T′ let iτ , 1 ≤ iτ < s, be such that

T(iτ , s) = τ.

Then

GM := GM∗ ∪ iτ , s : τ ∈ T \ T′is a Gebauer–Moller set for G.

Proof. Let i < s, τ := T(i, s). Then:

• if there exists τ ′ ∈ T such that T(iτ ′ , s) = τ ′ | T(i, s) = τ ′, then sinceiτ ′ < s, S(i, s) is redundant;

• if i = iτ and T(iτ )T(s) = T(iτ , s), then (M is an ideal and) S(iτ , s) has aweak Grobner representation in terms of G by Buchberger’s First Criterion;

• if i = iτ and T(iτ )T(s) = T(iτ , s) then iτ , s ∈ GM ;


Fig. 25.1. Gebauer–Moller S-pair management

GM := SyzygyBasis(G,GM∗)where

G := (g1, . . . , gs) ⊂ P \ 0,G∗ = g1, . . . , gs−1,GM∗ ⊂ i, j, 1 ≤ i < j < s is a Gebauer–Moller set for G∗,GM ⊂ i, j, 1 ≤ i < j ≤ s is a Gebauer–Moller set for G

For each i, j ∈ GM∗ doIf T(i, j, s) = T(i, j), T(i, s) = T(i, j) = T( j, s), do

GM∗ := GM∗ \ i, j,S := i, s, 1 ≤ i < s, S∗ := ∅For each i, 1 ≤ i < s do

If there is j, 1 ≤ j < s : T( j, s) | T(i, s) = T( j, s), doS := S \ i, s,

T := T(i, s) : i, s ∈ SFor each τ ∈ T do

S(τ ) := i, s ∈ S : T(i, s) = τ ,If T(i, s) = T(i)T(s) for each i, s ∈ S(τ ), then

Choose i, s ∈ S(τ )S∗ := S∗ ∪ i, s

GM := GM∗ ∪ S∗

• if i = iτ then

S(i, s) = T(i, iτ , s)

T(i, iτ )S(i, iτ ) + S(iτ , s)

where S(i, iτ ) has the required term-bounded representation in terms ofGM∗.

Algorithm 25.1.10. The results of these two lemmata allowed Gebauer andMoller to devise the algorithm of Figure 25.1 which guarantees the neededmanagement of the set of the pairs in B, disposing of the corresponding bot-tleneck.

This algorithm became the central tool of an improved version of Buch-berger’s Algorithm implemented by Gebauer and Moller and which is sketchedin Figure 25.2.

The comparison between the original Buchberger algorithm (Figure 22.4)and Gebauer and Moller’s improvement (Figure 25.2) is dramatic: the maximalcardinality of B in the algorithm of Figure 25.2 is on average 10–20% of thatof Figure 22.4.5

5 It is worth noting that, in connection with Remark 22.6.2 and using the same assumptions andnotation, the trimming of B discussed there is automatically performed by SyzygyBasis: in factT( j, s) | T(i, j), for each j = i, j < s, and therefore either S(i, j) or S( j, s) is removedfrom B.The only thing to take care of is to remove i from J .


Fig. 25.2. Gebauer–Moller Scheme for Buchberger Algorithm

(G) := GrobnerBasis(F)where

F := g1, . . . , gs ⊂ P \ 0,lc(gi ) = 1, for each i,I is the ideal generated by (F),G is a Grobner basis of I;

G := g1, g2, B := ∅If T(1)T(2) = T(1, 2) then B := B ∪ 1, 2For each r, 3 ≤ r ≤ s do

G := G ∪ gr B := SyzygyBasis(G, B)

While B = ∅ doChoose i, j ∈ BB := B \ i, jh := S(i, j)(h,

∑mi=1 ci ti gi ) := NormalForm(h, G)

If h = 0 thens := s + 1, gs := lc(h)−1h, G := G ∪ gsB := SyzygyBasis(G, B)

Remark 25.1.11. In connection with Remark 24.3.5, the reader must be awarethat in the module case, since Buchberger’s First Criterion does not hold inFigure 25.1, the lines

For each τ ∈ T doS(τ ) := i, s ∈ S : T(i, s) = τ ,If T(i, s) = T(i)T(s) for each i, s ∈ S(τ ), then

Choose i, s ∈ S(τ )

S∗ := S∗ ∪ i, smust be replaced with

For each τ ∈ T doS(τ ) := i, s ∈ S : T(i, s) = τ ,Choose i, s ∈ S(τ )

S∗ := S∗ ∪ i, s


We are now able to present (in Figure 25.3) what essentially is the ‘standard’structure of Buchberger’s algorithm as can be found in most implementationsand to show its behaviour in an easy, but not trivial, example. The reader is


encouraged at least to consider the variation of the cardinality of B duringthe computation in order to appreciate the crucial role of Gebauer–Moller’simprovement.

Example 25.2.1. Let us consider the polynomial ring k[X, Y, Z , W, V ] and theideal

I := (V 2 − X Z , Y 2 − X3, Y Z V − X2W ),

and let us compute its Grobner basis under the lexicographical ordering <

induced by X < Y < Z < W < V .All through the computation, we will perform the Choose instruction o

choosing as i, j any pair such that T(i, j) is minimal under < and the leadingterm of the polynomial is marked in bold.

After renumbering the basis as

g1 := V2 − X Z , g2 := Y2 − X3, g3 := YZV − X2W,

since T(1, 2) = T(1)T(2), at the beginning we have

G := (g1, g2), B := ∅, J := 1, 2;considering g3 we obtain

G := (g1, g2, g3), B := 1, 3, 2, 3, J := 1, 2, 3.The computation of S(2, 3) gives

S(2, 3) = Y g3 − Z V g2 = X3ZV − X2Y W =: g4,

and B is modified – removing 2, 3 and adding 1, 4, 3, 4.6Then we have

S(3, 4) = Y g4 − X3g3 = −X2Y2W + X5W = −X2Wg2,

S(1, 4) = V g4 − X3 Zg1 = −X2YWV + X4 Z2 =: −g5,

so that B := 1, 3, 1, 5, 2, 5, 3, 5.7The computation

S(2, 5) = Y g5 − X2W V g2 = X5WV − X4Y Z2 =: g6

gives a new basis element and enlarges B adding 1, 6, 4, 6, 5, 6,8 sothat we have J := 1, 2, 3, 4, 5, 6, G := (gi , i ∈ J ) and

B := 1, 3, 1, 5, 3, 5, 1, 6, 4, 6, 5, 6.

6 T(2, 4) = T(2)T(4).7 T(4, 5) = XT(3, 5).8 Since T(2, 6) = Y T(5, 6), and T(3, 6) = Y T(4, 6).


Fig. 25.3. Buchberger’s Algorithm

(G) := GrobnerBasis(F)where

F ⊂ P \ 0,I is the ideal generated by (F),G is a Grobner basis of I;

While there exist g, h ∈ F : T(g) | T(h) doF := F \ h ∪ S(h, g)

G := F \ 0Re-order G =: g1, . . . , gs so that T(i) < T( j) ⇐⇒ i < j.For each i, 1 ≤ i ≤ s do

G := G \ gi , h := gi , gi := 0,While h = 0 do

If there exist t ∈ T , γ ∈ G : tT(γ ) = T(h) doh := h − lc(h)

lc(γ )tγ

Elseh := h − M(h), gi := gi + M(h)

gi := lc(gi )−1gi , G := G ∪ gi ,

G := g1, g2, B := ∅If T(1)T(2) = T(1, 2) then B := B ∪ 1, 2For each r, 3 ≤ r ≤ s do

G := G ∪ gr For each i, j ∈ B do

If T(i, j, r) = T(i, j), T(i, r) = T(i, j) = T( j, r), doB := B \ i, j,

S := i, r, i ∈ J , S∗ := ∅For each i ∈ J do

If there is j ∈ J : T( j, r) | T(i, r) = T( j, r), doS := S \ i, r,

T := T(i, r) : i, r ∈ SFor each τ ∈ T do

S(τ ) := i, r ∈ S : T(i, r) = τ ,If T(i, s) = T(i)T(s) for each i, s ∈ S(τ ), then

Choose i, r ∈ S(τ )S∗ := S∗ ∪ i, r

B := B ∪ S∗,For each i ∈ J do

If T(r) | T(i) doJ := J \ i, G := G \ gi ,

J := J ∪ rWhile B = ∅ do

o Choose i, j ∈ BB := B \ i, j, h := S(i, j)While T(h) ∈ T(G) doi Choose t ∈ T , γ ∈ G : tT(γ ) = T(h)

h := h − lc(h)tγIf h = 0 then

s := s + 1, gs := lc(h)−1h, G := G ∪ gsFor each i, j ∈ B do

If T(i, j, s) = T(i, j), T(i, s) = T(i, j) = T( j, s), doB := B \ i, j,


S := i, s, i ∈ J , S∗ := ∅For each i ∈ J do

If there is j ∈ J : T( j, s) | T(i, s) = T( j, s), doS := S \ i, s,

T := T(i, s) : i, s ∈ SFor each τ ∈ T do

S(τ ) := i, s ∈ S : T(i, s) = τ ,If T(i, s) = T(i)T(s) for each i, s ∈ S(τ ), then

Choose i, s ∈ S(τ )S∗ := S∗ ∪ i, s

B := B ∪ S∗, J := J ∪ sFor each i ∈ J do

If T(s) | T(i) doJ := J \ i, G := G \ gi .

The next computations give

S(5, 6) = Y g6 − X3g5 = −X4Y2Z2 + X7 Z2 = −X4 Z2g2

and

S(4, 6) = Zg6 − X2Wg4 = X4YW2 − X4Y Z3 =: g7;we have therefore to update B, adding 2, 7 and 5, 7;9

S(2, 7) = Y g7 − X4W 2g2 = X7W2 − X4Y 2 Z3 = X4 Z3g2 + X7W2 − X7 Z3

so that we set

g8 := X7W2 − X7 Z3, J := i, 1 ≤ i ≤ 8, G := (gi , i ∈ J )

and10

B := 1, 3, 1, 5, 3, 5, 1, 6, 5, 7, 6, 8, 7, 8.Then we have

S(7, 8) = 0

and

S(3, 5) = Zg5 − X2Wg3 = X4W2 − X4 Z3 =: g9.

Since T(7) = Y T(9) and T(8) = X3T(9) we have to trim J , getting

J := 1, 2, 3, 4, 5, 6, 9, G := gi : i ∈ J ,

9 T(1, 7) = V T(5, 7), T(3, 7) = T(4, 7) = ZT(5, 7) and T(6, 7) = XT(5, 7).10 T(1, 8) = V T(6, 8), T(2, 8) = Y T(7, 8), T(3, 8) = Y ZT(6, 8), T(4, 8) = ZT(6, 8), T(5, 8) =

V T(7, 8).


and to modify B by adding 11 4, 9, 6, 9, 7, 9, 8, 9 and removing6, 8.12

The insertion in B of the pairs 7, 9 and 8, 9 is needed, since we have tocompute the normal forms of g7 and g8 w.r.t. G;13 not withstanding we aresimply following our strategy to choose the S-pairs to be treated, the first onesto be considered are just those pairs:

S(8, 9) = X3g9 − g8 = 0,

S(7, 9) = Y g9 − g7 = 0.

The next S-pairs have the required representation:

S(6, 9) = X V g7 − Wg6 = −X5Z3V + X4Y Z2W = −X2 Z2g4,

S(5, 7) = V g7−X2Wg5 = −X4YZ3V+X6 Z2W = −XY Z2g4+X3 Z2Wg2,

while

S(4, 9) = Z V g9 − X W 2g4 = X3YW3 − X4 Z4V

= −X Z3g4 + X3YW3 − X3Y Z3W

gives the new basis element

g10 := X3YW3 − X3Y Z3W

and requires us to add to B = 1, 3, 1, 5, 1, 6 the new pairs 14 2, 10,5, 10, 9, 10 which give no new basis element, since

S(9, 10) = 0

and

S(2, 10) = Y g10 − X3W 3g2 = X6W3 − X3Y 2 Z3W = +X2Wg9 − X3 Z3Wg2,

S(5, 10) = V g10 − X W 2g5 = −X3YZ3WV+ X5 Z2W 2 = −X Z3g5 + X Z2g9.

It is now time to deal with the oldest S-pair listed, 1, 3, computing

S(1, 3) = V g3 − Y Zg1 = −X2WV + XY Z2 =: −g11,

11 T(1, 9) = T(1)T(9), T(2, 9) = Y T(7, 9), T(3, 9) = Z V T(7, 9), T(5, 9) = V T(7, 9).12 T(6, 8, 9) = T(6, 8), T(6, 9) = T(6, 8) = T(8, 9).13 In fact, all the computations performed up to now give us that each S-pair treated has a weak

Grobner representation in terms of G′ := gi : 1 ≤ i ≤ 9; we now need to be given that eachsuch S-pair also has a weak Grobner representation in terms of the subset G′′ := gi : 1 ≤ i ≤9, i = 7, 8.This is granted if we have a Grobner representation in terms of G′′ for both g8 and g9: infact, in order to produce the required weak Grobner representation in terms of G′′, it is suf-ficient to substitute within such representations each occurrence of g8 and g9 with a Grobnerrepresentation in terms of G′.

14 T(1, 10) = T(1)T(10), T(3, 10) = T(4, 10) = ZT(5, 10), T(6, 10) = X2T(5, 10).


which allows us to remove two other redundant elements – g5, g6 – from Gand also empty B,15 so that we have16

B := 1, 11, 4, 11, 5, 11, 6, 11, 9, 11.The first S-pair computations give

S(6, 11) = X3g11 − g6 = 0

and

S(5, 11) = Y g11 − g5 = −X Z2g2;the next one gives a new basis element:

S(4, 11) = X Zg11 − Wg4 = X2YW2 − X2Y Z3 =: g12,

allowing us to remove g10 from G, while B must be enlarged, giving 17

B := 9, 12, 2, 12, 10, 12, 9, 11, 11, 12, 1, 11.We leave to the reader the task of checking that these last S-pairs have therequired weak Grobner representation in terms of the computed Grobner basis

G := g1, g2, g3, g4, g9, g11, g12.

I chose this example (taken from an old hand computation I did around1986) because, while being a short example, it perfectly illustrates the com-binatorial behaviour of the algorithm: the constant increase of the basis sizeuntil the dramatical collapse at the latest stage of the computation,18 the erraticbehaviour of the size of B and the role of Gebauer–Moller management.19

15 Since

T(1, 5, 11) = T(1, 5), T(1, 11) = T(1, 5) = T(5, 11)

and

T(1, 6, 11) = T(1, 6), T(1, 11) = T(1, 6) = T(6, 11).

16 T(2, 11) = T(2)T(11), T(3, 11) = ZT(5, 11), T(10, 11) = X W 2T(5, 11).17 T(1, 12) = T(1)T(12), T(3, 12) = ZT(11, 12), T(4, 12) = X ZT(11, 12).18 The first 8 S-pair computations performed added 5 new basis elements; the next 13 S-pair

computations were needed to replace 4 of these elements (and a new fifth one) with the stillmissing 3 elements; at this stage we needed 6 further S-pair tests to conclude the algorithm.

19 We have dealt with 27 S-pairs but the largest size reached by B is just 8, achieved at the intro-duction of g9.

If we apply Buchberger criteria only instead of Gebauer–Moller management, after havingperformed the 10th S-pair computation – S(7,9) – the size of B is 16 and we have just applied theFirst Criterion 6 times and never applied the Second Criterion. We would only use that criterionjust before performing the next computation – S(6, 9) – in order to avoid the computation ofS(4, 5), and S(3, 6).


The reader however must be conscious that the computation we have pre-sented is quite idyllic in comparison with reality: we have only in fact pre-sented computations of Grobner bases of binomial ideals.20 In a normal casewhere the input basis consists of polynomials (even if sparse) and the coeffi-cients of each term are not trivial, we meet an obvious size explosion effect notdissimilar to the one I discussed about the Euclidean algorithm (Section 1.6)and due to the same reasons: the intermediate computations produce denserand denser polynomials with larger and larger coefficients.

Example 25.2.2. I do not have the faintest idea why I did that computation, butI know why I kept it: in fact realizing the huge number of redundant elementsproduced by that computation, I wondered whether it was possible to deal withthe S-pairs using a strategy different from the ‘standard’ one 21 in order toobtain fewer redundant elements.

Suitably re-ordering the computation was completely trivial;22 the conclu-sion, instead, was quite astonishing: I was computing a Grobner basis w.r.t.the lexicographical ordering induced by X < Y < Z < W < V and a min-imal computation – as I will show below – was obtained by choosing thosepairs i, j for which T(i, j) was minimal w.r.t. the lexicographical orderinginduced by X > Y > Z > W > V !

Well, I informed the friends who were working on improving the algorithm,I filed this curious example and moved to another computation.

In order to show this example, let us now perform the same computationusing this different strategy.

We start with the basis

f1 := V2 − X Z , f2 := Y2 − X3, f3 := YZV − X2W,

and the set 23 B := 1, 3, 2, 3 24

Then we have

• S(1, 3) = V f3 − Y Z f1 = −X2WV + XY Z2 =: − f4 and 25

B := 2, 3, 1, 4, 3, 4;20 That is ideals just generated by binomials, the polynomials having the shape t1 − t2, t1, t2 ∈ T .

Of course, the Grobner basis of a binomial ideal consists of binomials and the computationalgorithm produces binomials only.

21 Dealing with an S-pair i, j for which T(i, j) is minimal w.r.t. the ordering for which theGrobner basis is sought.

22 It is clear that we must choose any strategy which picks (1, 3) as the first S-pair, so that thefirst element added to the basis is g11 and when g5 and g6 are produced they are reduced to 0,thus also avoiding the production of g8 and g9; note also that the preliminary production of g11allows a fast production of g12, thus also avoiding the production of g10.

23 To help the reader we keep the pairs ordered according to the new strategy.24 Remember that T(1, 2) = T(1)T(2).25 T(2, 4) = T(2)T(4).

25.3 Traverso’s Choice 271

• S(2, 3) = Y f3 − Z V f2 = X3ZV − X2Y W =: f5, and 26

B := 1, 4, 3, 4, 1, 5, 4, 5, 3, 5;• S(1, 4) = −X Z f3;• S(3, 4) = Y Z f4−X2W f3 = X4W2−X4 Z3+X Z3 f2, f6 := X4W2−X4 Z3

and 27

B := 1, 5, 4, 5, 3, 5, 4, 6;• S(1, 5) = −Y f4 − X Z2 f2;• S(4, 5) = W f5 − X Z f4 = X2YW2 − X2Y Z3 =: f7 and 28

B := 4, 7, 2, 7, 3, 5, 4, 6, 6, 7;• all the remaing pairs – as the reader can easily check – have the required

representation.

The comparison between this and the previous computation – we dealt with11 (respectively 27) S-pairs producing no redundant element (respectively 5)– is a good introduction to the next section.

25.3 Traverso’s Choice

In Section 22.6, we mainly discussed the two While-loops of Algorithm 22.6.3and Figure 22.5, in order to deduce termination and complexity but wegave no thought to the corresponding Choose instructions controlling theloops.

The catastrophic effect of an unsuitable strategy for implementing theseChoose instructions has already been illustrated by Example 25.1.1 but thatexample was in fact a concocted one aimed at introducing Gebauer–Moller;instead, the example discussed in the section above dealt with a real compu-tation using a valid strategy, for which however an effective improvement wasavailable.

While such short, hand computations on binomial ideals can be easily per-formed by suitably adapting the strategies during the computation on the ba-sis of the partial outputs, non-trivial real-life computations – which are, aswe said above, vulnerable to polynomial densification and coefficient growth

26 T(2, 5) = T(2)T(5).27 T(i, 6) = T(i)T(6), 1 ≤ i ≤ 3 and T(5, 6) = ZT(4, 6).28 T(1, 7) = T(1)T(7), T(3, 7) = ZT(4, 7), T(5, 7) = X ZT(4, 7).


explosion – necessarily require machine computation and therefore a prioridetermination of the strategies for performing the Choose instruction.29

In order to determine a heuristically good strategy for the Choose instruc-tions, one needs extensive experiments on a large set of significant examples,being aware that the choices ‘can interract, the optimal choice may depend onthe term ordering and on the special form of the original basis’.30

In the late 1980s, Traverso designed and built a software system (AlPI)mainly aimed at allowing the performance of such deep experimental inves-tigation on a large (and increasing) set of test cases; the published conclusionsof this analysis have remained unchallenged.

• For theoretical reason we have always assumed that a given basis G := gi satisfies, for each i, lc(gi ) = 1. If we force this assumption in practice whenwe are given a basis G ⊂ Z[X1, . . . , Xn], the consequence is that all thecomputations must be performed over Q. Mutatis mutandis the situationis not dissimilar to that of the PRS computation (Section 1.6.1) and it isworth verifying whether it is better to perform all the computations over Z

by slightly modifying the basic instructions related to S-polynomials andrewriting rule reduction, that is replacing

S(g, f ) := lcm(T( f ), T(g))

T( f )f − lcm(T( f ), T(g))

T(g)g,

g := h − c(tT( f ), h)

lc( f )t f,

with

S(g, f ) := lcm(M( f ), M(g))

M( f )f − lcm(M( f ), M(g))

M(g)g,

g := lcm(c(tT( f ), h), lc( f ))

c(tT( f ), h)h − lcm(c(tT ( f ), h), lc( f ))

lc( f )t f,

29 The dream of producing a software adapting its strategies according to the partial outputs is ascience-fiction fantasy which, wisely, nobody has ever pursued.There are, however, specialized improved implementations for specific classes of ideals, forexample homogeneous or binomial ideals, which take advantage of the structure and propertiesof such classes.

30 From C. Traverso and L. Donato Experimenting the Grobner Basis Algorithm with AlPI Sys-tem. Proc. ISSAC ’89, ACM (1989), pp. 192–198, where these experiments are reported; furtherexperiments are discussed in A. Giovini et al. ‘One Sugar Cube, Please’ OR Selection Strategiesin the Buchberger Algorithm. Proc. ISSAC ’91, ACM (1991) pp. 49–54.The quotations of this section are taken from these papers.

25.3 Traverso’s Choice 273

suitably simplifying the output polynomials or some intermediate compu-tation polynomials by dividing them by the gcd of their coefficient.31 ‘Therational arithmetic is bad compared to integer arithmetic. This can be ex-plained since with rational arithmetic almost all coefficients of a polynomialhave often the same large denominator, wasting space and time.’

• Once an S-polynomial h is treated and a non-zero normal form g is pro-duced, is it better to perform a complete reduction in order to obtain thecanonical form of h to be added to the basis or limit oneself to adding theobtained normal form g to the basis? The latter ‘is bad compared to totalreduction, since it often causes more pairs to process, and especially highcoefficient growth’.

• Once a new polynomial is added to the basis it is better not to reduce the oldelements using the new one.

• The Choose instruction i (i.e. the choice of the basis element to be usedin a reduction step) was studied, keeping in mind particularly the potentialgrowth of the coefficient and the densification of the polynomials; Traversoconsidered several strategies:

• choosing the polynomial with a lesser number of monomials,• choosing the polynomial with the smallest (or largest) leading term,• choosing the polynomial according to its ‘age’ 32

and other more esoteric choices. The conclusions point towards the use ofthe polynomial with a lesser number of terms, but also confirm the wideconsensus for using the oldest polynomial.

• As regards the other central Choose instruction o (i.e. which S-pair to se-lect from B) Traverso supported the strategy implemented within the systemCoCoA and called there ‘sugar strategy’. To introduce it, we need a prelim-inary discussion: probably following Macaulay, the software dedicated tohim restricted itself to homogeneous ideals only; Grobner bases of the idealgenerated by F are then computed by applying, in increasing degree, theBuchberger algorithm to the homogeneous ideal h f : f ∈ F, thus obtain-ing a homogeneous basis G and returning ag : g ∈ G. The good aspectis that ‘it is experimentally known that in this [homogeneous] setting Buch-berger algorithm is less sensible to strategies’. The negative one is that ‘theGrobner basis of the ideal generated by the homogenized polynomials (thatis not a Grobner basis of the homogenized ideal 33) can be much larger than

31 Or by suitable, predetermined integers which have a high probability of dividing such gcd.32 With respect to its inclusion in the basis.33 Compare the relation between h I and I discussed in Section 23.1. Author’s Note.


the Grobner basis of the original ideal and can have components at infin-ity of large dimension (consider the extreme case 1 ∈ I)’. The basic ideaof the ‘sugar strategy’ is to ‘simulate the homogeneous algorithm in whatconcerns the selection strategy.’ This is performed by introducing a ‘phan-tom’ homogenization of all the polynomials in the Buchberger algorithm,defining for each polynomial f its Sugar S f , in the following way:

for the initial fi , S fi := deg( fi ) [ . . . ]if f is a polynomial and t a term, then St f := deg(t) + S f

if f = g + h, then S f := max(Sg, Sh).

To every polynomial ‘with sugar’ we can associate a homogeneous polyno-mial of degree equal to the sugar, homogenizing with an additional variableand multiplying with a suitable power of the same variable”. The ‘sugarstrategy’ chooses S-pairs in order to minimize the sugar of the correspond-ing S-polynomial and breaking ties with some other strategy; a good oneis the ‘normal selection strategy’ proposed by Buchbgerger, that is choos-ing a pair i, j which minimizes T(i, j) w.r.t. the ordering under which theGrobner basis is computed.

25.4 Gebauer–Moller’s Staggered Linear Bases and Faugere’s F5

As we noted in Remark 22.3.13, Gebauer and Moller proposed and expoundedthe argument, which I borrowed in my presentation of the Buchberger algo-rithm, as a tool to produce Grobner bases avoiding as much as possible reduc-tion of useless S-polynomials.

Definition 25.4.1 (Gebauer–Moller). Let I ⊂ P be an ideal, where T isordered by the well-ordering <.

A staggered linear basis B of I is the assignment of

• a finite basis g1, . . . , gs ⊂ I and• for each i a monomial ideal Ti ⊂ T

such that

B := tgi : t ∈ T \ Ti , 1 ≤ i ≤ s

is a Gauss basis of I.In particular,

(1) I = Spank(B),(2) for each f, g ∈ B, T<( f ) = T<(g) ⇒ f = g.

25.4 Staggered Linear Bases and F5 275

In order to describe their argument we need to temporarily relax their re-quirements, removing assumption (2):

Definition 25.4.2. Let I ⊂ P be an ideal.A Gebauer–Moller linear basis B of I is the assignment of

• a finite basis g1, . . . , gs ⊂ I and• for each i a monomial ideal Ti ⊂ T

such that B := tgi : t ∈ T \ Ti , 1 ≤ i ≤ s is a generating set of I.In particular I = Spank(B).

Note that all the generating sets produced in the discussion of Section 22.3are Gebauer–Moller linear bases.

Algorithm 25.4.3 (Gebauer–Moller). The algorithm by Gebauer and Moller(Figure 25.4), given a basis G := g1, . . . , gs of I, produces the staggeredlinear basis by:34

• starting with the Gebauer–Moller linear basis obtained by the assignment of

g1, . . . , gs, Ti := T(g j ), 1 ≤ j i

• whose normal form is computed only if 36

T(i, j)

T( j)/∈ T j and

T(i, j)

T(i)/∈ Ti ;

• moreover, the normal form computation is restricted so that g is replaced by

g − lc(g)

lc(gi )tgi , gi ∈ G, t ∈ T : T(g) = tT(gi )

only if t ∈ T \ Ti ;

34 This presentation is a polished version of their original result and some improvements are in-fluenced by Faugere’s ideas.

35 This algorithm must consider each S-polynomial: the approach is mutually exclusive withBuchberger’s Criteria or Gebauer–Moller sets.Or, better, there has never been research to clarify in which way and under which conditionsthe two approaches can be merged; the only known result is that merging the two approacheswithout a suitable restriction gives wrong answers.

36 In other words the S-polynomials S(i, j), j > i , for which either T(i, j)/T( j) ∈ T j orT(i, j)/T(i) ∈ Ti , are considered to be useless.


Fig. 25.4. Staggered Basis Algorithm

(g1, . . . , gs, T1, . . . , Ts) := Staggered Basis(F)where

F := (g1, . . . , gs) ⊂ P \ 0,I is the ideal generated by F ;B := tgi : t ∈ T \ Ti , 1 ≤ i ≤ s is a staggered basis of I.

G := F , T1 := ∅,For i = 2..s do

Ti := T(g j ), 1 ≤ j < iB := i, j, 1 ≤ i < j ≤ s, T(i, j)

T( j) /∈ T j While B = ∅ do

Choose i, j ∈ BB := B \ i, jτ := T(i, j)

T( j)

If τ ∈ T j and T(i, j)T(i) /∈ Ti then

h := S(i, j)While there exist l ≤ s, t ∈ T \ Tl : T(g) = tT(gl ) do

h := h − lc(g)lc(gl )

tgl

%% T(S(i, j)) ≥ T(h) and S(i, j) − h has a Gauss representation%% in terms of the generating set%% tgi : t ∈ T \ Ti , 1 ≤ i ≤ s

If h = 0 thens := s + 1, gs := lc(h)−1h, G := G ∪ gsTs := (

T j : τ) + (T(gi ) : 1 ≤ i < s)

B := B ∪ i, s, 1 ≤ i < s, T(i,s)T(s) /∈ Ts

T j := T j + (τ ).

• any time a normal form computation of S(i, j) is performed, giving a non-zero result h, gs+1 := h is included in G, associating to it the monomialideal

Ts+1 :=(

T j :T(i, j)

T( j)

)+

(T(g) : g ∈ G

);

• T j is enlarged with the inclusion of the generator T(i, j)/T( j) also in casethe normal form is 0;

• any time a new element gs+1 is added to the basis, the set B of theS-polynomials is enlarged by adding not the whole set i, s+1, 1 ≤ i ≤ sbut only the subset i, s + 1, 1 ≤ i ≤ s, T(i, s + 1)/T(s + 1) /∈ Ts+1.

The correctness of the algorithm is based on the following.


Lemma 25.4.4. Let I ⊂ P be an ideal; let (g1, . . . , gs) be a basis of it.The following hold:

(1) if Ti = T(g j ), 1 ≤ j < i for all i , then

B := tgi : t ∈ T \ Ti , 1 ≤ i ≤ sis a generating set of I;

(2) if

• Ti , i ≤ s, are monomial ideals,• B := tgi : t ∈ T \ Ti , 1 ≤ i ≤ s is a generating set of I,• gs+1 ∈ P is such that

S(i, j)) − gs+1 has a Gauss representation∑

h chth gih in termsof B,

T(S(i, j)) ≥ T(gs+1) /∈ tT(gi ) : t ∈ T \ Ti , 1 ≤ i ≤ s,• T(i, j)/T(i) /∈ Ti ,• τ := T(i, j)/T( j) ∈ T j ,

• Uh :=⎧⎨⎩

Th if 1 ≤ h ≤ s, h = j,Th + (τ ) if h = j,(T j : τ

) + (T(gi ) : 1 ≤ i ≤ s) if h = s + 1,

then

tgi : t ∈ T \ Ui , 1 ≤ i ≤ s + 1is a generating set of I;

(3) if Ti , 1 ≤ i ≤ s, are such that

B := tgi : t ∈ T \ Ti , 1 ≤ i ≤ s

is a Gauss basis of I, and there are j < l, ω ∈ T such that T(gl) =ωT(g j ), then, setting

Uh :=

Th if 1 ≤ h ≤ s, h = l, h = j,

T \(τ /∈ T j ∩ τω, τ /∈ Tl

)if k = j ,

B′ := tgi : t ∈ T \ Ui , 1 ≤ i ≤ s, i = l is a Gauss basis of I;(4) if Ti , 1 ≤ i ≤ s, are such that B := tgi : t ∈ T \ Ti , 1 ≤ i ≤ s is a

Gauss basis of I, then

gi : 1 ≤ i ≤ s : T(g j ) T(gi ), j < i

is a Grobner basis of I.


Proof.

(1) Setting gi = T(gi )+ri for each i , from T(g j )gi = gi g j −r j gi we deduce,for each t ∈ T , the Gauss representation

tT(g j )gi =∑τ∈T

c(τ, gi )tτg j −∑τ∈T

c(τ, r j )tτgi .

(2) From the Gauss representation

T(i, j)

T( j)g j − T(i, j)

T(i)gi =

∑k

cktk gik + gs+1

we deduce, for each t ∈ (T j : τ

), the Gauss representation

tT(i, j)

T( j)g j = t

T(i, j)

T(i)gi −

∑k

ckt tk gik + tgs+1.

(3) B′ is obtained from the Gauss basis B by substituting each element

τgl ∈ B, τ ∈ T \ Tl

with the element τωg j which satisfies τT(gl) = τωT(g j ) =: v.Moreover, each τgl − τωg j has a Gauss representation in terms of

γ ∈ B, T(γ ) < v.Therefore, denoting B′′ := tgi : t ∈ T \ Ti , 1 ≤ i ≤ s,i = l theinductive argument already used in Example 22.3.11 allows to deducethat, for each τ ∈ T \ Tl , the set

B′′ ∪ tgl : t ∈ T \ Tl , t > τ ∪ tωg j : t ∈ T \ Tl , t ≤ τ is a Gauss basis; thus proving the claim

(4) A direct consequence of Lemma 22.2.2

Example 25.4.5. Let P := k[X, Y, Z ] and T be ordered by the degrevlex or-dering < induced by X > Y > Z and let us compute a Grobner basis of theideal (g1, g2, g3) ∈ k[X, Y, Z ] where

g1 := X2Y − Z2, g2 := XZ2 − Y 2, g3 := YZ3 − X2

so that

T1 := ∅, T2 := X2Y , T3 := X2Y, X Z2.Following Gebauer and Moller’s proposal, we perform the Choose instruc-

tion by means of Buchberger’s normal selection strategy, that is choosing apair i, j which minimizes T(i, j) w.r.t. <.


B := 1, 2, 1, 3, 2, 3;2,3 : S(2,3) = Y3Z − X3 =: g4, T3 := X, T4 := XY, Z2;B := 1, 2, 1, 3, 1, 4, 2, 4;

1,2 : −S(1, 2) = XY3 − Z4 =: g5, T2 := XY , T5 := X, Y Z3, Y 3 Z;B := 1, 3, 1, 4, 2, 4, 2, 5, 3, 5, 4, 5;

4,5 : −S(4, 5) = Z5 − X4 =: g6, T5 := X, Z, T6 := X, Y 3, Y Z2;B := 1, 3, 1, 4, 2, 4, 2, 5, 3, 5, 3, 6;

3,6 : S(3, 6) + X2g1 = 0,37 T6 := X, Y ;1,3 : T(1, 3)/T(3) = X2 ∈ T3;2,4 : S(2, 4) = Y5 − X4 Z =: g7, T4 := XY, Z2, X Z, T7 := Y, Z;

B := 1, 4, 2, 5, 3, 5, 1, 7, 5, 7;2,5 : T(2, 5)/T(5) = Z2 ∈ T5;1,4 : −S(1, 4) = X5 − Y 2 Z3 =: g8, T4 := XY, Z2, X Z , X2,

T8 := Y, Z;B := 3, 5, 1, 7, 5, 7;

5,7 : −S(5, 7) = X5Z − Y 2 Z4 =: g9,38 T7 := X, Y, Z, T9 := Y, Z;B := 3, 5, 1, 7, 8, 9;

8,9 : T(8, 9)/T(8) = Z ∈ T8;39

3,5 : T(3, 5)/T(5) = Z3 ∈ T5;1,7 : T(1, 7)/T(7) = X2 ∈ T7.

In conclusion we have obtained

• the staggered basis

B := tg1, t ∈ T ∪ tg2, t ∈ T , XY t ∪ tg3, t ∈ T , X t∪

tg4, t ∈ 1, X, Z ∪ Y i , Y i Z , i ∈ N∪ Y i g5, i ∈ N ∪ Zi g6, i ∈ N ∪ g7∪ Xi g8, i ∈ N ∪ Xi g9, i ∈ N,

where we can replace Xi g9, i ∈ N with Xi Zg8, i ∈ N, and• the Grobner basis gi : 1 ≤ i ≤ 8by computing 1 useless S-pair, 5 useful S-pairs, 1 S-pair giving a redundantelement and performing just 1 reduction 40 to get a G-basis.

If we had performed Buchberger’s algorithm, we would have computed5 useful S-pairs and 7 useless S-pairs (whose reduction to zero requires 8

37 X2 /∈ T1 := ∅ so the reduction must be performed.38 Note that g9 = Zg8; however, the reduction is forbidden because Z ∈ T8 and Zg8 is not a

member of the Gebauer–Moller linear basis.39 Note that, for the first time, we are discarding this computation using the condition

T(i, j)/T(i) ∈ Ti , instead of T(i, j)/T( j) ∈ T j .40 S(3, 6) → S(3, 6) + X2g1 = 0; the reduction g9 → g9–Zg8 = 0 is made useless by the

theoretical argument of Lemma 25.4.4(3).


reductions); 8 pairs are removed via Buchberger’s Second Criterion and 8 byhis First Criterion.

In both cases we would need 2 further reductions in order to replace g8 withg8 + Y g3 + g1 = X5 − Z2 and to make the basis reduced.

Remark 25.4.6 (Faugere). If the basis (g1, . . . , gs) is a regular sequence,meaning that all the syzygies are generated by the trivial ones gi g j − g j gi =0, i < j, no useless S-pair normal form is performed.

More recently, Faugere, motivated by Remark 25.4.6, independently dis-covered, in the same frame of investigation, a completely different algorithm,which can be easily described as a variation of Algorithm 25.4.3, consisting oftwo crucial modifications of Gebauer and Moller’s proposal.

The first modification performs the Choose instruction with a completelydifferent strategy: while Gebauer and Moller proposed to perform the Chooseinstruction using what the general consensus of that time (1986) consideredthe best strategy,41 Buchberger’s normal selection strategy, Faugere’s strategycomputes, iteratively, a Grobner basis for each ideal 42 generated by the basis(g1, . . . , gσ ), 2 ≤ σ ≤ s, and performs the Choose instruction by picking upan S-pair i, j which minimizes deg(T(i, j)).43

Before presenting the other modification,44 which is the real turning-pointof Faugere’s algorithm, it is better to consider what happens if we perform thestaggered-bases algorithm with Faugere’s Choose strategy; I therefore performon the same example the variant presented in Figure 25.5.

Example 25.4.7. Let us therefore perform this algorithm on Example 25.4.5;we begin with h1 := g1 and h2 := g2 so that:

B := 1, 2, T2 := X2Y ;1,2 : −S(1, 2) = XY3 − Z4 =: h3 = g5; T3 := X; T2 := XY ,

B := 2, 3;2,3 : −S(2, 3) = Z6 − Y 5 =: h4 = Zg6 − g7; T3 := X, Z2; T4 := X,B := ∅.

41 And this consensus was confirmed by Traverso’s investigation. I have the impression thatapplying the sugar strategy, would not dramatically improve Gebauer and Moller’s algo-rithm.

42 This choice is obviously suggested by the aim of taking full advantage of Remark 25.4.6.43 The rationale of this choice will be clear when I discuss the more crucial modification of

Algorithm 25.4.3 performed by Faugere.44 There is also another modification which is needed in order to ensure the effectiveness of the

whole algorithm but that, in itself, has no proper effect. Namely, the algorithm must be appliedto the homogenization of the input basis.


Fig. 25.5. Staggered Basis Algorithm with Faugere’s Strategy.

(g1, . . . , gs, T1, . . . , Ts) := Staggered Basis++(F)where

F := (g1, . . . , gs) ⊂ P \ 0,I is the ideal generated by F ;Gσ is a Grobner basis of the ideal (g1, . . . , gσ ), 2 ≤ σ ≤ s,G := Gs is a Grobner basis of I.

h1 := g1, G1 = h1, r := 1, T1 := ∅,For σ = 2..s do

r := r + 1,hr := gσ , Gσ := Gσ−1 ∪ hr ,Tr := T(h j ), 1 ≤ j < r,B := i, r, 1 ≤ i < r, T(i,r)

T(r)/∈ Tr ,

While B = ∅ doChoose i, j ∈ B : deg(T(i, j)) = mindeg(T(l, k)) : (l, k) ∈ BB := B \ i, jτ := T(i, j)

T( j)

If τ ∈ T j and T(i, j)T(i) /∈ Ti then

h := S(i, j)

While exist l ≤ r, t ∈ T \ Tl : T(g) = tT(gl ) do h := h − lc(g)lc(gl )

tgl

If h = 0 thenr := r + 1, hr := lc(h)−1h, Gσ := Gσ ∪ hr Tr := (

T j : τ) + (T(gi ) : 1 ≤ i < r)

B := B ∪ i, r, 1 ≤ i < r, T(i,r)T(r)

/∈ Tr T j := T j + (τ ).

G := Gs

We have therefore obtained the Grobner basis h1, h2, h3, h4 of the sub-ideal (g1, g2). Then we add h5 := g3 and we obtain:

T5 := X2Y, X Z2, XY 3, Z6, B := 1, 5, 2, 5, 3, 5, 4, 5;2,5 : S(2, 5) = Y3Z− X3 =: h6 = g4; T5 := X, Z6, T6 := XY, Z2, Y 3;B := 1, 5, 3, 5, 4, 5, 1, 6, 2, 6, 3, 6;

3,6 : S(3, 6) = Z5 − X4 =: h7 = g6; T6 := X, Z2, Y 3; T7 := Y, Z2;B := 1, 5, 3, 5, 4, 5, 1, 6, 2, 6, 2, 7, 4, 7.

We have now five S-pairs i, j which minimize deg(T(i, j)) = 6, namely1, 5, 1, 6, 2, 6, 2, 7, 4, 7; while we can easily dispose of some of themby remarking that

1,5 : T(1, 5)/T(5) = X2 ∈ T5;1,6 : T(1, 6)/T(6) = X2 ∈ T6;2,6 : T(2, 6)/T(6) = X Z ∈ T6;

B := 3, 5, 4, 5, 2, 7, 4, 7.


we still need to make a choice between 2, 7 and 4, 7 and, as we will seesoon, such a choice produces different scenarios.

Let us begin with the wrong choice:

4,7 : S(4, 7) = −h4 + Zh7 = Y5 − X4 Z =: h8 = g7; T7 := Y, Z,T8 := Y, Z;

B := 3, 5, 4, 5, 2, 7, 1, 8, 3, 8;2,7 : −S(2, 7) = X5 − Y 2 Z3 =: h9 = g8; T7 := X, Y, Z, T9 := Y, Z;3,8 : −S(3, 8) = X5Z − Y 2 Z4 =: h10 = g9,45 T8 := X, Y, Z,

T10 := Y, Z;B := 3, 5, 4, 5, 1, 8, 9, 10;

9,10 : T(9, 10)/T(9) = Z ∈ T9;4,5 : S(4, 5) = Y6 − X2 Z3 =: h11,46 T5 := X, Z3, T11 := X, Z3;

B := 3, 5, 1, 8, 6, 11, 8, 11;8,11 : T(8, 11)/T(8) = Y ∈ T6;3,5 : T(3, 5)/T(5) = XY 2 ∈ T5;6,11 : T(6, 11)/T(6) = Y 3 ∈ T6;1,8 : T(1, 8)/T(8) = X2 ∈ T8.

In conclusion we have obtained the Grobner basis hi : 1 ≤ i ≤ 9, i = 4by performing no reduction 47 and computing 5 useful S-pairs, 2 S-pairs givingredundant elements and 1 giving a redundant element which is irredundant forthe sub-ideal (h1, h2).

If we instead make the good choice, the computation behaves as follows:

2,7 : −S(2, 7) = X5 − Y 2 Z3 =: h′8 = g8; T7 := X, Y, Z2, T8 := Y, Z2;

4,7 : S(4, 7) = −h4 + Zh7 = Y5 − X4 Z =: h′9 = g7;48

T7 := X, Y, Z, T9 := X, Y, Z;

45 As we have already remarked, h10 = Zh9, but the reduction is forbidden because Z ∈ T9 andZg9 is not a member of the Gebauer–Moller linear basis.

46 Again T(h11) = Y T(h8) but Y ∈ T8 and Y h8 is not a member of the Gebauer–Moller linearbasis.

47 We still need 2 reductions in order to replace h9 with h9 + Y h5 + h1 = X5 − Z2 and make thebasis reduced.

48 Note that the redundant element

h4 = S(h2, h3) = S(h2, S(h1, h2))

which is an irredundant element for the sub-ideal (h1, h2), and thus necessarily produced byFaugere’s strategy, is now disposed of in this computation which performs a Buchberger reduc-tion and produces the irredundant element g7.Within Gebauer and Moller’s strategy the corresponding computation

S(g2, g5) = S(g2, S(g1, g2))

is avoided since Z2 has been inserted in T5 by the previous computation of S(g4, g5) = g6because ZT(5) ∈ (T(4)).


B := 3, 5, 4, 5;4,5 : S(4, 5) = Y6 − X2 Z3 =: h′

10, T5 := X, Z3, T10 := X, Z3;B := 3, 5, 6, 10, 9, 10;

9,10 : T(9, 10)/T(9) = Y ∈ T9;6,10 : T(6, 10)/T(6) = Y 3 ∈ T6;3,5 : T(3, 5)/T(5) = XY 2 ∈ T5;

producing the Grobner basis hi : 1 ≤ i ≤ 9, i = 4 by performing no reduc-tion 49 and computing 5 useful S-pairs, 1 S-pair giving a redundant element and1 giving a redundant element which is irredundant for the sub-ideal (h1, h2).

As a consequence choosing the pair 2, 7 before 4, 7 allows us to avoid

In other words

within Faugere’s strategy: S(g2, g5) = S(h2, h3) = h4 = Zg6 + g7 and g7 is produced bythe reduction S(h4, h7) = h4 − Zg6 = g7;

within Gebauer and Moller’s strategy: g7 is produced as g7 = S(g2, g4) and

S(h2, h3) = S(g2, g5)

is avoided since

S(g2, g5) = Z2g5 − Y 3g2

= Z(Zg5 − Xg4) + X Zg4 − Y 3g2

= Z S(g4, g5) + S(g2, g4)

= −Zg6 + g7.

Let us finally remark that, in Faugere’s strategy S(h2, h6) = S(g2, g4) is avoided since XT(6) ∈(T(3)); the formula in this case is

S(g2, g4) = S(h2, h6)

= X Zh6 − Y 3h2

= Z(Xh6 − Zh3) + Z2h3 − Y 3h2

= Z S(h3, h6) + S(h2, h3)

= Z S(g5, g4) + S(g2, g5)

= Zg6 − (Zg6 − g7)

= g7.

The moral is that both algorithms apply differently the same relation

0 = S(g2, g5) − Z S(g4, g5) − S(g2, g4)

= S(h2, h3) + Z S(h3, h6) − S(h2, h6):

Faugere’s strategy computes S(h2, h3) and S(h3, h6) and uses them to avoid the useless compu-tation of S(h2, h6);

Gebauer and Moller’s strategy computes S(g2, g4) and S(g4, g5) and uses them to avoid the use-less computation of S(g2, g5).

49 We still have to count the 2 reductions needed to make the basis reduced.


the useless computation of the redundant element

h10 := −S(h3, h8) = −S(h3, S(4, 7)) = −S(g5, g7) = −S(h3, h′9).

This happens because the previous computation of S(2, 7) inserts X in T7 andso (when computing h′

9 := S(4, 7)) X in T9 thus not inserting in B the uselesspair 3, 9.

If, on the other hand, we first compute h8 := S(4, 7) then X is not yeta member of T7; therefore it is not inserted in T8 and we cannot detect theuselessness of 3, 8. When, in the next computation of S(2, 7), X is insertedin T7, is there a way to insert it in T8 also?

Faugere’s strategy provides an indirect way for doing that.

Faugere’s approach which aims to compute iteratively a Grobner basis ofeach sub-ideal Iσ := (g1, . . . , gσ ) has a direct consequence; when the Grobnerbasis of (g1, . . . , gσ−1) is computed and the next generator gσ is taken intoconsideration, each Gaussian reduction performed by Buchberger reduction isapplied only to elements

tgσ , t ∈ T , t /∈ T(Iσ ).It is therefore possible, for each new element hr , to track down the correspond-ing element tr gσ of which it is the Gaussian reduction.

Example 25.4.8. In the two computations of Example 25.4.7 we have:

• In the ‘good’ choice:

2,5 : h6 = Xh5 − Y Zh2 so that Xg3 → h6;3,6 : h7 = Xh6 − Zh3 so that X2g3 → Xh6 → h7;2,7 : h′

8 = −Xh7 + Z3h2 so that X3g3 → Xh7 → h′8;

4,7 : h′9 = Zh7 − h4 so that X2 Zg3 → Zh7 → h′

9;4,5 : h′

10 = Z3h5 − Y h4 so that Z3g3 → h′10.

It is possible to illustrate pictorially the situation in a similar way to that inExample 21.2.4 by giving two planes,50 the left one representing the mono-mials Xi Y j , (i, j) ∈ N

2, the right one the monomials Xi Y j Z , (i, j) ∈N

2, where each terms is marked by

if t ∈ T(I2), if tg3 is a member of the staggered basis of Iσ+1 produced by the algo-

rithm,r if, in the staggered basis computation, r is, equivalently,

50 We can of course give just a partial picture, omitting the terms Xi Y j Zh , h ≥ 2.


• the value such that tg3 has been Gaussian reduced to the stag-gered basis element (t/tr )hr ,

• the maximal value such that tr divides t ,• the single value such that t/tr ∈ Tr :

......

......

...

· · · · · · 6 · · · 6 · · · 6 7 8 8 · · ·

......

......

...

· · · · · · 6 · · · 6 · · · 6 9 9 9 · · ·

• In the ‘wrong’choice, we obtain

2,5 : h6 = Xh5 − Y Zh2 so that Xg3 → h6;3,6 : h7 = Xh6 − Zh3 so that X2g3 → Xh6 → h7;4,7 : h8 = Zh7 − h4 so that X2 Zg3 → Zh7 → h8;2,7 : h9 = −Xh7 + Z3h2 so that X3g3 → Xh7 → h9;3,8 : h10 = −Xh8 + Y 2h3;4,5 : h11 = Z3h5 − Y h4 so that Z3g3 → h11;

and the following picture, necessarily restricted to r ≤ 9:...

......

......

· · · · · · 6 · · · 6 · · · 6 7 9 9 · · ·

......

......

...

· · · · · · 6 · · · 6 · · · 6 8 · · ·

It is easy to realize that the picture does not have the same properties as inthe previous case, the crucial points are the term t = X3+i Z (marked by ),for which

• tg3 has been Gaussian reduced to the staggered basis element (t/t8)h8,• the maximal value such that tr divides t is r = 9,• both t/t8 ∈ T8 and t/t9 ∈ T9.

As a consequence this pictorial approach allows us to deduce a priori that

h10 ← Xh8 ← X3 Zg3 → Zh9

and so to mark as useless the S-pair 3, 8.Algorithm 25.4.9 (F5). I am now able to describe Faugere’s algorithm. As Ihave already said:


• the algorithm computes, iteratively, a Grobner basis (and the related stag-gered basis) of Iσ := (g1, . . . , gσ );

• in the σ th loop, the staggered basis

Bσ−1 := thi , t ∈ Ti , 1 ≤ i ≤ ρof Iσ−1, where Gσ−1 = h1, . . . , hρ, is enlarged to the Gebauer–Mollerlinear basis of Iσ B′ := Bσ−1 ∪ thρ+1, t ∈ N(Iσ−1) where hρ+1 := gσ ;

• any new element hr inserted in Gσ is the result of the Gaussian reduction ofan element tr hρ+1, tr ∈ N(Iσ−1);

• the input of the algorithm consists of homogeneous polynomials and theChoose instruction picks up an S-pair i, j minimizing deg(T(i, j));

• therefore, the algorithm produces a sequence of polynomials

hρ+1, . . . , hr , . . .

and a corresponding sequence of terms tρ+1 = 1, . . . , tr , . . . in N(Iσ−1)

which satisfies ti | t j ⇒ i < j, for each i, j ≥ ρ + 1;• this, in itself, is not sufficient, as Example 25.4.8 shows; what one needs is

just to renumber all polynomials of the same degree in order to be grantedthat ti < t j ⇐⇒ i < j, for each i, j ≥ ρ + 1;

• therefore if we set, for each i , i ≥ ρ + 1,

Fi := t ∈ N(Iσ−1) : t ti ∈ (ti+1, . . . , tr )we have

Fi ⊆ Ti , for i < r , whileFr ⊇ Tr ,⋃

i Fi = N(Iσ−1),Bσ−1 ∪ thi , t ∈ Fi , ρ + 1 ≤ i ≤ r is a Gebauer–Moller linear basis

of Iσ .

In conclusion, Faugere’s algorithm instead of explicitly constructing and us-ing the sets Ti makes implicit use of the sets Fi .

His algorithm is presented in Figure 25.6.

Example 25.4.10. Let us perform this algorithm on Examples 25.4.5and 25.4.7. Introducing T as homogenizing variable we impose on hT theordering <h which coincides with the degrevlex ordering < induced by X >

Y > Z > T and we consider in hP := K [T, X, Y, Z ], the ideal (hg1,hg2,

hg3).We begin by setting H1 := hg1 and H2 := hg2 so that:

B := 1, 2;1,2 : −S(1, 2) = XY3T − Z4T =: H3 = T hh3; t3 := XY, e3 = 2;


Fig. 25.6. F5 Algorithm

G := F5(F)where

F := (g1, . . . , gs) ⊂ hP \ 0,gi homogeneous;I is the ideal generated by F ;Gσ is a Grobner basis of the ideal Iσ (g1, . . . , gσ ), 2 ≤ σ ≤ s,G := Gs is a Grobner basis of I.

h1 := g1, G1 = h1, r := 1,For σ = 2..s do

r := r + 1hr := gσ , Gσ := Gσ−1 ∪ hr , tr := 1, er := σ

B := i, r, 1 ≤ i < r, T(i,r)T(r)

tr ∈ N(Iσ )While B = ∅ do

Choose i, j ∈ B :deg(T(i, j)) = mindeg(T(l, k)) : (l, k) ∈ BT(i, j)T( j) t j is <-minimal

B := B \ i, jτ := T(i, j)

T( j)If

τ t j ∈ (t j+1, . . . , tr ),T(i, j)T(i) ti ∈ (tι : ι > i, eι = ei ) and

T(i, j)T(i) ti ∈ T(Iσ )

thenh := S(i, j)While exists l ≤ r, t ∈ T \ Tl :

T(g) = tT(gl )t tl ∈ (tι : ι > l, eι = el ) ,

do h := h − lc(g)lc(gl )

tgl

If h = 0 thenr := r +1, hr := lc(h)−1h, Gσ := Gσ ∪hr tr := τ t j , er := σ

B := B ∪ i, r, 1 ≤ i < r, T(i,r)T(r)

tr ∈ N(Iσ )G := Gs

B := 2, 3;51

2,3 : −S(2, 3) = TZ6 − T 2Y 5 =: H4 = T hh4; t4 := XY Z2, e4 = 2;B := ∅;52

We have therefore obtained the Grobner basis H1, H2, H3, H4 of the sub-ideal (hg1,

hg2). Then we add H5 := hg3 and we obtain:

B := 1, 5, 2, 5, 3, 5, 4, 5;

51 (T(1, 3)/T(3))t3 = X2Y ∈ T(I1).52 T(i, 4)/T(4) ∈ (X), (T(i, 4)/T(4))t4 ∈ (X2Y ) ⊂ T(I1).


2,5 : S(2, 5) = TY3Z − T 2 X3 =: H6 = T hh6; t6 := X, e6 = 3;B := 1, 5, 3, 5, 4, 5, 1, 6, 2, 6, 3, 6;

1,5 : T(1,5)T(5)

t5 = X2 = Xt6;

3,6 : S(3, 6) = TZ5 − T 2 X4 =: H7 = T hh7; t7 := X2, e7 = 3;B := 3, 5, 4, 5, 1, 6, 2, 6, 2, 7, 4, 7;

2,6 : T(2,6)T(6)

t6 = X2 Z = Zt7;B := 3, 5, 4, 5, 1, 6, 2, 7, 4, 7;

4,7 :53 S(4, 7) = Y5T2 − X4 Z T 2 =: H8 = T 2 hh8; t8 := X2 Z , e8 = 3;B := 3, 5, 4, 5, 1, 6, 2, 7, 1, 8, 3, 8;

2,7 : −S(2, 7) = X5T2 − Y 2 Z3T 2 =: H9 = T 2 hh9; t9 := X3, e9 = 3;1,6 : T(1,6)

T(6)t6 = X3 = Xt7;

3,8 : T(3,8)T(8)

t8 = X3 Z = Zt9;54

3,5 : T(3,5)T(5)

t5 = XY 2 = Y 2t6;

4,5 : S(4, 5) = Y6T2 − X2 Z3T 3 =: H10 = T 2 hh11; t10 := Z3T, e10 = 3;B := 1, 8, 6, 10, 8, 10;

8,10 : (T(8, 10)/T(8))t8 = X2Y Z ∈ T(I2);1,8 : (T(1, 8)/T(8))t8 = X4 Z = X Zt9;6,10 : (T(6, 10)/T(6))t6 = XY 3T ∈ T(I2).

Thus, in comparison with the Gebauer–Moller Algorithm, which in this ex-ample computes 7 S-pairs (5 useful, 1 useless, 1 giving a redundant element)and 1 reduction, Faugere’s F5 computes 7 S-pairs (5 useful, 1 giving a re-dundant element and 1 giving a redundant element which is irredundant for asub-ideal) and no reduction.

53 4, 7 has been chosen before 2, 7 because X2 Z < X3.54 Thus the useless S-pair is detected and avoided.

26

Spear

Buchberger’s results, which are dated 1965 (his Ph.D. thesis) and 1970 (hisjournal publication), became known within the computer algebra commu-nity around 1976; at the same time David A. Spear was implementing, inMACSYMA, a package allowing the solution of ideal (and subring) theoreticalproblems within commutative rings.

This package was already ahead of most of the recent commonly used, spe-cialized software in commutative algebra, covering classes of rings which areeven only partially available in modern software: the classes of rings availablecovered at least quotient rings of a polynomial ring over any field representedin the Kronecker Model!1

While the report2 of this package contains no documentation, fortunatelymany of the ideas embedded there soon became available within the researchcommunity.3

In particular, Zacharias’ results (Section 26.1) hint that Spear’s notion of ad-missible rings required at least algorithms for syzygy computation, member-ship test and membership representation, the tool for lifting such algorithmsfrom R to R[X1, . . . , Xn] being essentially Grobner technology.4

The relation between Buchberger’s result and Spear’s own is presented bySpear in his report as follows:

1 The notion of ‘admissible ring’ introduced by Spear requires, among other things, that

• if R is admissible so is R[X ];• if R is admissible and I is a finitely generated ideal in R, then R/I is admissible.

2 D. A. Spear, A constructive approach to commutative ring theory, Proc. of the 1977 MACSYMAUsers’ Conference, (NASA CP-2012) (1977) 369–376.

3 Mainly through the MIT researchers with whom Spear cooperated while building his package.4 In his report, among the axioms defining the notion of admissible ring Spear quoted polynomial

extension and ideal quotienting. Zacharias’ results cover polynomial extension; as regards idealquotienting, Proposition 24.7.3 was ‘folklore knowledge’ from the 1980s, but I suspect that theresult stemmed from Spear.

289

290 Spear

The solution to each of the problems described above depends on a fundamental algo-rithm for expressing ideals in a canonical form. This algorithm appears to have beenfirst discovered by Buchberger . . . . My own version, independently obtained, is onlyslightly different from Buchberger’s; however, the difference is crucial – it is the key tosolving most of the problems listed in the previous section.

The list included, among other things, ideal operations, prime testing, syzygycomputation and subalgebra membership. The ‘crucial difference’ is explainedby Zacharias:5

Buchberger has shown how to construct a Grobner basis for any given ideal ink[X1, . . . , Xn] and how to use it to decide membership in an ideal.

David A. Spear has achieved most of these results independently. His initial defini-tion of Grobner bases differed from Buchberger’s principally in that it ordered polyno-mials lexicographically rather than by total degree.’

In more recent lingo, while Buchberger originally introduced his notion forthe deg-rev-lex term ordering case, Spear introduced it for the lexicographicalterm ordering.

The advantage is that the application of the lexicographical ordering allowedcomputation of the elimination ideals I ∩ k[X1, . . . , Xi ] (Section 26.2); whilethis achieved the computational aspects of Grobner’s proof of the Nullstel-lensatz (Section 20.3), Spear used it to compute ideal theoretical operations(Section 26.3) applying formulas such as:

∀I := ( f1, . . . , fs), J := (g1, . . . , gt ) ⊂ k[X1, . . . , Xn],

I ∩ J = L ∩ k[X1, . . . , Xn]

where

L = (f1T, . . . , fs T, g1(T − 1), . . . , gt (T − 1)

) ⊂ k[X1, . . . , Xn, T ].

Another crucial idea which was included in Spear’s software is the ‘tag-variable’ technique (Section 26.3): given a set of polynomials f1, . . . , fs ∈k[X1, . . . , Xn], we can consider the ideal

I := ( f1 − T1, . . . , fs − Ts) ⊂ k[X1, . . . , Xn, T1, . . . , Ts]

and any ordering < on k[X1, . . . , Xn, T1, . . . , Ts] such that

Xi > t, for each i and each term t ∈ k[T1, . . . , Ts];then any reduction of a polynomial g ∈ k[X1, . . . , Xn] by the Grobner basis

5 In G. Zacharias, Generalized Grobner bases in commutative polynomial rings B.Sc. thesis, MIT(1978), where a ‘private communication’ by Spear is listed in the bibliography.

26.1 Zacharias Rings 291

of I w.r.t. < will have the effect of replacing each instance of fi with Ti ; as aconsequence if the normal form of g is a polynomial

h(T1, . . . , Ts) ∈ k[T1, . . . , Ts]

then we can deduce that

g(X1, . . . , Xn) = h( f1, . . . , fs) ⊂ k[ f1, . . . , fs],

that is that g is a member of the subalgebra k[ f1, . . . , fs] ⊂ k[X1, . . . , Xn].Moreover, I ∩ k[T1, . . . , Ts] gives the ideal of all the relations among the fi s.We report also other applications of the tag-variable technique in a similar

mood presented by Shannon and Sweedler. Finally (Section 26.6), we discussa recent result by Traverso and Caboara which revives Spear’s technique oftag-variables as a tool to compute syzygies and resolutions.

26.1 Zacharias Rings

Definition 26.1.1. A ring Z with identity is called a Zacharias ring if it satisfiesthe following properties

(1) Z is a Noetherian ring;6

(2) there is an algorithm which, for each c ∈ Z , C := c1, . . . , ct ⊂Z \ 0, allows us to decide whether c ∈ (C) in which case it produceselements di ∈ Z : c = ∑t

i=1 ci di ;(3) there is an algorithm which, given c1, . . . , ct ⊂ Z \ 0, computes a

finite set of generators for the syzygy Z-module,(d1, . . . , dt ) ∈ Zt :

t∑i=1

di ci = 0

.

Following the work of Spear and Zacharias, we will consider a polynomialring P := R[X1, . . . , Xn] where R is a ring (not necessarily a field) withidentity, and we will adapt the definition of Grobner basis in this setting. Asbefore

T := Xa11 . . . Xan

n : (a1, . . . , an) ∈ Nn

6 A ring with identity Z is called Noetherian iff every strict ascending chain

a1 ⊂ a2 ⊂ · · · ⊂ ai ⊂ ai+1 ⊂ · · ·of ideals in Z is finite.

292 Spear

will denote the set of terms of P , which we will assume to be ordered by aterm ordering < and, for each polynomial f ∈ P ,

f :=∑

t∈=Tc( f, t)t, c( f, t) ∈ R,

we will set

T( f ) := max<

t ∈ T : c( f, t) = 0,lc( f ) := c( f, T( f )),

M( f ) := lc( f )T( f ).

Then we will call G ⊂ P a Grobner basis 7 w.r.t. < of the ideal I which itgenerates iff

MG := M(g) : g ∈ G generates M(I) := (M(g) : g ∈ I).

Moreover we will say that f ∈ P has a Grobner representation in terms ofG if there exist h1, . . . , hm ∈ P such that

f =∑

hi gi , T(hi gi ) ≤ T(h), for each i.

Our aim is to prove Zacharias’ result that if R is a Zacharias ring so isP; moreover not only has each ideal given by a finite basis a finite Grobnerbasis, but there is an algorithm which computes such a Grobner basis foran ideal presented via a finite basis; finally, both ideal membership andsyzygy computation will be computed using such Grobner bases in a stylenot dissimilar to the one discussed here and obviously linked to the LiftingTheorem (Theorem 23.7.3).

I will give here only a sketch of Zacharias’ results: the reader, with a goodunderstanding of the results discussed in Chapters 22 and 24, should be able tocomplete the arguments easily.

Let us begin with an elementary remark, which a fortiori holds in the ‘clas-sical’ set of Grobner theory:

Proposition 26.1.2 (Zacharias). Let R be any ring with identity.Let (g1, . . . , gm) ⊂ R, ( f1, . . . , fn) ⊂ R be such that

(g1, . . . , gm) = ( f1, . . . , fn).

Let, then

• xi j ∈ R be such that, for each i, gi = ∑nj=1 xi j f j ;

• y ji ∈ R be such that, for each j, f j = ∑mi=1 y ji gi ;

7 The definition we gave, in a setting in which R was a field, used the notion of leading term (T)while here, in a setting in which R is a ring, it uses the notion of leading monomial (M).

Of course, if R is a field the two notions of Grobner basis coincide since wlog lc(g) = 1, andT(g) = M(g), for each g ∈ G.


• (d(1), . . . , d(r)) ⊂ Rm, d(k) := (d(k)1 , . . . , d(k)

m ), be a basis of the syzygymodule

(d1, . . . , dm) ∈ Rm :m∑

i=1

di gi = 0

.

Using δi j , the Kronecker symbol, and

• ∂( j) := (∑mi=1 y ji xi1 − δ j1, . . . ,

∑mi=1 y ji xin − δ jn

) ∈ Rn, 1 ≤ j ≤ n,

• D(k) :=(∑m

i=1 d(k)i xi1, . . . ,

∑mi=1 d(k)

i xin

)∈ Rn, 1 ≤ k ≤ r ,

then

∂( j), 1 ≤ j ≤ n ∪ D(k), 1 ≤ k ≤ ris a basis of the syzygy module

(d1, . . . , dn) ∈ Rn :n∑

j=1

d j f j = 0

.

Proof. One has, for each j, 1 ≤ j ≤ n,

n∑l=1

(m∑

i=1

y ji xil − δ jl

)fl =

m∑i=1

y ji

n∑l=1

xil fl − f j =m∑

i=1

y ji gi − f j = 0,

and, for each k, 1 ≤ k ≤ r ,n∑

l=1

m∑i=1

d(k)i xil fl =

m∑i=1

d(k)i

n∑l=1

xil fl =m∑

i=1

d(k)i gi = 0.

Conversely let (d1, . . . , dn) ∈ Rn:∑n

j=1 d j f j = 0; then

0 =n∑

j=1

d j f j =n∑

j=1

m∑i=1

d j y ji gi =m∑

i=1

(n∑

j=1

d j y ji

)gi ,

and, by assumption, there exists (a1, . . . , ar ) ∈ Rr such that, for each i ,r∑

k=1

akd(k)i =

n∑j=1

d j y ji .

As a consequence, for each l, 1 ≤ l ≤ n,

dl = dl −m∑

i=1

(n∑

j=1

d j y ji −r∑

k=1

akd(k)i

)xil

=n∑

j=1

−d j

(m∑

i=1

y ji xil − δ jl

)+

r∑k=1

ak

(m∑

i=1

d(k)i xil

)

=n∑

j=1

−d j∂( j)l +

r∑k=1

akD(k)l .

294 Spear

Let R be any ring with identity. Of course, the proposition can be stated inmatrix terms as

Corollary 26.1.3. Let G := (g1, . . . , gm) ∈ Rm, F := ( f1, . . . , fn) ∈ Rn besuch that (g1, . . . , gm) = ( f1, . . . , fn).

Let, then, X, Y be the matrices such that X F = G, F = Y G.Let P be an m × r matrix such that

D ∈ Rm : DG = 0 = AP : A ∈ Rr .Then

D ∈ Rn : DF = 0 = B(Y X − I ) + AP X, A ∈ Rr , B ∈ Rn.

Let R be a Zacharias ring, let G = g1, . . . , gm ⊂ P\0 and let us considerthe modules Rm and Pm both of whose canonical bases we will denote bye1, . . . , em.

Let us now define a set S(G) as follows:

• consider the set T of all the least common multiples of the leading terms ofelements contained in any subset of G:

T := lcmT(h) : h ∈ H, H ⊆ G;• for any m ∈ T, define

• v(m) = (v(m)1, . . . , v(m)m) ∈ Rm the vector such that

v(m)i :=

lc(gi ) if T(gi ) | m0 otherwise;

• for each i, 1 ≤ i ≤ m, ti (m) :=

m/(T(gi )) if T(gi ) | m1 otherwise;

• C(m) ⊂ Rm a finite basis of the syzygy module(c1, . . . , cm) ∈ Rm :

m∑i=1

civ(m)i = 0

,

• S(m) := (c1t1(m), . . . , cmtm(m)) : (c1, . . . , cm) ∈ C(m);• S(G) := ⋃

m∈T S(m);• R(G) := ∑

i hi gi : (h1, . . . , hm) ∈ S(G).

With this notation:

Theorem 26.1.4 (Zacharias). Let R be a Zacharias ring.Let G = g1, . . . , gm ⊂ P \0, and let I be the ideal generated by G. Then

the following conditions are equivalent:


(1) G is a Grobner basis;(2) for each h ∈ P , either

• h ∈ I and h has a Grobner representation in terms of G, or• h ∈ I and there is g ∈ P \ 0 : M(g) ∈ M(I) and h − g has a

Grobner representation in terms of G;

(3) for each (h1, . . . , hs) ∈ S(G),∑

j h j g j has a Grobner representationin terms of G.

Proof.

(1) ⇒ (2) Let us prove the statement, by induction on T(h).

• If T(h) = 1, then M(h) = lc(h) ∈ R.

Setting H := i : gi ∈ G : T(gi ) = 1 we have

M(h) ∈ M(I) ⇐⇒ lc(h) ∈ (lc(gi ) : i ∈ H) .

By assumption, ideal membership and representation are solvablein the Zacharias ring R; therefore it is possible to decide whether

• lc(h) ∈ (lc(gi ) : i ∈ H) and M(h) ∈ M(I), in which case h /∈ Iand we are through setting g := h, or

• there is a representation lc(h) = ∑i∈H di lc(gi ) = ∑

i∈H di gi

from which, setting di := 0, for each i ∈ H , we obtain

h = M(h) = lc(h) =∑i∈H

di lc(gi ) =m∑

i=1

di gi .

• If T(h) = t > 1, let us inductively assume the claim holds for eachh′ : T(h′) < t.Let us now set H := i : gi ∈ G : T(gi ) | t, and, for each i ∈ H,

ti := t/(T(gi )).

Then we have

lc(h) ∈ lc(gi ), i ∈ H ⇐⇒ ∃ di ∈ R : lc(h) =∑i∈H

di lc(gi )

⇐⇒ ∃ di ∈ R : M(h)=∑i∈H

di ti M(gi )

⇐⇒ M(h) ∈ M(I).

Therefore, since ideal membership and representation are solvablein the Zacharias ring R, we can decide whether

• M(h) ∈ M(I), so that h /∈ I and we are through setting g := h,or

296 Spear

• M(h) ∈ M(I), in which case, setting di = 0, ti := 0, for eachi ∈ H and

h′ := h −m∑

i=1

di ti gi

we have

M(h) =m∑

i=1

di ti M(gi ) and T(h′) < T(h),

so that by induction h satisfies the required property. In particu-lar, either

h′ ∈ I and has the Grobner representation h′ := ∑mi=1 hi gi in

terms of G, so that h ∈ I too, having the Grobner representa-tion h := ∑m

i=1(di ti + hi )gi in terms of G, or h′ ∈ I, and there are g : M(g) ∈ M(I) and a Grobner repre-

sentation h′ − g = ∑mi=1 hi gi in terms of G, in which case

h ∈ I and h − g := ∑mi=1(di ti + hi )gi is the required Grobner

representation in terms of G.

(2) ⇒ (3) Obvious since for each (h1, . . . , hs) ∈ S(G),∑

j h j g j ∈ I.(3) ⇒ (1) Assume that G is not a Grobner basis; then there is a polynomial

h ∈ I such that M(h) ∈ M(G); since h ∈ I we know that there are hi

for which h = ∑mi=1 hi gi .

Among all possible representations

m∑i=1

hi gi such that M

(m∑

i=1

hi gi

)∈ M(G)

we choose one which minimizes maxT(hi )T(gi ). For such a repre-sentation let us write

h :=m∑

i=1

hi gi , t := maxT(hi )T(gi ), H := i : T(hi )T(gi ) = t.

Since M(h) ∈ M(G), necessarily T(h) < t and∑

i∈H M(hi )M(gi )

= 0.

If we set m := lcmT(gi ) : i ∈ H,

ci :=

lc(hi ) if i ∈ H ,0 otherwise

and ti (m) :=

m/(T(gi )) if T(gi ) | m,1 otherwise,


we have m | t,m∑

i=1

civ(m)i =m∑

i=1

ci lc(gi ) =∑i∈H

lc(hi ) lc(gi ) = 0,

m∑i=1

ci ti (m)T(gi ) =m∑

i=1

civ(m)i m = 0,

so that (t

m

)−1 ∑i∈H

M(hi )ei =(

tm

)−1 m∑i=1

ci T(hi )ei

=(

tm

)−1 m∑i=1

cit

T(gi )ei

=m∑

i=1

ci ti (m)ei

is a linear combination of the elements

σ :=m∑

i=1

cσ i ti (m)ei ∈ S(m)

and there are cσ ∈ R such that, for each i ,

ci T(hi ) = tm

ci ti (m) = tm

∑σ∈S(m)

cσ cσ i ti (m).

Then, if we set, for each i ,

ki := hi + tm

∑σ∈S(m)

cσ (hσ i − cσ i ti (m)) ,

where, for each σ ∈ S(m),∑m

i=1 hσ i gi is the Grobner representationof

∑mi=1 cσ i ti (m)gi , and

m∑i=1

(hσ i − cσ i ti (m)) gi = 0,

we havem∑

i=1

ki gi =m∑

i=1

hi gi +∑

σ∈S(m)

cσ

tm

m∑i=1

(hσ i − cσ i ti (m)) gi

=m∑

i=1

hi gi = h

298 Spear

and, for each i ,

M(hi ) = tm

∑σ∈S(m)

cσ cσ i ti (m) and T(hσ i ) < ti (m)

so that

maxT(ki )T(gi ) < maxT(hi )T(gi ),

contradicting the minimality of the representation h := ∑mi=1 hi gi .

Note that S(G) is a (not necessarily minimal) basis of the syzygy module(h1, . . . , hm) ∈ P :

∑j

h j M(g j ) = 0

;

the result however holds (and was stated by Zacharias) for any minimal basisS ′ ⊂ S(G) of such a syzygy module, as one can easily check by adapting theproof of (3) ⇒ (1).

This allows us to read condition (3) in Theorem 26.1.4 as another instanceof the syzygy lifting property. Analogously condition (2) is another instanceof the classical normal form property and its use for testing membership. As aconsequence:

Corollary 26.1.5. Let R be a Zacharias ring. If there is an algorithm which,for any finite set F = f1, . . . , fn ⊂ P \ 0, allows us to compute

• a Grobner basis G = g1, . . . , gm ⊂ P \ 0 such that (F) = (G), and• elements xi j ∈ R such that, for each i, gi = ∑

j xi j f j ,

then P is a Zacharias ring.

Proof.

(1) It is well known (see Lemma 27.1.5) that if R is Noetherian so is P =R[X1, . . . , Xn].

(2) By condition (2) of Theorem 26.1.4, given any polynomial h ∈ P , it ispossible to check whether h ∈ (F) = (G), in which case one computesa representation

h =m∑

i=1

hi gi =m∑

i=1

hi

n∑j=1

xi j f j =n∑

j=1

(m∑

i=1

hi xi j

)f j .

(3) For each σ = (c1t1, . . . , cmtm) ∈ S(G) one has∑

i ci ti gi ∈ (G), and,by condition (2) of Theorem 26.1.4, there are hi such that

∑i ci ti gi =

26.1 Zacharias Rings 299∑i hi gi ; therefore

∑i (ci ti − hi )gi = 0 and Σ(σ) := ∑

i (ci ti − hi )ei

is a syzygy among the elements of G.The proof of (3) ⇒ (1) of the Theorem 26.1.4 directly implies thatΣ(σ) : σ ∈ S(G) is a basis of the syzygy module

(h1, . . . , hs) :∑

i

hi gi = 0

.

Finally, Proposition 26.1.2 allows us to deduce the syzygies among theelements of F from those among the elements of G.

Noting that the implication (3) ⇒ (1) of Theorem 26.1.4 is essentiallythe formulation in this setting of Buchberger’s S-pair criterion, one can directlystate

Lemma 26.1.6. Let R be a Zacharias ring. There is an algorithm which, forany finite set F = f1, . . . , fn ⊂ P \ 0, allows us to compute

• a Grobner basis G = g1, . . . , gm ⊂ P \ 0 such that (F) = (G), and• elements xi j ∈ R such that, for each i, gi = ∑

j xi j f j .

Proof. The construction performed in the proof of (1) ⇒ (2) in Theo-rem 26.1.4 once it is applied to a polynomial h ∈ P using a (not necessarilyGrobner) basis F allows us to produce a ‘normal form’ N F(h, F) := g ∈ Psuch that

• h − g has a Grobner representation in terms of F ,• g = 0 ⇒ T(g) ∈ TF.

As a consequence if we set G0 := F and, for each i > 0,

Gi := Gi−1 ∪ N F(h, Gi−1) : h ∈ R(Gi−1) \ 0,then we obtain a sequence

G0 ⊆ G1 ⊆ · · · ⊆ Gi−1 ⊆ Gi ⊆ · · ·of bases of the ideal (F) and, at the same time, the sequence of ideals

M(G0) ⊆ M(G1) ⊆ · · · ⊆ M(Gi−1) ⊆ M(Gi ) ⊆ · · · ;since R, and so P , is Noetherian, there is a value ν such that, for each i > ν,M(Gi ) = M(Gν).

As a consequence, for each h ∈ R(Gν), N F(h, Gν) = 0, and, by condition(3) of Theorem 26.1.4, Gν is the required Grobner basis of (F); note thatGν = Gi , i > ν.

Corollary 26.1.7. If R is a Zacharias ring, so is R[X1, . . . , Xn].

300 Spear

26.2 Lexicographical Term Ordering and Elimination Ideals

Let R be a Zacharias ring 8 and let us consider the polynomial rings

R[Y] := R[Y1, . . . , Yd ],

R[Y][Z] := R[Y, Z] := R[Y1, . . . , Yd , Z1, . . . , Zr ]∼= R[X1, . . . , Xn] = P,

and the corresponding monomial semigroups

Y := Y a11 . . . Y ad

d : (a1, . . . , ad) ∈ Nd,

Z := Zb11 . . . Zbr

r : (b1, . . . , br ) ∈ Nr ,

T := Xc11 . . . Xcn

n : (c1, . . . , cn) ∈ Nn

= tY tZ : tY ∈ Y, tZ ∈ Z,where n = d + r and we identify P and R[Y, Z] by

Xi :=

Yi if i ≤ dZi−d if i > d.

Let us denote <Z a term ordering on Z and <Y a term ordering on Y andlet us consider the block ordering < on T inducing Y < Z, that is the onewhich, for each t (1), t (2) ∈ T , t (i) := t (i)Y t (i)Z , t (i)Y ∈ Y, t (i)Z ∈ Z, i = 1, 2, is de-fined by

t (1) < t (2) ⇐⇒ t (1)Z <Z t (2)

Z or t (1)Z = t (2)

Z and t (1)Y <Y t (2)

Y .

Note immediately that R[Y][Z] = R[Y, Z] can be interpreted as

(1) the polynomial ring in the variables Y1, . . . , Yd , Z1, . . . , Zr with coef-ficients in the ring R, or as

(2) the polynomial ring in the variables Z1, . . . , Zr with coefficients in thering R[Y]

and, according to those interpretations, we have, for a monomial

m := ctY tZ ∈ R[Y][Z] = R[Y, Z], c ∈ R, tY ∈ Y, tZ ∈ Z

(1) M<(m) = m, lc(m) = c, T<(m) = tY tZ ,(2) M<Z (m) = m, lc(m) = ctY , T<Z (m) = tZ ;

as shorthand we will denote this ring by

8 Most applications just require that R is a field or can be easily reduced to that setting.For instance (see Section 34.5) if p ⊂ P is a prime and R is the integral domain R = P/p

it is sufficient to consider its quotient field Q, apply the results to Q[Y, Z] and interpret it inR[Y, Z].

26.2 Elimination Ideals 301

(1) R[Y, Z] or(2) R[Y][Z]

according to the interpretation we use.

Example 26.2.1. We interpret f = 2Y Z + Z + Y ∈ Z[Y, Z ] as

(1) f = 2Y Z + Z + Y ∈ Z[Y, Z ], T<( f ) = Y Z , lc( f ) = 2,

(2) f = (2Y + 1)Z + Y ∈ Z[Y ][Z ], T<Z ( f ) = Z , lc( f ) = 2Y + 1.

Then:

Theorem 26.2.2. With the notation above, if I ⊂ R[Y][Z] is an ideal and Gis a Grobner basis of I w.r.t. < then

(1) G is a Grobner basis of I ⊂ R[Y][Z] w.r.t. <Z ;(2) G ∩ R[Y] is a Grobner basis of I ∩ R[Y] ⊂ R[Y] w.r.t. <Y .

Proof.

(1) Remarks that, for any f ∈ R[Y, Z] one has M<(M<Z ( f )) = M<( f ),

so that as a consequence we have

M<(M<Z (G)) = M<(G) = M<(I) = M<(M<Z (I)).

Let f ∈ I, and let

f =m∑

j=1

c j t jτ j gi , ci ∈ R\0, t j ∈ Y, τ j ∈ Z, gi , ∈ G,

be a Grobner representation of f ∈ R[Y, Z] in term of G, where wlog

T( f ) = t1τ1T(gi1) > t2τ2T(gi2) > . . . > tmτmT(gim );denoting µ ≤ m the highest value for which T<Z ( f ) = τ1T<Z (gi1) =τµT<Z (giµ), we have, in R[Y][Z],

M<Z ( f ) = lc( f )T<Z ( f ) =µ∑

j=1

c j t jτM<Z (gi j ).

(2) It is sufficient to remark that, according to the definition of <, for eachg ∈ R[Y, Z], T<(g) ∈ R[Y] ⇒ g ∈ R[Y].Therefore

T<(G ∩ R[Y]) = T<(G) ∩ R[Y] = T<(I) ∩ R[Y] = T<(I ∩ R[Y]).

Therefore G ∩ R[Y] is a Grobner basis of I ∩ R[Y] w.r.t. <. The con-clusion follows by the remark that < and <Y coincide in R[Y].

302 Spear

Corollary 26.2.3. With the notation above, G is a Grobner basis w.r.t. <Z of∑i

fi gi : fi ∈ R(Y)[Z], gi ∈ I

=: IR(Y)[Z] ⊂ R(Y)[Z].

If < represents the lexicographical ordering < on T induced by X1 < X2

< · · · < Xn and its restriction to each subset T [1, i] ⊂ R[X1, . . . , Xi ], then:

Corollary 26.2.4. If I ⊂ R[X1, . . . , Xn] is an ideal and G is a Grobner basisof I w.r.t. < then for each i, 1 ≤ i ≤ n, Gi := G ∩ R[X1, . . . , Xi ] is a Grobnerbasis of I ∩ R[X1, . . . , Xi ].

This result allows, through the computation of a Grobner basis w.r.t. thelexicographical ordering, the computation of all elimination ideals; however,practical experience indicates that the Grobner bases computation is much lessefficient w.r.t. the lex ordering than with other orderings (degrevlex being, alsofor theoretical reasons, one of the most efficient: see Theorem 38.5.12); tech-niques for producing indirectly a lex Grobner basis from the most efficientdegrevlex one have therefore been proposed (see Sections 29.2 and 29.5).

If one is interested in a single elimination ideal the following alternativeapproach can be applied.

Proposition 26.2.5 (Bayer–Stillman). Let I ⊂ k[X1, . . . , Xn] =: P be anideal, < be any term ordering on T , w := (w0, . . . , wn) ∈ R

n+1 \ 0 be theweight function where

w j :=

0 iff j ≤ i ,1 iff j > i ,

vw : hP −→ R be the valuation induced by vw(Xi ) = wi for each i , ≺ therefinement of vw with <h, G the Grobner basis of h I w.r.t. ≺ and H := ag :g ∈ G, vw(g) = 0.

Then H is the Grobner basis of I ∩ k[X1, . . . , Xi ] w.r.t. the restriction of <

to T [1, i].

Proof. In fact I ∩ k[X1, . . . , Xi ] = ah : h ∈ h I, vw(h) = 0.A stronger characterization of the lexicographical ordering < has been

pointed out by Kalkbrener. In order to present it, I need to introduce a suit-able notation:

for each polynomial f ∈ R[X1, . . . , Xi ],

f =δ∑

d=0

hd(X1, . . . , Xi−1)Xdi , hδ = 0,

26.2 Elimination Ideals 303

we write degi ( f ) := δ, Lp( f ) := hδ;9

for any set G ∈ R[X1, . . . , Xn] and each i, 1 ≤ i ≤ n, δ ∈ N,

Giδ := g ∈ G, g ∈ R[X1, . . . , Xi ], degi (g) ≤ δand

Lpiδ(G) := Lp(g), g ∈ Giδ.Theorem 26.2.6 (Kalkbrener). With the notation above, if

I ⊂ R[X1, . . . , Xn]

is an ideal and G a basis of I, then the following conditions are equivalent:

• G is a Grobner basis of I w.r.t. <,• Lpiδ(G) is a Grobner basis of Lpiδ(I) w.r.t. <, for each i, 1 ≤ i ≤ n, δ ∈ N.

Proof. If G is a Grobner basis of I w.r.t. <, then, for each i, 1 ≤ i ≤ n, δ ∈ N

and each f ∈ Iiδ there is a Grobner representation f = ∑j h j g j , T(h j g j ) ≤

T( f ); necessarily we have

h j g j ∈ R[X1, . . . , Xi ],h j = 0 ⇒ degi (g j ) ≤ degi (g j h j ) ≤ degi ( f ) ≤ δ,h j = 0 ⇒ g j ∈ Giδ ,Lp( f ) = ∑

j∈J Lp(h j ) Lp(g j ) where J := j : degi (g j h j ) = degi ( f ),which proves that each Lpiδ(G) is a Grobner basis of Lpiδ(I).

Conversely, assuming that each Lpiδ(G) is a Grobner basis of Lpiδ(I), let usconsider any f ∈ I and let i ≤ n, δ ∈ N be the values such that

f ∈ R[X1, . . . , Xi ] \ R[X1, . . . , Xi−1], δ := degi ( f ) > 0.

Then, by assumption, there is a Grobner representation

Lp( f ) =∑

j

Lp(h j ) Lp(g j ), g j ∈ Giδ ⊂ G,

and f ′ := f − ∑j h j g j ∈ I, degi ( f ′) < δ. An inductive argument therefore

allows us to produce a Grobner representation f ′ = ∑l hl gl , gl ∈ Giδ−1, and

the required Grobner representation f = ∑j h j g j + ∑

l hl gl .

This discussion will be taken further in Section 34.6.

9 Note that the notation does not assume f ∈ R[X1, . . . , Xi−1]; if f ∈ R[X1, . . . , Xi−1] wesimply write degi ( f ) := 0, Lp( f ) := f .

304 Spear

26.3 Ideal Theoretical Operation

The most direct application of Spear’s result is the production of techniques tocompute (the Grobner basis of) the result of ideal operations.

Definition 26.3.1. Let R be a ring with identity, a, b ⊂ R be ideals and f ∈R. The following ideal operations, whose results are ideals, are defined anddenoted as follows:

sum: a + b := a + b ∈ R : a ∈ a, b ∈ b;intersection: a ∩ b := c ∈ R : c ∈ a, c ∈ b;product: ab := ∑s

i=1 ai bi ∈ R : ai ∈ a, bi ∈ b;quotient: a : b := c ∈ R : cb ⊆ a = c ∈ R : for each b ∈ b, cb ∈ a;colon: a : f := a : ( f ) = c ∈ R : c f ∈ a;saturation: a : f ∞ := c ∈ R : there exists ρ ∈ N, c f ρ ∈ a;ideal saturation: a : b∞ := c ∈ R : there exists ρ ∈ N, cbρ ⊆ a =

∪∞i=1

(a : bi

).

The ideal definitions above are generalizations of the classical operationsamong the (generators of the) ideals of a principal ideal domain. Let us notethat in a principal ideal domain

• ( f ) + (g) = gcd( f, g);• ( f ) ∩ (g) = lcm( f, g);• ( f )(g) = ( f g);• ( f ) : (g) = ( f/ gcd( f, g)).

Note that the principal ideal domain formula

gcd( f, g) lcm( f, g) = f g

does not hold in this generalization since we have just the inclusion (see con-dition (14) in Theorem 26.3.1 below)

(a ∩ b)(a + b) ⊆ ab.

When R := k[X1, . . . , Xn], all these operations can be interpreted by takinginto consideration the effect they have on the associated varieties:

• The sum (respectively: intersection) of two ideals is associated with the va-riety which is the intersection (respectively: union) of the correspondingassociated varieties:

Z(a + b) = Z(a) ∩ Z(b),Z(a ∩ b) = Z(a) ∪ Z(b).

26.3 Ideal Theoretical Operation 305

• The colon operation a : f has a geometrical meaning when a is radical, inwhich case it removes from the variety Z(a) those components contained inthe hypersurface f = 0.

• In the same way, when a is radical, the quotient a : b removes from thevariety Z(a) the components contained in the variety Z(b).

• The saturation a : f ∞ and the ideal saturation a : b∞ produce the sameeffects as the colon (respectively: quotient) without the requirement of rad-icality, that is Z(a : f ∞) (respectively: Z(a : b∞)) is the variety obtainedwhen all the components contained in the hypersurface f = 0 (respectively:in the variety Z(b)) are removed from the variety Z(a).

Let us begin by recalling the elementary relations between these operations:

Theorem 26.3.2. Let R be a ring with identity, f ∈ R, and a, b, c ⊂ R beideals.

Let a1, . . . , am and b1, . . . , bn be bases of (respectively) a and b. Then

(1) a1, . . . , am, b1, . . . , bn is a basis of a + b;(2) a + b = b + a;(3) (a + b) + c = a + (b + c);(4) a ∩ b = b ∩ a;(5) (a ∩ b) ∩ c = a ∩ (b ∩ c);(6) (a ∩ b) + c ⊆ (a + c) ∩ (b + c);(7) (a + b) ∩ c ⊇ (a ∩ c) + (b ∩ c);(8) b ⊆ c ⇒ (a ∩ c) + b = (a + b) ∩ c;(9) ai b j , 1 ≤ i ≤ m, 1 ≤ j ≤ n is a basis of ab;

(10) ab = ba;(11) (ab)c = a(bc);(12) a(b + c) = ab + ac;(13) ab ⊆ a ∩ b;(14) (a ∩ b)(a + b) ⊆ ab;(15) b(a : b) ⊆ a; (ab) : b ⊇ a;(16) bc ⊆ a ⇒ b ⊆ a : c, c ⊆ a : b;(17) a : (b + c) = (a : b) ∩ (a : c); (a + b) : c ⊇ (a : c) + (b : c);(18) a : b = ⋂n

i=1(a : bi );(19) (a ∩ b) : c = (a : c) ∩ (b : c);(20) a : (bc) = (a : b) : c;(21) (a : b) + c ⊆ (a + bc) : b;(22) (a : f ) + c = (a + f c) : f ;(23) writing d := a : (a : b), we have (a : b) = (a : c) ⇒ d ⊃ c.

Proof. We focus on the non-trivial statements:

306 Spear

(8) Since (7) implies

(a + b) ∩ c ⊇ (a ∩ c) + (b ∩ c) = (a ∩ c) + b

we have to prove only the converse inclusion.So let a ∈ a, b ∈ b ⊆ c be such that d := a + b ∈ c. Then a =d − b ∈ c, so that a ∈ a ∩ c and d = a + b ∈ (a ∩ c) + b.

(14) If d ∈ (a ∩ b)(a + b), there exist ci ∈ a ∩ b, ai ∈ a, bi ∈ b such that

d =∑

i

ci (ai + bi ) =∑

i

ai ci +∑

i

ci bi ∈ ab.

(17) d ∈ (a : b) and d ∈ (a : c) iff for each b ∈ b, c ∈ c, db, dc ∈ a iffd ∈ a : (b + c).

(18) This is a direct consequence of an iterative application of (17).(20) For d ∈ R we have

d ∈ a : (bc) ⇐⇒ for each b ∈ b, c ∈ c, dbc ∈ a

⇐⇒ for each c ∈ c, dc ∈ a : b

⇐⇒ d ∈ (a : b) : c;

(21) (17) – with b := b and c := b – and (15) – with a := c – imply

(a + bc) : b ⊇ (a : b) + (bc : b) ⊇ (a : b) + c.

(22) We have just to prove (a + f c) : f ⊆ (a : f ) + c.

So let d ∈ (a+ f c) : f and let c ∈ c, a ∈ a be such that d f = a +c f .Then

(d − c) f = a ∈ a, d − c ∈ a : f, d = (d − c) + c ∈ (a : f ) + c.

(23) Let c ∈ c; then for each f ∈ (a : b) = (a : c), f c ∈ a and c ∈ a :(a : b) = d.

We intend now to discuss how, given a Zacharias ring R, two ideals a, b ⊂ Rthrough a basis and an element f ∈ R, to compute the result of the idealoperations listed above.

As regards sum and product operations, the basis structures are already de-scribed in Theorem 26.3.2(1) and (9).

The computation of the intersection can be obtained by:

Lemma 26.3.3 (Spear). Let R be a Zacharias ring. Let a, b ⊂ R be idealsand let a1, . . . , am and b1, . . . , bn be bases of, respectively, a and b.


Let c ⊂ R[T ] be the ideal generated by the basis

(a1T, . . . , am T, b1(T − 1), . . . , bn(T − 1)) .

Then

a ∩ b = c ∩ R.

Proof. Let f ∈ a ∩ b; then there are ci , d j ∈ R such that f = ∑mi=1 ci ai =∑n

j=1 d j b j . Therefore

f = f T − f (T − 1) =m∑

i=1

ci ai T −n∑

j=1

d j b j (T − 1) ∈ c ∩ R.

Conversely let f ∈ c ∩ R and let ci , d j ∈ R, ei , f j ∈ R[T ] be such that

f =m∑

i=1

(ci + (T − 1)ei )ai T +n∑

j=1

(d j + T f j )b j (T − 1).

Then equating the quotient and the remainder of both sides of the equation byT we get f = −∑n

j=1 d j b j and

0 =m∑

i=1

(ci + (T − 1)ei )ai +n∑

j=1

d j b j +n∑

j=1

f j b j (T − 1).

It is then sufficient to take the remainder by the division of this expression by(T − 1), to obtain

m∑i=1

ci ai = −n∑

j=1

d j b j = f

and to prove that f ∈ a ∩ b.

A different algorithm had already been proposed by Hilbert:10

Lemma 26.3.4 (Hilbert). With the notation above, let

s1, . . . , st , sk := (ck1, . . . , ckm, dk1, . . . , dkn)

be a basis of the syzygy module

S :=

(ci , . . . cm, d1, . . . dn) :m∑

i=1

ci ai −n∑

j=1

d j b j = 0

.

Then∑m

i=1 cki ai , 1 ≤ k ≤ t

is a basis of a ∩ b.

10 In D. Hilbert, Uber die Theorie der algebraicschen Formen, Math. Ann. 36 (1890), p. 517.

308 Spear

Proof. It is sufficient to remark that

a ∩ b =

m∑i=1

ci ai : (ci , . . . , cm, d1, . . . , dn) ∈ S

.

The quotient operation can be reduced to the colon one in different ways:

Proposition 26.3.5. Let R be a Zacharias ring. Let a, b ⊂ R be ideals and leta1, . . . , am and b1, . . . , bn be bases of, respectively, a and b.

The following hold:

(1) a : b = ⋂ni=1(a : bi ).11

(2) Let a′ ⊂ R[T ] be the ideal generated by a1, . . . , am and let

b :=n∑

i=1

bi Ti−1 ∈ R[T ];

then

a : b = (a′ : b

) ∩ R.

(3) Let b be a (random) element of b and let c := a : b. Then

cb ⊆ a ⇒ c = a : b.

Proof.

(1) Compare Theorem 26.3.2(18).(2) Let g ∈ R be such that gb = ∑n

i=1 gbi T i−1 ∈ a′. Then there arec j (T ) ∈ R[T ] such that

n∑i=1

gbi Ti−1 =

m∑j=1

ci (T )a j .

11 If one computes a : b by performing the recursive computation

N⋂i=1

(a : bi ) =(

N−1⋂i=1

(a : bi )

)∩ (a : bN ),

it is better to avoid the useless and time consuming computation of (a : bN ) when

N−1⋂i=1

(a : bi ) ⊆ (a : bN ).

Therefore it is advisable to first test the easier condition

bN

N−1⋂i=1

(a : bi ) ⊆ a,

which is equivalent to⋂N−1

i=1 (a : bi ) ⊆ (a : bN ), thereby skipping the useless computation of(a : bN ) if the test is positive.


Since a j ∈ R, it is sufficient to equate each coefficient of the powersof T in the equation above, in order to prove that, for each i, gbi ∈ a,that is g ∈ (a : bi ).

(3) The claim follows, since trivially a : b ⊆ a : b = c, while the testcb ⊆ a implies a : b ⊇ c.

As regards the colon operation, there are different ways to perform it. Spearreduced the problem to computing the intersection, which can be done eitherby elimination or by syzygy computation:

Lemma 26.3.6. Let R be a Zacharias ring. Let a ⊂ R be an ideal and letf ∈ R,. Then, for each g ∈ R,

g ∈ a : f ⇐⇒ g f ∈ a ∩ ( f ).

Corollary 26.3.7. Let R be a Zacharias ring. Let a ⊂ R be the ideal generatedby a1, . . . , am and let f ∈ R. Then

(1) Let c1 f, . . . , ct f be a basis of a∩ ( f ). Then c1, . . . , ct is a basis ofa : f.

(2) Let s1, . . . , st , sk := (ck, dk1, . . . , dkm) be a basis of the syzygymodule

S :=

(c, d1, . . . , dm) : c f −m∑

i=1

di ai = 0

.

Then c1, . . . , ct is a basis of a : f.

In a similar mood one has also

Lemma 26.3.8. With the same notation as above, let c ⊂ R[T ] be the idealgenerated by a1T, . . . , am T, 1 − f T . Then a : f = c ∩ R.

Proof. If g ∈ a : f , then there are di such that g f = ∑i di ai ; therefore

g = g(1 − f T ) + g f T = g(1 − f T ) +∑

i

di ai T .

Conversely, if g ∈ R and g = c(1 − f T ) + ∑i di (T )ai T, we obtain, by

equating the coefficients of T , g = c, c f = ∑i di (0)ai , that is g ∈ a : f .

Saturations can be reduced to colon and quotient operations by means of

Lemma 26.3.9. Let R be a Zacharias ring. Let a, b ⊂ R be ideals and writeci := a : bi . Then

310 Spear

(1) for each i, ci+1 = ci : b,(2) there exists i such that ci = c j , for j ≥ i ,(3) for the minimal such i , one has a : b∞ = ci .

Proof.

(1) Compare Theorem 26.3.2(20).(2) Clearly, for each i, ci ⊆ ci+1. Therefore there is an infinite sequence

c1 ⊆ c2 ⊆ · · · ⊆ ci ⊆ ci+1 ⊆ · · ·

Since R is Noetherian, this implies the existence of i such that ci = c j ,

for each j ≥ i.(3) Therefore a : b∞ = ⋃

j c j = ci .

On the basis of the proof above, there are obvious ways to reduce the com-putation of a : b∞ to the quotient/colon computation:

• repeatedly compute c1, c2, . . . , ci , ci+1, . . . until ci = ci+1 in which casea : b∞ = ci = ci+1;

• repeatedly compute cn1 , cn2 , . . . , cni , cni+1 , where n1, n2, . . . , ni , . . . is anincreasing sequence of integers, until cni = cni+1 in which case a : b∞ =cni = cni+1;

• repeatedly choose two large values N1, N2 and compute cN1 and cN2 untilthey are equal, in which case a : b∞ = cN1 = cN2 .

A more direct approach to compute saturation by f is to reduce it to thelocalization at f and then apply directly Rabinowitch’s Trick (see the proof ofLemma 20.1.10).

We recall that, given a ring R and an element f ∈ R, the localization of Rat f is the ring R f := a/ f i , a ∈ R, i ∈ N with the obvious generalizationof the ring operations.

Lemma 26.3.10. Let R be a Zacharias ring. Let a ⊂ R be the ideal generatedby a1, . . . , am and let f ∈ R.

Let a′ ⊂ R f be the ideal in R f generated by a1, . . . , am.Then we have a′ ∩ R = a : f ∞

Proof. Let g ∈ a : f ∞ ⊂ R. Then there exists i : f i g ∈ a and there is arepresentation f i g = ∑

j d j a j such that g = ∑j (d j/ f i )a j ∈ a′ ∩ R.

Let g ∈ a′ ∩ R; then g = ∑j (d j/ f i )a j for suitable i ∈ N and d j ∈ R;

therefore f i g = ∑j d j a j ∈ a, that is g ∈ a : f i ⊆ a : f ∞.


Let us now consider the morphism φ : R[T ] −→ R f defined, for eachg(T ) ∈ R[T ], by φ(g) = g(1/ f ), for which ker(φ) = (1− f T ), implying theexistence of an isomorphism π : R[T ]/(1 − f T ) −→ R f .

Applying π we obtain

a′ ∩ R = (a1, . . . , am)R f ∩ R∼= (a1, . . . , am)R[T ]/(1 − f T ) ∩ R∼= (a1, . . . , am, 1 − f T )R[T ] ∩ R.

As a consequence we have

Corollary 26.3.11. Let R be a Zacharias ring. Let a ⊂ R be the ideal gener-ated by a1, . . . , am and let f ∈ R.

Let d ⊂ R[T ] be the ideal generated by a1, . . . , am, 1 − f T . Then

a : f ∞ = d ∩ R.

When R = k[X1, . . . , Xn], this computation requires a lexicographicalGrobner basis computation; Bayer suggested using alternatively Grobner basesw.r.t. the reverse lexicographical (rev-lex) ordering.

Lemma 26.3.12 (Bayer). Impose on k[T, X1, . . . , Xn] the rev-lex ordering <

induced by T < X1 < · · · < Xn. Then, for each f ∈ k[T, X1, . . . , Xn],

T d | T<( f ) ⇒ T d | f.

Proof. Let t1 := T d1 Xa11 . . . Xan

n and t2 := T d2 Xb11 . . . Xbn

n . If t1 > t2 thend1 ≤ d2.

Therefore, if T<( f ) is divisible by T d , the same happens for each term inf , whence the thesis.

Corollary 26.3.13. Let < be the rev-lex ordering on k[T, X1, . . . , Xn] in-duced by T < X1 < · · · < Xn.

Let I ⊂ k[T, X1, . . . , Xn] and let g1, . . . , gs be the Grobner basis of Iw.r.t. <. Then, we have:

(1) Writing, for each i,

fi :=

gi/T if T | gi

gi otherwise,

then f1, . . . , fs is the Grobner basis of I : T .

(2) Expressing each gi as

gi = T di hi , T hi ,

h1, . . . , hs is the Grobner basis of I : T ∞.

312 Spear

Proof.

(1) Let g ∈ I : T , that is gT ∈ I and there is i such that T(gi ) | T T(g). Asa consequence the claim is proved since T( fi ) | T(g) is independent ofT | gi .

(2) Let g ∈ I : T ∞; then, for some ρ ∈ N, gT ρ ∈ I; therefore there is isuch that

T di T(hi ) = T(gi ) | T(gT ρ) = T ρT(g) ⇒ T(hi ) | T(g).

Let then R := k[X1, . . . , Xn], < be the reverse lexicographical orderingon R[T ] = k[T, X1, . . . , Xn] induced by T < X1 < · · · < Xn . Let a ⊂ Rbe the ideal generated by a1, . . . , am and let f ∈ R. If g1, . . . , gs is theGrobner basis of (a1, . . . , am, T − f ) in R[T ] w.r.t. <, each gi can be uniquelyexpressed as

gi = T di hi , T hi , hi ∈ k[T, X1, . . . , Xn].

Defining, for each i :

fi (T, X1, . . . , Xn) :=

gi/T if T | gi ,gi otherwise,

Hi (X1, . . . , Xn) := hi ( f, X1, . . . , Xn),

Fi (X1, . . . , Xn) := fi ( f, X1, . . . , Xn),

by the corollary above we know that

• (a1, . . . , am, T − f ) : T is generated by f1, . . . , fs, and• (a1, . . . , am, T − f ) : T ∞ is generated by h1, . . . , hs.

The morphism φ : R[T ] −→ R defined, for each g(T ) ∈ R[T ], byφ(g) = g( f ) being such that ker(φ) = ( f − T ), there is an isomorphismπ : R[T ]/( f − T ) −→ R; it is then sufficient to apply it in order to deducethat

• π( f1), . . . , π( fs) is a basis of π(a1, . . . , am, T − f ) : π(T ), and• π(h1), . . . , π(hs) is a basis of π(a1, . . . , am, T − f ) : π(T )∞,

that is:

Corollary 26.3.14. With the notation above we have:

• F1, . . . , Fs is a basis of a : f and• H1, . . . , Hs is a basis of a : f ∞.

26.4 *Multivariate Chinese Remainder Algorithm 313

It is worth quoting the illuminating comment of Caboara and Traverso12

on the relation between this algorithm and the application of Rabinowitch’sTrick described in Lemma 26.3.8 for performing the computation a : f whenf := Xr is any variable of R := k[X1, . . . , Xn]:

The ‘special remark by Bayer’ algorithm [Corollary 26.3.14] just describes what hap-pens when performing the tag variable algorithm [Lemma 26.3.8] in that special situa-tion; . . . we assume that [a1, . . . , am] is already a Grobner basis [of a]. We multiplyall the ai by T , add the equation T Xr − 1, then compute the Grobner basis; this com-putation, using the fact that the input is Grobner:

(1) substitutes T Xr with 1 in all polynomials having T Xr in the [leading term](S-polynomial between T ai and T Xr − 1 when Xr | T(ai )),

(2) multiplies by Xr all the polynomials not having Xr in the [leading term], thensubstitutes T Xr with 1 (S-polynomial between T ai and T Xr − 1 when Xr

T(ai )); this means, putting in the result all the polynomials ai not divisible byXr unchanged,

(3) performs a Buchberger algorithm on the polynomials not having T in the head,just to discover that they form a Grobner basis.

26.4 *Multivariate Chinese Remainder Algorithm

Let

P = k[X1, . . . , Xn],I1, . . . , Im ⊂ P be ideals given by bases F1, . . . , Fm ,I := ⋂m

i=1 Ii ,f1, . . . , fm ∈ P be polynomials.

Let us also consider

a subset of variables in P which wlog we assume to be X1, . . . , Xr ;the polynomial ring k[X1, . . . , Xn, Y1, . . . , Ym];any term ordering < on it under which, for any term t ∈ k[X1, . . . , Xr ],

Y j > t and Xi > t for any j and any i, r < i ≤ n;the ideal J generated by

1 −m∑

i=1

Yi

m⋃

i=1

f Yi : f ∈ Fi ;

the polynomial f := ∑mi=1 Yi fi .

Proposition 26.4.1 (Becker–Weispfenning). Let G be the Grobner basis of Jand let f ′ := N F( f, G) be the normal form of f w.r.t. G. Then:

12 In C. Traverso and M. Caboara, Efficient Algorithms for Module Operation (2002), unpublished.I modify their notation to adapt it to the notation used here.

314 Spear

(1) G ′ := G ∩ k[X1, . . . , Xr ] is the Grobner basis of I ∩ k[X1, . . . , Xr ];(2) The following conditions are equivalent

(a) f ′ ∈ k[X1, . . . , Xr ](b) f ′ ∈ k[X1, . . . , Xr ] and f ′ ≡ fi mod Ii for each i(c) there is g ∈ k[X1, . . . , Xr ] such that g ≡ fi mod Ii for each i;

(3) If f ′ ∈ k[X1, . . . , Xr ] then f ′ = N F(g, G ′) for each g ∈k[X1, . . . , Xr ] such that g ≡ fi mod Ii for each i .

Proof.

(1) Let g ∈ J ∩ k[X1, . . . , Xr ] and let q, qi j ∈ k[X1, . . . , Xr , Y1, . . . , Yd ]such that

g = q

(1 −

m∑i=1

Yi

)+

m∑i=1

mi∑j=1

qi j fi j Yi

where Fi = fi1, . . . , fimi ; then the evaluations

Y j := 1 if j = i ,

0 otherwise

prove that g = ∑mij=1 qi j fi j ∈ Ii . Therefore

J ∩ k[X1, . . . , Xr ] =m⋂

i=1

Ii ∩ k[X1, . . . , Xr ] = I ∩ k[X1, . . . , Xr ]

and the claim follows from Theorem 26.2.2.13

(a) ⇒ (b) Since f − f ′ ∈ J then there are q, qi j ∈ k[X1, . . . , Xr ,

Y1, . . . , Yd ] such that

m∑i=1

Yi fi − f ′ = q

(1 −

m∑i=1

Yi

)+

m∑i=1

mi∑j=1

qi j fi j Yi

where Fi = fi1, . . . , fimi and the evaluations

Y j := 1 if j = i ,

0 otherwise

give fi − f ′ = ∑mij=1 qi j fi j ∈ Ii .

(3) Let g ∈ k[X1, . . . , Xr ] be such that g ≡ fi mod Ii for each i ; then

f − g =m∑

i=1

Yi ( fi − g) − g

(1 −

m∑i=1

Yi

)∈ J

13 The other inclusion is trivial: if g ∈ ∩mi=1Ii , let qi j be such that g = ∑mi

j=1 qi j fi j for each i .Then

g = g

(1 −

m∑i=1

Yi

)+

m∑i=1

gYi = g

(1 −

m∑i=1

Yi

)+

m∑i=1

mi∑j=1

qi j fi j Yi .

26.4 *Multivariate Chinese Remainder Algorithm 315

so that f ′ = N F(g, G); because of our choice of < the assumptionthat g ∈ k[X1, . . . , Xr ] implies that f ′ = N F(g, G ′) and

(c) ⇒ (a) f ′ ∈ k[X1, . . . , Xr ].

Setting r = n we have

Corollary 26.4.2. Let G be the Grobner basis of J and let f ′ := N F( f, G)

be the normal form of f w.r.t. G. Then:

(1) G ′ := G ∩ k[X1, . . . , Xn] is the Grobner basis of I;(2) f ′ ≡ fi mod Ii for each i;(3) for each g ∈ k[X1, . . . , Xn] such that g ≡ fi mod Ii for each i , f ′ :=

N F(g, G ′).

This corollary is a Lagrangian formulation of the solution of the Chi-nese remainder problem; the Newtonian solution is a direct corollary ofLemma 26.1.6: let

h ∈ k[X1, . . . , Xn] be such that h ≡ fi mod Ii for each i < m,a1, . . . , as be a basis of

⋂m−1i=1 Ii ,

b1, . . . , bt be a basis of Im ,

g1, . . . , gm be a Grobner basis of the ideal(∩m−1

i=1 Ii)

+ Im generated by

a1, . . . , as, b1, . . . , bt ,xi j , yil be such that gi = ∑s

j=1 xi j a j + ∑tl=1 yilbl , for each i ;

then

Proposition 26.4.3 (Becker–Weispfenning). The following conditions

• there is f ′ ∈ P such that f ′ ≡ fi mod Ii for each i ≤ m

• fm − h ∈(⋂m−1

i=1 Ii)

+ Im

are equivalent; if they are satisfied and fm − h = ∑mi=1 zi gi , then the required

solution is

f ′ = fm −t∑

l=1

(m∑

i=1

zi yil

)bl = h +

s∑j=1

(m∑

i=1

zi xi j

)a j

Proof. The existence of f ′ ∈ P such that f ′ ≡ fi mod Ii for each i ≤ m isequivalent to the existence of elements p1 ∈ ⋂m−1

i=1 Ii and p2 ∈ Im such thatf ′ − h = p1 and f ′ − fm = p2; this in turn is equivalent to

fm − h ∈(

m−1⋂i=1

Ii

)+ Im .

316 Spear

The value of f ′ is a consequence of the relations

f ′ = h + p1 = fm + p2

and

fm − h = p1 − p2 =m∑

i=1

zi gi =s∑

j=1

(m∑

i=1

zi xi j

)a j +

t∑l=1

(m∑

i=1

zi yil

)bl .

26.5 Tag-Variable Technique and Its Application to Subalgebras

In order to compute saturation and localization at f over R, Spear’s proposalis to consider the kernel (1 − f T ) of the map φ : R[T ] −→ R f defined byφ(T )=1/ f . This proposal has also been used in order to describe subalgebras.

Let us assume we are given the homomorphism

ω : k[Y1, . . . , Yd ] −→ k[X1, . . . , Xn]

defined by ω(Yi ) = fi , for each i.Then the image of ω,

Im(ω) = k[ f1, . . . , fd ] ⊂ k[X1, . . . , Xn]

is the subalgebra generated by f1, . . . , fd.The elementary question is, given g ∈ k[X1, . . . , Xn], to decide whether

g ∈ k[ f1, . . . , fd ]. The solution to solving this problem allows us also todescribe the structure of k[ f1, . . . , fm].

Let us consider the map

γ : k[Y1, . . . , Yd , X1, . . . , Xn] −→ k[X1, . . . , Xn] : γ (Xi ) = Xi , γ (Y j ) := f j ,

whose kernel is ker(γ ) = ( f j − Y j , 1 ≤ j ≤ d) so that

k[ f1, . . . , fd ] ∼= k[X1, . . . , Xn, Y1, . . . , Yd ]/( f1 − Y1, . . . , fd − Yd),

and let G be the reduced Grobner basis of ker(γ ) w.r.t. any term ordering <

under which X j > t for any j and any term t ∈ k[Y1, . . . , Yd ].14

Under this notation, we have:

Proposition 26.5.1 (Shannon–Sweedler). Let

h ∈ k[Y1, . . . , Yd , X1, . . . , Xn]

be the canonical form of g w.r.t. G. Then

g ∈ k[ f1, . . . , fd ] ⇐⇒ h ∈ k[Y1, . . . , Yd ].

Moreover, if this happens g = h( f1, . . . , fd).

14 A good choice is the lexicographical ordering < induced by Y1 < · · · < Yd < X1 < · · · < Xn .

26.5 Tag Variables Technique 317

Proof. Since g reduces to h, g − h ∈ ker(γ ) and γ (g) = γ (h).

If h ∈ k[Y1, . . . , Yd ] then

g = γ (g) = γ (h) = h( f1, . . . , fd).

Conversely, assume that g ∈ k[ f1, . . . , fd ] and let p ∈ k[Y1, . . . , Yd ] bea polynomial such that g = p( f1, . . . , fd), that is γ (p) = γ (g) = γ (h),

h − p ∈ ker(γ ) and h is the canonical form of p ∈ k[Y1, . . . , Yd ] w.r.t. G. Bythe assumption on < any reduction (such as h) of a polynomial (such as p) ink[Y1, . . . , Yd ] necessarily is a member of k[Y1, . . . , Yd ].

In order to prove that g ∈ k[ f1, . . . , fd ] it is sufficient to perform reductionuntil an element (not necessarily the canonical form) h ∈ k[Y1, . . . , Yd ] isfound, in which case the same argument proves also g = h( f1, . . . , fm).

If such an element is not found during the reduction, then the canonical formis not in k[Y1, . . . , Yd ] and g is not a member of k[ f1, . . . , fd ].

Let us write GT := G ∩ k[Y1, . . . , Yd ] and G M := G \ GT .Spear already used GT to determine the relations among the fi s:

Corollary 26.5.2 (Spear). The ideal of the polynomial relations among thefi s is generated by GT . Moreover, if k[ f1, . . . , fd ] = k[X1, . . . , Xn] and d =n, then GT = ∅.

Proof. We know that GT is the basis of ker(γ ) ∩ k[Y1, . . . , Yd ].The assumptions that k[ f1, . . . , fd ] = k[X1, . . . , Xn] and d = n imply that

the fi s are algebraically independent.

In the same mood, G M can be used to investigate whether k[ f1, . . . , fd ] =k[X1, . . . , Xn].

Theorem 26.5.3 (Shannon–Sweedler). With the same assumptions and nota-tions, the following conditions are equivalent:

• k[ f1, . . . , fd ] = k[X1, . . . , Xn];• G M = Xi − φi , 1 ≤ i ≤ n for some φi ∈ k[Y1, . . . , Yd ].

Proof. If G M has the described shape, then, for each i , φi is the canonical formof Xi ; therefore, by the result above, Xi ∈ k[ f1, . . . , fd ], for each i .

Conversely, let us assume k[ f1, . . . , fd ] = k[X1, . . . , Xn].By assumption each Xi must be reduced by G to a polynomial φi ∈

k[Y1, . . . , Yd ], which wlog can be assumed to be reduced w.r.t. GT .Therefore each Xi − φi ∈ ker(γ ) and necessarily is a member of G M ;

therefore (X1, . . . , Xn) ⊆ TG M .As a consequence, GT ∪ Xi − φi is the Grobner basis of ker(γ ).

318 Spear

Let σ : k[Y1, . . . , Yd ] −→ k[X1, . . . , Xn] be the restriction of γ , that is themap such that, for each i , σ(Yi ) = fi ; of course

Im(σ ) = k[ f1, . . . , fd ] ⊆ k[X1, . . . , Xn].

Corollary 26.5.4. With the notation above we have

(1) σ is injective iff GT = ∅,(2) σ is surjective iff G M = Xi − φi , 1 ≤ i ≤ n for some φi ∈

k[Y1, . . . , Yd ].

If both conditions are satisfied (so that σ is an isomorphism), let µ :k[X1, . . . , Xn] −→ k[Y1, . . . , Yd ] be the homomorphism defined by µ(Xi ) =φi . Then σµ is the identity.

Proof.

(1) Since ker(σ ) = ker(γ ) ∩ k[Y1, . . . , Yd ] and G is a Grobner basis ofker(γ ), then GT = G ∩ k[Y1, . . . , Yd ] is a Grobner basis of ker(σ ),whence the claim.

(2) If σ is surjective, then k[ f1, . . . , fd ] = k[X1, . . . , Xn] and G M has therequired shape.

Conversely, if G M has the required shape, then, for each i

Xi = φi ( f1, . . . , fd) = σ(φi )

and σ is surjective.

Moreover Xi = σ(φi ) = σµ(Xi ) so that σµ is the identity.

A further analysis allows us to study the algebraic/transcendental nature ofthe field extension k( f1, . . . , fd) ⊆ k(X1, . . . , Xn), allowing us also to char-acterize in terms of G M the case k( f1, . . . , fd) = k(X1, . . . , Xn).

To do so, we need to assume that < satisfies the stronger property that Xi >

t , for each i and for any term t ∈ k[Y1, . . . , Yd , X1, . . . , Xi−1],15 and wepartition G M as G M = G1 · · · Gn where

Gi := G ∩ k[Y1, . . . , Yd , X1, . . . , Xi ] \ Gi−1.

For each i such that Gi = ∅ let gi ∈ Gi be the element which minimizesT(gi ) and let us write

gi =:ei∑

j=0

ai j Xei − ji , ai j ∈ k[Y1, . . . , Yd , X1, . . . , Xi−1], ai0 = 0.

Let us moreover set M := k(X1, . . . , Xn), L := k( f1, . . . , fd). Then:

15 Again the lexicographical ordering < induced by Y1 < · · · < Yd < X1 < · · · < Xn is a goodchoice.

26.5 Tag Variables Technique 319

Theorem 26.5.5 (Shannon–Sweedler). Using the notation and assumptionsabove, we have:

(1) Gi = ∅ iff Xi is algebraic over L(X1, . . . , Xi−1), in which case[L(X1, . . . , Xi ) : L(X1, . . . , Xi−1)

] = ei .

(2) M is algebraic over L iff Gi = ∅, for each i , in which case

[M : L] =∏

i

ei .

(3) The transcendency degree of M over L equals the number such thatGi = ∅.

Proof. (2) and (3) being direct consequence of (1), let us prove (1).Assume Gi = ∅. Since G is a reduced Grobner basis, we know that ai0 ∈

ker(γ ).Let us now define P(Z) ∈ L(X1, . . . , Xi−1)[Z ] as

P(Z) :=ei∑

j=0

γ (ai j )Zei − j

so that P(Xi ) = γ (gi ) = 0. This is sufficient to prove that Xi is algebraic overL(X1, . . . , Xi−1) and that[

L(X1, . . . , Xi ) : L(X1, . . . , Xi−1)] ≤ ei .

Conversely, assume Xi is algebraic over L(X1, . . . , Xi−1) and consider aminimal polynomial in L(X1, . . . , Xi−1)[Z ] for Xi over L(X1, . . . , Xi−1).

It is then sufficient to multiply out the denominator to have a polynomial

Q(Z) :=t∑

j=0

c j Z t− j ∈ k[ f1, . . . , fd , X1, . . . , Xi−1][Z ],

with the same degree in Z which is satisfied by Xi and whose coefficients arein k[ f1, . . . , fd , X1, . . . , Xi−1].

For each ci let us choose ai ∈ k[Y1, . . . , Yd , X1, . . . , Xi−1] such thatγ (ai ) = ci , and T(ai ) ∈ T(G).16

Then h(Xi ) := ∑tj=0 a j Xt− j

i satisfies γ (h) = Q(Xi ) = 0 and h ∈ ker(γ ).

Remarking that T(h) = T(a0)Xti ∈ T(G) and that T(a0) ∈ T(G), we can

deduce that there is g ∈ Gi such that m Xui =: T(g) | T(a0)Xt

i , so that t ≥ u.Therefore Gi = ∅ and, by the minimality of gi , we have[

L(X1, . . . , Xi ) : L(X1, . . . , Xi−1)] = t ≥ u ≥ ei .

16 This requirement is completely elementary; it just needs us to avoid poor choices!

320 Spear


k(X1, . . . , Xn) = k( f1, . . . , fd) ⇐⇒ G M ⊇ δi Xi − φi : 1 ≤ i ≤ d,where δi ∈ k[Y1, . . . , Yd ], φi ∈ k[Y1, . . . , Yd , X1, . . . , Xi−1].

Proof. k(X1, . . . , Xn) = k( f1, . . . , fd) is equivalent to[k(X1, . . . , Xn) : k( f1, . . . , fd)

] = 1,

that is ei = 1 for each i, which in turn is equivalent to the assumption on thestructure of G M .

Let us now consider P = k[X1, . . . , Xn] and the quotient ring B := P/I,where I = (g1, . . . , gm) ⊂ P is an ideal and ψ : P −→ B is the canonicalprojection, and a subalgebra

A := k[a1, . . . , ad ] ⊂ B where ai = ψ( fi ), fi ∈ P;writing

J = (g1, . . . , gm, Y1 − f1, . . . , Yd − fd) ⊂ k[X1, . . . , Xn, Y1, . . . , Yd ]

and

Jc := J ∩ k[Y1, . . . , Yd ]

we have

B ∼= k[Y1, . . . , Yd , X1, . . . , Xn]/J and A ∼= k[Y1, . . . , Yn]/Jc.

Denoting by G the reduced Grobner basis of J w.r.t. any term ordering < underwhich X j > t for any j and any term t ∈ k[Y1, . . . , Yd ], then:

Proposition 26.5.7 (Conti–Traverso). The notation is the same as above.

(1) The following conditions are equivalent:

• B is an integral extension of A,• for each i, 1 ≤ i ≤ n there is gi ∈ G such that T(gi ) = Xdi

i .

In this case B is generated as an A-module by the set

ae11 . . . aed

d : Xe11 . . . Xed

d ∈ N(Jc).(2) Let h ∈ k[Y1, . . . , Yd , X1, . . . , Xn] be the canonical form of g ∈ P

w.r.t. G. Then ψ(g) ∈ A ⇐⇒ h ∈ k[Y1, . . . , Yd ].(3) B = A iff for each i, 1 ≤ i ≤ n, there is φi ∈ k[Y1, . . . , Yd ] such that

Xi − φi ∈ G.

26.6 Caboara–Traverso Module Representation 321

(4) If < is such that

Xi > t for each i, 1 ≤ i ≤ n, t ∈ k[Y1, . . . , Yd , X1, . . . , Xi−1]

then B = A are birational17 iff for each i, 1 ≤ i ≤ n there are δi ∈k[Y1, . . . , Yd ], φi ∈ k[Y1, . . . , Yd , X1, . . . , Xi−1] such that δi Xi −φi ∈G.

Proof.

(1) It is sufficient to recall that B is an integral extension of A iff for eachi, 1 ≤ i ≤ n, there is a monic polynomial gi ∈ k[Y1, . . . , Yd ][T ] suchthat gi ( f1, . . . , fd , Xi ) ∈ I.

(2) If ψ(g) ∈ A, then there is p ∈ k[Y1, . . . , Yd ] such that ψ(g) −p ∈ J; therefore h is the canonical form of p, which implies h ∈k[Y1, . . . , Yd ].

(3) If B = A then, for each i , there is φi ∈ k[Y1, . . . , Yd ] such thatψ(Xi ) = φi (a1, . . . , ad) whence ψ(Xi − φi ( f1, . . . , fd)) = 0,Xi − φi ( f1, . . . , fd) ∈ I, Xi − φi (Y1, . . . , Yd) ∈ J, Xi ∈ T(J),Xi ∈ T(G).

(4) It is sufficient to apply the same argument as in Corollary 26.5.6

26.6 Caboara–Traverso Module Representation

An interesting application of the tag-variable technique was proposed byCaboara and Traverso who applied it in order to interpret Buchberger’s al-gorithm for modules as an instance of the one for ideals, their aim being ‘forobvious implementation reasons [ . . . ] to avoid the coming into being of twovery similar-but-yet-different algorithms’.

Let us consider (see Section 24.318) P := k[Z1, . . . , Zr ], endowed with aterm ordering <Z on Z := Zb1

1 · · · Zbrr : (b1, . . . , br ) ∈ N

r , and the free-module Pm – whose canonical basis will be denoted by e1, . . . , em – whichis a k-vectorspace generated by the basis

Z(m) := tei , t ∈ Z, 1 ≤ i ≤ mon which we impose a well-ordering < satisfying, for each t1, t2 ∈ Z, τ1, τ2 ∈Z(m),

t1 ≤Z t2, τ1 ≤ τ2 ⇒ t1τ1 ≤ t2τ2.

17 That is they have the same quotient field.18 But note that we have changed variables and parameters to adapt the situation to Section 26.2.

322 Spear

Their proposal is simply to embed Pm into k[Z1, . . . , Zr , e1, . . . , em] andjust apply the usual Buchberger algorithm, performing only two small modifi-cations:19

• S-pairs ( f ei , ge j ) are considered only if i = j ;• those S-pairs ( f ei , gei ), such that T( f )T(g) = lcm(T( f ), T(g)), are not to

be discarded.20

It is quite interesting to see how their proposal applies to another of theirsuggestions about syzygy computation:

Algorithm 26.6.1 (Caboara–Traverso). If we are given a basis

F := f1, . . . , fd ⊂ Pof an ideal, the computation of the Grobner basis G w.r.t. <Z of the sameideal allows us to produce the syzygies among G while one needs thoseamong F and some bookkeeping is therefore needed (see for example Propo-sition 26.1.2); they propose an easy trick for this bookkeeping, which theypresent as ‘very similar to obtain the Bezout identity from the Euclidean algo-rithm’.21

Their proposal is to consider the module P1+d , whose canonical basis wewill denote e0, . . . , ed, imposing on it the term ordering

mei < m′e j ⇐⇒

i > j ori = j and m <Z m′

and the submodule fi e0 + ei , 1 ≤ i ≤ d of which a Grobner basis G iscomputed w.r.t. <. The elements

(h0, . . . , hd) =d∑

i=0

hi ei ∈ G

19 They in fact consider the more general case which embeds Pm into a polynomial ring P ′ :=k[Z1, . . . , Zr , Y1, . . . , Yd ] by choosing a set t1, . . . , tm of terms in P ′ which are linearlyindependent on P and defining the embedding χ : Pm −→ P ′ by χ

(∑i fi ei

):= ∑

i fi ti .Two instances of such a choice, in connection with the ideal theoretic operations, are

(1) P ′ := P[T ], 1, T (see Lemma 26.3.8), and(2) P ′ := P[T ], 1, T, . . . , T n−1 (see Proposition 26.3.5(2)).

In this setup, one has to add to the modifications listed above a stricter notion of divisibilityaccording to which

χ(mei ) = mti | m′t j = χ(me j ) ⇐⇒ m | m′, i = j.

20 Remember that Buchberger’s First Criterion does not hold for modules.21 But I consider this to be more related to the classical Gaussian algorithm for computing the

inverse of a square matrix A performing the same row-operations on A and I , until the firstmatrix becomes I and the second A−1; within this algorithm, in each instance any row ofthe second matrix gives the representation of the corresponding row of the first matrix as acombination of their original rows.


can be partitioned into two classes:

• G0 := (h0, . . . , hd) ∈ G : h0 = 0 which is a Grobner basis of the syzygymodule among F ; and

• G1 := (h0, . . . , hd) ∈ G : h0 = 0.Since within the algorithm each computation performed on module elementsf e0 is simply performing the same operation that would have been performedby the application of Buchberger algorithm on F ⊂ P , then

G := h0 : (h0, . . . , hd) ∈ G1is the required Grobner basis of F w.r.t. <.

Moreover, for each (h0, . . . , hd) ∈ G one has h0 = −∑di=1 hi fi , and for

each (h0, . . . , hd) ∈ G0 one has 0 = ∑di=1 hi fi .

Let us now interpret the same computation within the Caboara–Traversomodule representation: let us therefore write

P ′ := k[Z1, . . . , Zr , Y1, . . . , Yd ],

Y := Y a11 . . . Y ad

d : (a1, . . . , ad) ∈ Nd,

Z := Zb11 . . . Zbr

r : (b1, . . . , br ) ∈ Nr ,

T := tY tZ : tY ∈ Y, tZ ∈ Z,and impose on T the block ordering < inducing Y < Z, which for eacht (1), t (2) ∈ T , t (i) := t (i)Z t (i)Y , t (i)Y ∈ Y, t (i)Z ∈ Z, i = 1, 2, is defined by

t (1) < t (2) ⇐⇒ t (1)Z <Z t (2)

Z or t (1)Z = t (2)

Z and t (1)Y <Y t (2)

Y ,

where <Y is the lexicographical ordering induced by Y1 < Y2 < · · · < Yd .

Finally we embed P1+d into P ′ via the map χ : P1+d −→ P ′ defined by

χ(∑d

i0gi ei

):= g0 + ∑d

iigi Yi .

Within this setting, their algorithm computing the syzygy module of F := f1, . . . , fd computes a Grobner basis22 w.r.t. < of the ideal

( fi − Yi : 1 ≤ i ≤ d).

22 With only the two following modifications:

• the S-pairs ( f, g) are taken into consideration only if

T( f ) = mYi , T(g) = m′Y j , m, m′ ∈ Z and i = j;• the reduction of a term mYi ∈ Z by a basis element f ∈ P is forbidden.

The intent of these restrictions is to avoid, during the computation and mainly in the output,the appearance of terms t ∈ (Y1, . . . , Yd )2.

324 Spear

The output H ⊂ k[Z1, . . . , Zr , Y1, . . . , Yd ] will consist of polynomials

h := h0 +d∑

i=1

hi Yi , hi ∈ P.

Moreover:

• h(Z1, . . . , Zr , 0, . . . , 0) : h ∈ H is a Grobner basis of (F) w.r.t. <Z ;• h(Z1, . . . , Zr , f1, . . . , fd) : h ∈ H : h(Z1, . . . , Zr , 0, . . . , 0) = 0 ‘repre-

sents’ the syzygy module among F ;• for each h ∈ H, h(Z1, . . . , Zr , f1, . . . , fd) ‘gives’ the representation in

terms of F of

h(Z1, . . . , Zr , 0, . . . , 0) = h0(Z1, . . . , Zr ) = −d∑

i=1

hi (Z1, . . . , Zr ) fi .

In other words, the computation suggested by Spear in order to compute therelations among the generators f1, . . . , fd of a subalgebra only require to be‘calibrated’ in order to split it into two steps:

(1) in a first step all computations introducing terms in (Y1, . . . , Yd)2 arepostponed; the output of this step is the knowledge of a Grobner basisof the ideal ( f1, . . . , fd), the knowledge of its syzygies and the repre-sentation of the Grobner basis in terms of the input basis;

(2) in the second step all postponed computations are performed producingthe subalgebra relations.

One wonders whether this ‘calibration’ was already present in the MAC-SYMA package and how much computer algebra lost with the disappearanceof Spear . . . .

The Caboara–Traverso module representation also had the explicit aim ofreducing ideal and module operations into a unified frame to describe and anal-yse them; the results are quite interesting.

Within this frame, the structure of (module) colon and intersection opera-tions23 can be described by

Lemma 26.6.2. Let P := k[X1, . . . , Xn]. Let a, b ⊂ Pm be modules gen-erated by the bases, respectively, a1, . . . , aµ and b1, . . . , bν. Let v1, v2 ∈Pm, f, g ∈ P and

I := (a : v1) ∩ ( f ) = h f, h ∈ P : h f v1 ∈ a ⊂ P.

23 For simplicity all the algorithms have been described in the case of ideals. All the algorithmsdescribed (and also the connected proofs) apply verbatim to the module case.


Then

(1) Let c ⊂ P2m = Pm ⊕ Pm be the module generated by

(ai , 0), 1 ≤ i ≤ µ ∪ ( f v1, f v2).Then

c ∩ (0 ⊕ Pm) = (a, b), a = 0, b ∈ Iv2

= (a, cv2), a = 0, c ∈ (a : v1) ∩ ( f ).(2) Let d ⊂ P2m = Pm ⊕ Pm be the module generated by

(ai , 0), 1 ≤ i ≤ µ ∪ ( f b j , gb j ), 1 ≤ j ≤ ν.Then

d ∩ (0 ⊕ Pm) = (a, b), a = 0, b ∈ g (b ∩ (a : f ))

= (a, gw), a = 0, w ∈ b ∩ (a : f ).Proof.

(1) For any vector v = (a, b) ∈ P2m, a = 0, we have

b = cv2 ∈ Iv2,

⇐⇒ there exists h ∈ P such that b = cv2, cv1 ∈ a, c = h f,

⇐⇒ there exist h, h1, . . . , hm ∈ P : b = h f v2, h f v1 =∑

i

−hi ai

⇐⇒ ∃ h, h1, . . . , hm ∈ P :∑

i

hi (ai , 0) + h( f v1, f v2) = (0, b)

⇐⇒ (0, b) ∈ c.

(2) For any vector v = (a, b) ∈ P2m, a = 0, we have

b = gw ∈ g (b ∩ (a : f ))

⇐⇒ there exists w ∈ Pm : gw = b, w ∈ b ∩ (a : f )

⇐⇒ there exist w ∈ Pm, hi , k j ∈ P : gw = b,

f w =∑

i

hi ai , w =∑

j

k j b j

⇐⇒ there exist hi , k j ∈ P :∑

i

hi ai = f∑

j

k j b j , b = g∑

j

k j b j

⇐⇒ ∃ hi , k j ∈ P :∑

i

−hi (ai , 0) +∑

j

k j ( f b j , gb j ) = (0, b)

⇐⇒ (0, b) ∈ d.

326 Spear

Remark 26.6.3. The interesting result is that once the intersection and colonalgorithms discussed in Section 26.3 are interpreted within the Caboara–Traverso module representation, it appears that the computations they requiredare essentially equivalent.

We can present this result by discussing the easier case of the computation

(a1, . . . , am) : f ⊂ P = k[X1, . . . , Xn].

The Rabinowitch Trick algorithm (Lemma 26.3.8) requires us to computea basis of a1T, . . . , am T, f T − 1 ∈ P[T ] and take its intersection withP . This can be performed in the module P2 embedded in P[T ] by the mapχ( f1, f0) = f1T − f0. Therefore one has to compute a Grobner basis of themodule generated by

(a1, 0), . . . , (am, 0), ( f, 1)and consider the elements in 0 ⊕ P .

The syzygy algorithm (Corollary 26.3.7(2)) would require us to compute abasis B of the syzygies among a1, . . . , am, f and then to consider the coef-ficients of f in each element in B. Using the Caboara–Traverso algorithm thesyzygy computation requires us to compute the basis of

a1e0 + e1, . . . , ame0 + em, f e0 + em+1 ∈ Pm+2;however, since the coefficients of the ai s in the syzygy elements will be dis-carded at the end of the computation, it is useless to compute them. Thereforethe syzygy algorithm computation requires us to compute a basis of

a1e0, . . . , ame0, f e0 + em+1 ∈ P2.

Both algorithms, therefore perform the same computations in order to obtaina Grobner basis G of the module generated by

(a1, 0), . . . , (am, 0), ( f, 1)and their output is

b : (0, b) ∈ G = f1T − f0 ∈ χ−1(G) ∩ P.

Let us assume we are given

a ring A,two free A-modules M and N , whose canonical bases will be respectively

denoted e1, . . . , er and ε1, . . . , εs ,an A-module homomorphism Φ : M −→ N given through a matrix (ai j ) ∈

Ars such that Φ(ei ) = ∑j ai jε j for each i ,


a submodule K ⊂ N generated by χ1, . . . , χt , χl = ∑j bl jε j , (bi j ) ∈ Ats

and let us denote

L := At the module whose canonical basis is η1, . . . , ηt ,Ψ : L −→ N the morphism defined by Ψ (ηl) = ∑

j bl jε j ,π : Ar+t = M ⊕ L −→ M the projection.

Then, if A = P := k[X1, . . . , Xn] the computation of the Grobner basis ofthe submodule generated by G := Φ(e1), . . . , Φ(er ), χ1, . . . , χt allows usto compute the inverse image Φ−1(K ) ⊂ M since

Proposition 26.6.4. (Conti–Traverso) If A = P := k[X1, . . . , Xn], thenΦ−1(K ) = π(H) where

H := Syz(G) = ker(Φ ⊕ Ψ )

=

(α1, . . . , αr , β1, . . . , βt ) :r∑

i=1

αiΦ(ei ) +t∑

l=1

βlχl = 0

Proof. For any α = ∑ri=1 αi ei ∈ M ,

α ∈ π(H)

⇐⇒ exists (β1, . . . , βt ) ∈ L : (α1, . . . , αr , β1, . . . , βt ) ∈ H

⇐⇒ exists (β1, . . . , βt ) ∈ L : Φ(α) =r∑

i=1

αiΦ(ei ) = −t∑

l=1

βlχl ∈ K

⇐⇒ α ∈ Φ−1(K ).

If P := k[Y1, . . . , Yd ] and A := P/I , where I is an ideal and ψ :P −→ A is the canonical projection, then, with the same notation as above,denoting

Ai j , Bl j ∈ P any elements such that ψ(Ai j ) = ai j and ψ(Bi j ) = bi j ,M ′, N ′ and L ′ the free P-modules, whose canonical bases are e1, . . . , er ,

ε1, . . . , εs and η1, . . . , ηt ,Φ ′ : M ′ −→ N ′ the P-module homomorphism defined by Φ ′(ei ) =∑

j Ai jε j for each i ,K ′ ⊂ N ′ the submodule generated by χ ′

1, . . . , χ′t , χ ′

l = ∑j Bl jε j ,

Ψ ′ : L ′ −→ N ′ the morphism defined by Ψ ′(ηl) = ∑j Bl jε j ,

ψπ : Pr+t = M ′ ⊕ L ′ −→ M ′ −→ M the projection,24

we have

24 Where, with a slight abuse of notation, ψ here denotes the canonical projection ψ : M ′ −→ M .

328 Spear

Corollary 26.6.5. Φ ′−1(K ′) = ψπ(H ′) where

H ′ := ker(Φ ′ ⊕ Ψ ′)

=

(α1, . . . , αr , β1, . . . , βt ) :r∑

i=1

αiΦ′(ei ) +

t∑l=1

βlχ′l = 0

.

Algorithm 26.6.6 (Conti–Traverso). If we consider P = k[Y1, . . . , Yn], anideal I = (g1, . . . , gm) ⊂ P , the quotient ring B := P/I and a subalgebra

A := k[a1, . . . , ad ] ⊂ B

the computation discussed before Proposition 26.5.7 allows us

• to represent A as A ∼= k[Y1, . . . , Yn]/I for a suitable ideal I ⊂ k[Y1, . . . , Yn]of which we know a Grobner basis,

• to check whether B is an integral extension of A,• in which case, it returns a set γ1, . . . , γu of generators of B as A-module.

We recall that the conductor D of B ⊃ A is the ideal

D := a ∈ A : aB ⊆ A

and that if D = (0) then B is an integral extension of A and both are birational.Clearly for any b ∈ B, b ∈ D iff b ∈ A and bγi ∈ A for each i . Therefore if

Φi : A −→ B denotes the A-module homomorphism defined by Φi (a) := aγi

then D = ⋂Φ−1

i (A), and the results above allow us to explicitly compute theconductor of B ⊃ A.

If moreover we are given another subalgebra

C := k[c1, . . . , cδ] ⊂ B

and we want to compute, if any, the conductor D of C ⊃ A one needs to testwhether

• C ⊃ A by verifying, using Proposition 26.5.7(2), if each c j ∈ A;• A and C are birational, by verifying, using Proposition 26.5.7(4), whether

they are both birational to k[a1, . . . , ad , c1, . . . , cδ];• C ⊃ A is an integral extension, returning a set γ1, . . . , γu of generators of

C as A-module;

then we obtain again D = ⋂Φ−1

i (A), where Φi : A −→ C is the A-modulehomomorphism defined by Φi (a) := aγi .

26.7 *Homogeneous Minimal Resolutions 329

26.7 *Caboara Algorithm for Homogeneous Minimal Resolutions

The trick discussed above (Algorithm 26.6.1) in order to compute, for a givenbasis F , at the same time its Grobner basis G, the syzygy module, and therepresentation of G in terms of F , has been generalized and improved in orderto directly apply (a small modification of) Buchbgerger’s algorithm in order tocompute the minimal homogeneous resolution of a homogeneous submoduleof a graded free-module over the polynomial ring.

Before discussing this application, it is important to comment on the shapethat Buchberger’s algorithm has when applied to homogeneous ideals andmodules.

The central point is that the S-pair of homogeneous polynomials and thereduction of a homogeneous polynomial by homogeneous elements are bothhomogeneous and that reductions keep the degree constant.

This trivial remark and the trivial fact that each polynomial can be reducedonly by polynomials of lower degree, imply that, if the S-pairs are treated byincreasing degree, when all S-pairs of degree D have been treated, then the setof all interreduced polynomials of degree bounded by D, which are present inthe current basis set, is exactly the set of all the polynomials bounded by Dwhich will be present in the output Grobner basis.

A good version of Buchberger’s algorithm for homogeneous ideals (andmodules) will therefore, by increasing value D:

• compute and reduce all S-polynomials of degree D,• interreduce, between each other, all such normal forms and all the members

of the input basis having degree D,• insert these irreducible elements in the final basis.

Let P := k[X1, . . . , Xn] be endowed with the degree compatible termordering <X , Pr−1 be the graded free P-module whose canonical basis ise(−1)

1 , . . . , e(−1)r−1 , deg(e(−1)

i ) = 0 and let M ⊂ Pr−1 be a homogeneous sub-module.

Let then

0 −→ Prρδρ−→ Prρ−1

δρ−1−→ · · ·Pri+1δi+1−→ Pri

δi−→ Pri−1 · · ·Pr1

δ1−→ Pr0δ0−→ M (26.1)

be its homogeneous minimal resolution, with the same notation and assump-tion we used in Definition 20.6.8; in particular deg(e(σ )

i ) =: d(σ )i .

The first thing to do is to embed all modules Pri in some polynomial ex-tension of P; the task is made easier by the fact that all one needs is toembed

⊕ρi=−1 Pri , or, in other words, to mark each module basis element

330 Spear

e(i)j , −1 ≤ i ≤ ρ, 1 ≤ j ≤ ri , and this can be done by using two variables,

S, T ; in order that each polynomial has the correct degree, a homogenizingvariable D is used. Therefore the embedded morphism

ψ :ρ⊕

i=−1

Pri −→ P[S, T, D]

is defined by Ψ (e(i)j ) := Si+1 Dd(i)

j T j , −1 ≤ i ≤ ρ, 1 ≤ j ≤ ri .

We need also to impose a suitable ordering on P[S, T, D] which

• generalizes the degree-compatible ordering <X ,• preserves the degree imposed on each Pri ,• privilege, for technical reasons which will be justified later, components on

minor syzygies, that is module members, come before its syzygies, whichcome before syzygies of syzygies, und so weiter;

as a consequence we will set m1Ss1 Dd1 T t1 < m2Ss2 Dd2 T t2 iff⎧⎪⎪⎨⎪⎪⎩

deg(m1) + d1 < deg(m2) + d2,

deg(m1) + d1 = deg(m2) + d2, s2 < s1,

deg(m1) + d1 = deg(m2) + d2, s2 = s1, m1 <X m2

deg(m1) + d1 = deg(m2) + d2, s2 = s1, m1 = m2, t1 < t2.

For any term mSs Dd T t we will define

Deg(mSs Dd T t ) := deg(m) + d;over all the computation the algorithm will treat only homogeneous polyno-mials f in the sense that for each term t in its support the value Deg(t) is aconstant value which will be used as the definition of Deg( f ).

Throughout the computation all the treated polynomials are homogenousand have the shape

f = fi Si+1 − fi+1Si+2, fi , fi+1 ∈ P[T, D]

for some i . We will set, for a non-zero such polynomial

Head( f ) := fi Si+1,

Tail( f ) := fi+1Si+2,

Ecart( f ) :=

1 iff Tail( f ) = 00 iff Tail( f ) = 0.

Note that, if within

• Tail( f ), each instance of a term Si+2 Dd(i+1)j T j is replaced with e(i+1)

j , we

would obtain an element Ψ −1(Tail( f )) ∈ Pri+1 .

26.7 *Homogeneous Minimal Resolutions 331

• Head( f ), each instance of a term Si+1 Dd(i)j T j is replaced with e(i)

j we

would obtain the element Ψ −1(Head( f )) ∈ Im(δi+1) ⊂ Pri .

Moreover we have

δi+1(Ψ−1(Tail( f ))) = Ψ −1(Head( f )). (26.2)

Having completely described the notation and the setup, it is now time todescribe the modifications to be performed on Buchberger’s algorithm; theyare:

• When a monomial m in the support of f = fi Si+1 − fi+1Si+2 is underreduction:

• if Tail( f ) = 0 or the monomial m is in Tail( f ) the reduction is performedusing Head(g) : g ∈ G, where G is the current basis;

• if, instead, Tail( f ) = 0 and the monomial m is in Head( f ) the reductionis performed using the elements of the current basis G.

The practical effect of this reduction is that, in Equation (26.2), if the algo-rithm reduces the element

• Tail( f ), then Ψ −1(Tail( f )) is similarly reduced by the current basis ele-ments of ker(δi+1);

• Head( f ), then Ψ −1(Head( f )) ∈ Im(δi+1) is similarly reduced by thecurrent basis elements of Im(δi+1) while at the same time updating, ifTail( f ) = 0, its representation in terms of the basis of Im(δi+1) which isrecorded in Tail( f );

therefore Equation (26.2) is preserved by each reduction.Note that the reduction of Tail( f ) using only elements Head( · ) has also

the technical effect of not introducing terms t Si+3, so that each polynomialkeeps the required form f = fi Si+1 − fi+1Si+2.

• Remember that, when all S-polynomials and basis elements of degree Dhave been completely reduced, they are no longer simplifiable, so that eachreduced element of degree D is a normal form.

Then, for any such irreducible element f = fi Si+1 − fi+1Si+2 for which

• Ecart( f ) = 0 so that

f = Tail( f ), Head( f ) = 0 and δi+1(Ψ−1(Tail( f ))) = 0,

f is ‘marked’ – as a minimal basis element of ker(δi+1) – and is modifiedas f := f + Si+3 Dd T j where

i + 3 indicates that f is a minimal basis element of Im(δi+2),

332 Spear

j counts the number of current marked elements (including f ) in theminimal basis Im(δi+2),

d := Deg( f ).

• Ecart( f ) = 1 so that Head( f ) = 0, f is not ‘marked’ nor modified.

The element f is, in both cases, inserted in the current basis (which meansthat the corresponding S-pairs are produced, etc.) and, since no further re-duction is possible, it will be a member of the output basis.

In order to show the correctness of the algorithm, we need to discuss thestructure and the properties of the elements f = fi Si+1 − fi+1Si+2 which arethe normal forms of those elements h which are either members of the originalinput basis or a produced S-polynomial:

• If Ecart( f ) = 0, then Head(h) has been reduced to 0 and a new minimalbasis member

Ψ −1(Head( f )) = fi Si+1 = Ψ −1((Tail(h))

of Im(δi+1) = ker(δi ) has been produced; f has been modified by the ad-dition of Tail( f ) = Si+2 Dd T j and marked as a minimal basis element.The insertion of Tail( f ) = Si+2 Dd T j has the effect of introducing a tag-variable to denote this new minimal basis element Head( f ), preservingEquation (26.2).

• If Ecart( f ) = 1, then Ψ −1(Head( f )) is a Grobner basis element of Im(δi+1)

but δi+1(Ψ−1(Tail( f ))) gives a representation of Head( f ) in terms of the

heads of the marked elements, showing that Head( f ) is not a minimalelement.

In conclusion, when the computation is ended and an output basis G is pro-duced, then, for each i we have:

• the set Ψ −1(Head( f )) : f = fi Si+1 − fi+1Si+2 ∈ G is a Grobner basisof Im(δi+1) = ker(δi );

• the subset Ψ −1(Head( f )) : f = fi Si+1 − fi+1Si+2 ∈ G, f is marked isa minimal basis of Im(δi+1) = ker(δi );

• for any marked element f = fi Si+1 − fi+1Si+2 ∈ G, Tail( f ) being amonomial, Tail( f ) = Si+1 Dd T j = Ψ (e(i)

j ), and the resolution morphismδi+1 is defined by Equation (26.2).

Part four

Duality

And when he had opened the fourth seal, I heard the voice of the fourth beast say, Comeand see.

And I looked and behold a pale horse: and his name that sat on him was Death, andHell followed with him. And power was given unto them over the fourth part of theearth, to kill with sword, and with hunger, and with death, and with the beasts of theearth.Revelation (Authorized Verson)

The things depending from Venus: semen, copper, emerald, thyme, goat, swan, crane.E. C. Agrippa, De occulta phylosophia

A spectre is haunting Europe – the spectre of communism. All the powers of old Europehave entered into a holy alliance to exorcise this spectre: Pope and Tsar, Metternich andGuizot, French Radicals and German police-spies.Karl Marx and Fredrick Engels, Manifesto of the Communist Party

27

Noether

The Lasker–Noether Theorem which generalized polynomial factorization tothe multivariate case, stating that each ideal I ⊂ k[X1, . . . , Xn] =: P has an(essentially unique) decomposition I = ⋂r

i=1 qi into primary ideals is the the-oretical tool needed to extend the notion of ‘solving’ from the univariate to themultivariate case: ‘solving’ – and ‘computing’ Z(I) – now means to produce

each associated prime pi := √qi of I by means of an admissible sequence

f1, . . . , fr , thus producing, in the Kronecker Model, the ‘solution’(β1, . . . , βn) ∈ Ω(k)n – Ω(k) denoting the universal field (Defini-tion 9.4.1) of k – as

k[β1, . . . , βn] ∼= k[X1, . . . , Xn]/( f1, . . . , fr ) = P/√

q; a description of the ‘multiplicity’ of each such ‘root’, or, more formally, of

the corresponding primary qi .

This chapter is devoted to the Lasker–Noether Theorem: I begin withNoether’s intuition of interpreting Hilbert’s Basissatz as the non-existence ofa proper infinite increasing chain of ideals, that is with Noetherianity (Sec-tion 27.1), thus giving to Lasker’s result both finiteness and the strongest pos-sible uniqueness results.

I then introduce the terminology (prime, primary, radical, maximal ideals)needed to generalize factorization from the univariate to the multivariate caseand the properties related to this concept (Section 27.2).

I can then introduce the notion of Lasker–Noether decomposition and prove(Section 27.3) the strongest existence result given by Noether: each ideal hasa decomposition I = ⋂r

i=1 ii where each ii is an irredundant irreducible,reduced 1 primary ideal.

1 In the sense, that it is a maximal solution.

335

336 Noether

The non-uniqueness of reduced embedded primary components 2 led tothe weaker result according to which each ideal has an irredundant primarydecomposition I = ∩r

i=1qi where the primary components are irredundantand each associated to different primes, but gives stronger uniqueness results(Section 27.4): the associated primes and the isolated primaries are unique.

A crucial tool for algorithms computing such decomposition is the ability toconnect the decompositions of an ideal I ⊂ P and its extension

Ik(X1, . . . , Xd)[Xd+1, . . . , Xn] ⊂ k(X1, . . . , Xd)[Xd+1, . . . , Xn];I therefore discuss in Section 27.5 the connection between the ideal decompo-sitions within two commutative Noetherian rings R and S with identity whichare connected by a homomorphism Φ : R −→ S such that φ(1) = 1. Ialso extend the decomposition theory from affine to homogeneous ideals (Sec-tion 27.6) and I discuss the notion of closure of an ideal at the origin (Sec-tion 27.7).

Since both the geometry and the computational complexity of an idealstrongly depend on the frame of coordinates, after introducing the notationneeded to discuss a generic system of coordinates (Section 27.8) I introduce(Section 27.9) van der Waerden’s definition of dimension of a prime ideal p, –which is the transcendental degree of P/p – and the notion of Noether positionof an ideal I ⊂ P w.r.t. a frame of coordinates Y1, . . . , Yn – for each asso-ciated prime p, d := dim(p), P/p is integral over k[Y1, . . . , Yd ] – and I showthat a generic frame is a Noether position for I.

Aiming to give a complete characterization of the notion of dimension Idiscuss chains of prime ideals (Section 27.10) and Grobner characterizationof dimension that is the basis of the Kredel–Weispfenning algorithm whichdeduces the dimension of an ideal by the consideration of its Grobner basisw.r.t. any term ordering (Section 27.11).

I then limit to an analysis of the Grobnerian structure of a zero-dimensionalideal, thus allowing me (Section 27.12) to introduce Macaulay’s notion of mul-tiplicity of a zero-dimensional ideal I ⊂ P as the cardinality of N<(I) – where< is any term ordering.

Finally, I conclude this survey on decomposition by introducing (Sec-tion 27.13) the notions of unmixed ideal, equidimensional decomposition andtop-dimensional components.

This chapter is therefore the keystone of the book, introducing the terminol-ogy and preliminary results needed to discuss multivariate ‘solving’ in the nextParts:

in the Present part, we discuss linear algebra tools to describe and computethe multiplicity of both m-primary and m-closed ideals;

2 Which apparently depend on the frame of coordinates.

27.1 Noetherian Rings 337

in the next Part we will formalize (Section 34.5) the notion of ‘solving’hinted at here and introduce the techniques and algorithms aimed atproducing a Lasker–Noether decomposition.

27.1 Noetherian Rings

The classical proof (Lemma 1.5.5) of the existence of finite factorization forunivariate polynomials applies induction on degree; that is essentially a politeway of presenting the trivial argument:

Take a polynomial f := f0: either it is irreducible or it has a factor f1 | f0; repeatedlyconsider fi | fi−1 which either is irreducible or has a factor fi+1 | fi .Since no infinite sequence of polynomials with decreasing degree can exist, f := f0has an irreducible factor p0; repeatedly consider f1 := f0/p0 which in turns has anirreducible factor p1.

Since no infinite sequence f0, f1, . . . , fi | fi−1, of polynomials with decreasingdegree can exist, f = ∏r

i=0 pi .

In order to mimic the proof of the univariate factorization theorem in themultivariate case, one needs to guarantee that no infinite chain of ideals suchthat

a1 ⊂ a2 ⊂ · · · ⊂ ai ⊂ ai+1 · · ·can exist. Emmy Noether’s idea is to link this property with the Hilbert Basissatz.

Theorem 27.1.1. Let R be a commutative ring with unity.3 Then the followingconditions are equivalent:

(1) Given an infinite chain of ideals ai ⊂ R

a1 ⊆ a2 ⊆ · · · ⊆ ai ⊆ ai+1 · · ·there exists N ∈ N : aN = ai , for each i > N .

(2) Every chain of ideals ai ⊂ R such that ai ⊂ ai+1, for each i , is finite.(3) In every non-empty family of ideals in R there is a maximal ideal, that

is an ideal which is not contained in any other ideal in the family.(4) Every ideal a ⊂ R has a finite basis.

Proof.

(1) ⇐⇒ (2) This is trivial.(2) ⇒ (3) Consider a non-empty family F of ideals in R. Since F is not

empty there is an ideal a1 contained in it; if a1 is not maximal inF , there is another element a2 in F such that a2 ⊃ a1; if a2 is notmaximal in F , there is another element a3 in F such that a3 ⊃ a2 und

3 Throughout the whole chapter, the ring R should be assumed to be commutative and with unity,even where I forget to state it.

338 Noether

so weiter. By assumption, in a finite number of steps we find in F amaximal element.

(3) ⇒ (4) Let a ⊂ R and consider the family of all ideals b ⊆ a whichhave a finite basis; such a family, being non-empty because it containsat least (0), has a maximal element b.Now, for each a ∈ a, the ideal b + (a) ⊇ b also has a finite basis, sothat a ∈ b + (a) = b by the maximality of b; this implies that b = a

and a also has a finite basis.(4) ⇒ (1) Consider an infinite chain of ideals ai ⊂ R

a1 ⊆ a2 ⊆ · · · ⊆ ai ⊆ ai+1 ⊆ · · ·and consider the set b := ⋃∞

i=1 ai which is an ideal since

• if a ∈ b, there is n : a ∈ an , so that, for each r ∈ R, ra ∈ an ⊆ b;• if a1, a2 ∈ b, there are ni : ai ∈ ani , i = 1, 2; then, setting N :=

max(n1, n2), a1 − a2 ∈ aN ⊆ b.

Therefore b has a finite basis a1, . . . , ar and for each j there existsn j such that a j ∈ an j so that setting N := max(n j , 1 ≤ j ≤ r) wehave, for each i ≥ N

aN ⊆ ai ⊆ b ⊆ aN .

Definition 27.1.2. A commutative ring R with unity which satisfies all the con-ditions of the theorem above is called Noetherian.

Lemma 27.1.3. Let R be a Noetherian ring.If a property is valid for every ideal a ⊂ R whenever the property holds for

each ideal b ⊃ a,4 the property is valid for all ideals.

Proof. Consider the family of all ideals for which the property is not valid.Then, if there is an ideal for which the property does not hold, so that the familyis not empty, there is a maximal ideal a for which the property is not valid.

As a consequence, the property holds for each ideal b ⊃ a and by assump-tion also for a, thus contradicting the assumption that there is an ideal forwhich the property does not hold.

As a direct consequence of Hilbert’s Basissatz (Theorem 20.8.1) we have

Proposition 27.1.4. k[X1, . . . , Xn] is Noetherian.

For historical reasons, we gave as proofs of the Hilbert Basissatz essentiallythe original arguments by Hilbert and Gordan; this obliges me to give a moreelegant and stronger version of Hilbert’s original argument which holds forpolynomial rings over a Noetherian ring.

4 So that, in particular, it is necessarily valid for a = R.

27.1 Noetherian Rings 339

Lemma 27.1.5. If R is a Noetherian ring, so is R[X ].

Proof. Let A ⊂ R[X ] be an ideal and let a ⊂ R be the set of the leadingcoefficients lc( f ) := an of all the polynomials f = ∑n

i=0 ai Xi ∈ A, an = 0.The set a is an ideal since, if

a = lc( f ), b = lc(g), a = b, f =n∑

i=0

ai Xi , g =m∑

j=0

b j X j , n ≥ m, f, g ∈ A,

then

• ra = lc(r f ), for each r ∈ R and• a − b = lc(h), where h = f − Xn−m g.

Since R is Noetherian there are polynomials f1, . . . , fs ∈ A such that ai :=lc( fi ) are a basis of a = (a1, . . . , as). Let

di := deg( fi ), N := maxdi , 1 ≤ i ≤ s and G N := f1, . . . , fs ⊂ A.

For any polynomial f ∈ A : δ := deg( f ) ≥ N , we have lc( f ) ∈ a,and there are bi ∈ R : a = ∑s

i=1 bi ai . Therefore if we set g := f −∑si=1 bi X δ−di fi then deg(g) < δ.

Repeating this procedure we deduce the existence of h1, . . . , hs ∈ R[X ] andh ∈ A such that

f =s∑

i=1

hi fi + h, deg(h) < N , f − h ∈ (G N ).

In order to complete the proof we have to show what happens for polynomi-als f, deg( f ) = δ < N , and we will do this by decreasing induction on δ. Wewill therefore assume that we have a basis Gδ+1 := f1, . . . , ft such that

• G N ⊂ Gδ+1 ⊂ A and• for each f ∈ A, deg( f ) > δ, there exists h1, . . . , ht ∈ R[X ] and h ∈ A

such that

f =t∑

i=1

hi fi + h, deg(h) ≤ δ, f − h ∈ (Gδ+1),

and we will prove the existence of a basis Gδ := f1, . . . , fu such that

• Gδ+1 ⊂ Gδ ⊂ A and• for each f ∈ A, deg( f ) ≥ δ, there exists h1, . . . , hu ∈ R[X ] and h ∈ A

such that

f =u∑

i=1

hi fi + h, deg(h) < δ, f − h ∈ (Gδ).

340 Noether

Let us therefore consider the ideal b ⊂ R of the leading coefficients of allpolynomials f ∈ A, deg( f ) = δ; since R is Noetherian there are polynomials ft+1, . . . , fu ∈ A such that deg( fi ) = δ and, setting ai := lc( fi ), we haveb := (at+1, . . . , au).

Therefore if f ∈ A, deg( f ) = δ, there are bi ∈ R such that

lc( f ) =u∑

i=t+1

bi ai , h := f −u∑

i=t+1

bi fi ∈ A and deg(h) < δ.

The basis

Gδ := f1, . . . , fu = Gδ+1 ∪ ft+1, . . . , futherefore satisfies the required property.

Corollary 27.1.6. If R is Noetherian, the polynomial rings R[X1, . . . , Xn] arealso Noetherian.

Lemma 27.1.7. Let R be a Noetherian ring and d ⊂ R an ideal; then theresidue class ring R′ := R/d is also Noetherian.

Proof. Denote by Φ : R −→ R′ the canonical projection. For each ideala ⊂ R′,

Φ−1(a) := b ∈ R : Φ(b) ∈ a ⊂ R

is an ideal and, R being Noetherian, has a finite basis b1, . . . , bs. Thenψ(b1), . . . , ψ(bs) is a finite basis of a; in fact, let a ∈ a and b ∈ R : Φ(b) =a. Then there are di ∈ R such that b = ∑s

i=1 di bi , whence

a = Φ(b) =s∑

i=1

Φ(di )Φ(bi ).

27.2 Prime, Primary, Radical, Maximal Ideals

In order to generalize the factorization theorem from the univariate to the mul-tivariate case, one needs to generalize to ideals a ⊂ R the main notions of ir-reducible element, power of an irreducible element, squarefree and squarefreeassociate. The way of doing this is to consider at the same time the divisibilityproperty and the structure of the quotient ring R/a.

We begin by recalling and extending (see Definition 20.1.9)

Definition 27.2.1. An ideal a ⊂ R is called radical (or: squarefree) if,

for each f ∈ R, ρ ∈ N, f ρ ∈ a ⇒ f ∈ a.

27.2 Prime, Primary, Radical, Maximal Ideals 341

The radical√

a of an ideal a ⊂ R is the ideal√

a := f ∈ R : there exists ρ ∈ N, f ρ ∈ a.Proposition 27.2.2. Let p ⊂ R. The following conditions are equivalent:

(1) for each b, c ∈ R, bc ∈ p, b /∈ p ⇒ c ∈ p;(2) for each b, c ∈ R, b /∈ p, c /∈ p ⇒ bc /∈ p;(3) R/p is an integral domain, that is it has no zero-divisor;(4) for each ideal b, c ∈ R such that bc ⊆ p, we have

b ⊆ p ⇒ c ⊆ p.

Proof.

(1) ⇐⇒ (2) This is trivial.(2) ⇐⇒ (3) Denoting by Φ : R −→ R/p the canonical projection, one has

bc ∈ p ⇐⇒ Φ(b)Φ(c) = 0.

(2) ⇒ (4) If b ⊆ p there is b ∈ b such that b /∈ p. Then for each c ∈ c,bc ∈ bc ⊆ p and, since b /∈ p, c ∈ p, proving c ⊆ p.

(4) ⇒ (2) Assume there are b, c /∈ p : bc ∈ p and write b := p + (b),c := p + (c). Then bc ⊆ p, b ⊆ p, c ⊆ p.

Definition 27.2.3. An ideal p ⊂ R is called prime if it satisfies the equivalentconditions of the proposition above.

Corollary 27.2.4. Let p ⊂ R be a prime ideal; then

(1) if a ⊂ R is an ideal for which there exists ρ : aρ ⊆ p, then a ⊆ p;(2) for each a ∈ R for which there exists ρ : aρ ∈ p one has a ∈ p;(3) p = √

p.

Proof.

(1) If a ⊆ p, from aaρ−1 ⊆ p one deduces aρ−1 ⊆ p; repeating the sameargument one deduces that aρ−2 ⊆ p und so weiter until the deductiona2 ⊆ p is reached, from which we get the contradiction a ⊆ p.

(2) Follows from the statement above setting a := (a).

(3) Follows from (2) and the definition of radical.

If R := Z and p is prime, then Z/(p) is a field and the same happens in theunivariate case R := k[X ]. Of course, this is not true in the multivariate case:if we consider R := k[X, Y ] and the prime p := (Y ), the quotient R/p ∼= k[X ]is an integer domain but not a field.

342 Noether

Definition 27.2.5. An ideal m ⊂ R is called maximal if there is no ideal a

such that m ⊂ a ⊂ R.

Proposition 27.2.6. An ideal m ⊂ R is maximal iff R/m is a field.

Proof. Denote by Φ : R −→ R/m the canonical projection.Let us assume that m is maximal and let us prove that for each b ∈ R/m, b =

0, there exists c ∈ R/m such that cb = 1.

Let a ∈ R be such that Φ(a) = b so that a /∈ m and the ideal m + (a)

coincides with R. As a consequence there are m ∈ m, c′ ∈ R such that m +c′a = 1, that is Φ(c′)b = Φ(m + c′a) = 1.

Conversely, if R/m is a field, its only ideals are (1) = Φ(R) and (0) =Φ(m) which proves the maximality of m.

Corollary 27.2.7. Any maximal ideal m ⊂ R is prime.

Definition 27.2.8. An ideal q ⊂ R is called primary if

for each b, c ∈ R : bc ∈ q, b /∈ q ⇒ there exists ρ ∈ N : cρ ∈ q.

Corollary 27.2.9. Let q ⊂ R; the following conditions are equivalent:

(1) q is primary;(2) every zero-divisor of R/q is nilpotent;(3) for each b, c ∈ R, bc ∈ q, b /∈ q, c /∈ q ⇒ there exist ρ, σ ∈ N such

that bρ ∈ q, cσ ∈ q;(4) for each b, c ∈ R for which bc ∈ q, if cρ /∈ q for each ρ ∈ N, then

b ∈ q.

Proposition 27.2.10. Let q ⊂ R be a primary ideal and let p := √q. Then we

have:

(1) p is prime;(2) for each b, c ∈ R, bc ∈ q, b /∈ q ⇒ c ∈ p;(3) for each ideal b, c ⊂ R such that bc ⊆ q we have

b ⊆ q ⇒ c ⊆ p;

(4) for each ideal b ⊂ R, we have

b ⊆ p ⇒ q : b = q.

Proof.

(1) Assume bc ∈ p = √q and b /∈ p; since bc ∈ √

q, (bc)µ = bµcµ ∈ q

for some µ; since b /∈ p, then bµ /∈ q and there exists ν : (cµ)ν ∈ q sothat c ∈ √

q = p.

27.2 Prime, Primary, Radical, Maximal Ideals 343

(2) This follows directly from the definition.(3) If b ⊆ q there is b ∈ b such that b /∈ q. Then, for each c ∈ c, bc ∈ bc ⊆

q and, since b /∈ q, there exists ρ : cρ ∈ q, that is c ∈ √q. This proves

c ⊆ p.(4) By Theorem 26.3.2(15) we have b (q : b) ⊆ q; therefore, by the pre-

vious statement, q : b ⊆ q. The claim then follows, since the otherinclusion is trivial.

Proposition 27.2.11. Let q ⊂ R be a primary ideal and let p := √q.

For each ideal b ⊂ R, we have

(1) b ⊆ q ⇐⇒ q : b = R;(2) b ⊆ q, b ⊆ p ⇐⇒ q ⊂ q′ := q : b ⊂ R;(3) b ⊆ p ⇐⇒ q : b = q.

In case (2) we also have p = √q′ and pρ ⊆ q ⇒ pρ−1 ⊆ q′.

Proof.

(1) Obviously b ⊆ q ⇒ q : b = R.

(2) There is ρ such that pρ ⊆ q ⊆ p; if we take ρ minimal, then pρ−1 ⊆ q

and there exists c ∈ R : c ∈ pρ−1 \ q.

If b ⊆ p we have cb ⊆ pρ ⊆ q and c ∈ q′ := q : b; since c /∈ q wehave

b ⊆ q, b ⊆ p ⇒ q q′ := q : b ⊂ R.

Let b be any element such that b ∈ b \ q; then for each a ∈ q′ \ q, sinceab ∈ q, there exist ρ, σ : aρ, bσ ∈ q and a ∈ √

q = p. Since alsoq ⊂ p we have q′ ⊆ p. Moreover

pρ−1 · b ⊆ pρ−1 · p = pρ ⊆ q

and pρ−1 ⊆ q′. Therefore pρ−1 ⊆ q′ ⊆ p and p = √q′.

(3) Since (Theorem 26.3.2(15)) b (q : b) ⊆ q and

bc ⊆ q, b ⊆ q ⇒ c ⊆ p

we deduce

b ⊆ p ⇒ q : b = q.

The statement now follows by trichotomy.

Corollary 27.2.12. Let q ⊂ R be a primary ideal and let p := √q.

For each ideal b ⊂ R, we have

(1) b ⊆ p ⇒ q : b∞ = q,(2) b ⊆ p ⇒ q : b∞ = R.

344 Noether

Proof. If b ⊆ p then, inductively, q : bn = q, for each n ∈ N.

If b ⊆ p, since, for some ρ ∈ N, pρ ⊆ q, then bρ ⊆ pρ ⊆ q, that is1 ∈ q : bρ.

Definition 27.2.13. Let q ⊂ R be a primary ideal; the prime ideal p := √q

is called the associated prime ideal of q and we will say that q is a primarybelonging to p.

The minimal value, whose existence is proved below, ρ ∈ N such that pρ ⊆ q

is called the characteristic number or the exponent of q.

Proposition 27.2.14. Let R be Noetherian, q ⊂ R be a primary ideal and p

its associated prime. Then

(1) there is ρ ∈ N such that pρ ⊆ q;(2) for each ideal b, c ⊂ R,

bc ⊆ q, b ⊆ q ⇒ there exists ρ ∈ N : cρ ⊆ q.

Proof.

(1) Let P := p1, . . . , pr be a basis of p and, for each i let ρi ∈ N be suchthat pρi

i ∈ q; set ρ := 1+∑ri=1(ρi −1); pρ is generated by all products

of ρ instances of elements in P . In each such product b at least one pi

must occur at least ρi times, so that b ∈ q.

(2) Under the assumption one has c ⊆ p, whence

cρ ⊆ pρ ⊆ q.

Proposition 27.2.15. Let q, p ⊂ R be ideals such that:

(1) for each b, c ∈ R, bc ∈ q, b /∈ q ⇒ c ∈ p;(2) q ⊆ p;(3) for each c ∈ R, c ∈ p ⇒ there exists ρ ∈ N such that cρ ∈ q.

Then q is primary and p is its associated prime.

Proof. For each b, c ∈ R, bc ∈ q, b /∈ q ⇒ c ∈ p by (1), so that, by (3)there exists ρ, cρ ∈ q. This proves that q is primary.

In order to prove that p is its associated prime we must prove that for eachb ∈ R, if there exists ρ ∈ N such that bρ ∈ q then b ∈ p. Consider the minimalρ ∈ N such that bρ ∈ q: if ρ = 1 the claim follows by (2); if ρ > 1 thenbρ−1b ∈ q, bρ−1 /∈ q so that, by (1), b ∈ p.

Corollary 27.2.16. Let q1 and q2 be two primary ideals in R belonging to p.Then also q := q1 ∩ q2 is a primary ideal in R belonging to p.

27.3 Lasker–Noether: Existence 345

Proof. It is sufficient to prove that q satisfies the conditions of the above propo-sition.

(1) If bc ∈ q and b /∈ q = q1 ∩ q2, then wlog b /∈ q1 and c ∈ p.

(2) This is trivial.(3) For each c ∈ R, c ∈ p there exist ρ1, ρ2 : cρi ∈ qi , i = 1, 2. Then

setting ρ := maxρ1, ρ2 we have cρ ∈ q.

Corollary 27.2.17. Let q, p ⊂ R be ideals such that

(1) p is maximal,(2) q ⊆ p,(3) for each c ∈ R, c ∈ p ⇒ there exists ρ ∈ N such that cρ ∈ q.

Then q is primary and p is its associated prime.

Proof. We need to verify that for each b, c ∈ R,

bc ∈ q, b /∈ q ⇒ c ∈ p :

Assume c /∈ p. Then p + (c) ⊃ p so that, by the maximality of p, p + (c) = Rand exist m ∈ p, d ∈ R such that 1 = m + dc. By (3), there exists ρ ∈ N suchthat mρ ∈ q so that

1 = 1ρ = (m + dc)ρ = mρ + d ′c

for a suitable d ′ ∈ R. Hence b = mρb + d ′(bc) ∈ q.

Proposition 27.2.18. Let q ⊂ R be a primary ideal such that√

q is maximaland let m ⊂ R be any ideal such that m ⊂ √

q.Then q + m = R.

Proof. Let f ∈ m, f ∈ √q.

Then√

q + ( f ) √

q and, by maximality,√

q + ( f ) = R.

As a consequence, there are p ∈ √q, a, b ∈ R, ρ ∈ N such that pρ ∈ q and

ap + b f = 1 so that, for the suitable element c ∈ R

1 = 1ρ = (ap + b f )ρ = apρ + c f ∈ q + ( f ) ⊆ q + m.

27.3 Lasker–Noether Decomposition: Existence

Definition 27.3.1 (Noether). Let R be a commutative ring with unity and leta ⊂ R be an ideal.

346 Noether

Then a is said to be

• reducible if there are two ideals b, c ⊂ R such that

a = b ∩ c, b ⊃ a, c ⊃ a;• irreducible if it is not reducible.

Mimicking the univariate proof of Lemma 1.5.3, substituting Lemma 27.1.3as induction tool in place of the degree induction, we can easily prove:

Proposition 27.3.2 (Lasker–Noether). In a Noetherian ring R each ideal f ⊂R is a finite intersection of irreducible ideals: f = ⋂r

i=1 ii .

Proof. The property being obviously true for the irreducible ideal (1), in orderto prove the theorem it is sufficient to prove that the property holds for an idealf ⊂ R, provided that we have for each ideal f′ ⊃ f.

Let us consider any ideal f ⊂ R: either it is irreducible and we are through,or it has a decomposition

f = f1

⋂f2, f1 ⊃ f, f2 ⊃ f.

By inductive assumption both f1 and f2 are finite intersections of irreducibleideals:

f1 =r⋂

i=1

ri , f2 =s⋂

j=1

s j

whence also

f =(

r⋂i=1

ri

)∩

(s⋂

j=1

s j

)

is a finite intesection of irreducible ideals.

Definition 27.3.3 (Noether). Let R be a Noetherian ring and f ⊂ R an ideal.A representation f = ⋂r

i=1 ii , of f as intersection of finite irreducible ideals iscalled a reduced representation if, for each I, 1 ≤ I ≤ r ,

• iI ⊇ ⋂ri=1i =I

ii , and

• there is no irreducible ideal i′I ⊃ iI such that f =( ⋂r

i=1i =I

ii

)∩ i′I .

Proposition 27.3.4 (Noether). In a Noetherian ring R, each ideal f ⊂ R hasa reduced representation as intersection of finite irreducible ideals.

Proof. If in a decomposition f = ⋂ri=1 ii there is an irreducible component iI

such that iI ⊇ ⋂ri=1i =I

ii , then we obtain a better decomposition f = ⋂ri=1i =I

ii ,

removing the useless component iI .


Moreover, if in a decomposition f = ∩ri=1ii , it could happen that an ir-

reducible component iI can be replaced by another component i′I such that

i′I ⊃ iI and f =( ⋂r

i=1i =I

ii

)∩ i′I , this substitution can however be performed

only finitely many time, since otherwise we would have an infinite chain ofirreducible components

iI := iI 0 ⊂ iI 1 ⊂ · · · ⊂ iI j ⊂ iI j+1 ⊂ · · ·all satisfying

( ⋂ri=1i =I

ii

)∩ iI j thus contradicting Noetherianity.

Proposition 27.3.5. A prime ideal is irreducible.

Proof. In fact, if for the prime ideal p there were two ideals b, c ⊂ R such that

p = b ∩ c ⊇ bc, b ⊃ p, c ⊃ p,

this would contradict condition (4) of Proposition 27.2.2.

Example 27.3.6. Unlike prime ideals, primary ideals are not irreducible. Theeasiest example is the primary ideal

(X, Y )2 = (X2, XY, Y 2) ⊂ k[X, Y ]

whose associated prime is (X, Y ) and which has the decomposition

(X2, XY, Y 2) = (X2, Y ) ∩ (X, Y 2).

On the other hand the converse is true:

Lemma 27.3.7. In a Noetherian ring R, every irreducible ideal is primary.

Proof. Assume that f ⊂ R is not primary. Therefore there are b, c ∈ R suchthat

bc ∈ f, b /∈ f, cρ /∈ f for each ρ ∈ N.

Let us then consider the ideals bρ := f : cρ which form an infinite chain since,for eachρ, f : cρ ⊆ f : cρ+1. Therefore there existsρ such that f : cρ = f : cρ+1.

Now we intend to prove that

f ⊃ (f + (b)) ∩ (f + (cρ)

) ;this is sufficient to prove that f is reducible since

f ⊂ (f + (b)) ∩ (f + (cρ)

), f ⊂ f + (b) and f ⊂ f + (cρ).

Let us therefore consider an element

a ∈ (f + (b)) ∩ (f + (cρ)

).

348 Noether

Since a ∈ f + (cρ) there are f ∈ f, r ∈ R such that a = f + rcρ.

Moreover

a ∈ f + (b) ⇒ ac ∈ f + (bc) ⊆ f.

Hence

f c + rcρ+1 = ac ∈ f, rcρ+1 ∈ f, r ∈ f : cρ+1 = f : cρ, rcρ ∈ f

and, finally, a = f + rcρ ∈ f.

Definition 27.3.8. Let R be a Noetherian ring and f ⊂ R an ideal. A repre-sentation, f = ⋂r

i=1 qi , of f as intersection of finite primary ideals – where, foreach i , pi denotes the associate prime pi := √

qi of qi – is called an irredun-dant primary representation if

• for each I, 1 ≤ I ≤ r, qI ⊇ ⋂ri=1i =I

qi ;• for each i, j, 1 ≤ i < j ≤ r, pi = p j .

A component qI of such an irredundant primary representation is calledreduced if there is no primary ideal q′

I ⊃ qI such that f =( ⋂r

i=1i =I

qi

) ⋂q′

I .

Corollary 27.3.9. In a Noetherian ring R, each ideal f ⊂ R has an irredun-dant primary representation f = ⋂r

i=1 qi .Moreover each qi can be chosen to be reduced.

Proof. Let f = ⋂ri=1 ii be a reduced representation of f as an intersection of fi-

nite irreducible ideals and, for each i , pi be the associate prime pi := √ii of ii .

We can transform such a representation into an irredundant primary repre-sentation by

considering a subset J ⊆ i : 1 ≤ i ≤ r such that

pi , i ∈ J = pi , 1 ≤ i ≤ r,denoting, for each i ∈ J ,

Ji := j : p j = pi and qi := ⋂j∈Ji

i j

and setting f = ⋂i∈J qi

which is an irredundant primary representation because

• by construction,√

qi = √q j for each i, j ∈ J ;

• for each j ∈ J, q j ⊇ ⋂i∈Ji = j

qi , since otherwise we would get, for eachI ∈ J j , the contradiction

r⋂i=1i =I

ii ⊆⋂i∈Ji = j

qi ⊆ q j ⊆ iI ;


• for each j ∈ J, q j is a primary belonging to p j because it is the intersectionof primaries belonging to p j (Corollary 27.2.16),

Moreover, each component can be assumed to be reduced by the same Noethe-rianity argument as in Proposition 27.3.4.

Proposition 27.3.10. Let R be a Noetherian ring, f ⊂ R an ideal, f = ∩ri=1qi

an irredundant primary representation of f, pi the associated prime of qi foreach i . Then:

(1) for any ideal a ⊂ R, f : a = f ⇐⇒ a ⊆ pi , for each i;(2) for any a ∈ R, f : a = f ⇐⇒ a ∈ pi , for each i;(3) f = √

f ⇐⇒ qi = pi , for each i;(4) for a prime p ⊂ R, p ⊇ f ⇒ there exists i : p ⊇ pi ;(5) if r > 1, f is not primary.

Proof.

(1) Assume a ⊆ pi , for each i; then (Proposition 27.2.10(4)) qi : a = qi ,

for each i. As a consequence, using Theorem 26.3.2(19)

f : a =⋂

i

qi : a =⋂

i

qi = f.

Conversely, assume f : a = f and remark that, for any ideal d, we have

f : a = f, ad ⊆ f ⇒ d ⊆ f.

By contradiction, let us assume that a ⊆ pi ; therefore, for some ρ,aρ ⊆ qi .As a consequence, setting c := ⋂r

j=1i = j

q j we have

aρc ⊆ qi ∩ c = f.

Therefore, for each σ, 1 ≤ σ ≤ ρ, we have

aσ c ⊆ f ⇒ aσ−1c ⊆ f;in fact, setting d := aσ−1c we have

ad = a(aσ−1c

)= aσ c ⊆ f ⇒ d ⊆ f.

By decreasing induction on σ , we can therefore deduce the contradic-tion

r⋂j=1i = j

q j = c ⊆ f ⊂ qi .

350 Noether

(2) Set a := (a) in the statement above.(3) If each qi is prime, from aρ∈ f we can deduce aρ∈ qi , a ∈ qi and a ∈ f.

Conversely, assume f = √f and let x ∈ ⋂r

i=1 pi ; therefore there existsρ ∈ N such that xρ ∈ qi for each i , and xρ ∈ f; so x ∈ √

f = f; as aconsequence f = ⋂r

i=1 pi .Such a representation is irredundant: if, for some I , f = ⋂r

i=1i =I

pi , than

we have a contradiction with the irredundancy of the representationf = ⋂r

i=1 qi :

f =r⋂

i=1i =I

pi ⊇r⋂

i=1i =I

qi ⊇ f.

Now let us consider y ∈ pi and z ∈ ⋂rj=1j =i

p j , z /∈ pi ; then

zy ∈r⋂

i=1

pi = f ⊂ qi

and, since z /∈ pi , we deduce y ∈ qi , proving qi = pi .(4) In fact p ⊇ f ⊇ ∏r

i=1 qi implies p ⊇ qi , for some i .(5) Let p1 be minimal among the pi s, that is there is no i = 1 such that

pi ⊂ p1.Therefore, for each i, 1 < i ≤ r, exists ai ∈ pi , ai /∈ p1 so that thereexists ρ, aρ

i ∈ qi , for each i, 1 < i ≤ r.Also, since the representation is irredundant, there is q ∈ q1 such thatq /∈ f.

Then m := q∏r

i=2 aρi ∈ f, while q /∈ f; therefore if we assume f is

primary, we could deduce that there is σ such that∏r

i=2 aρσi ∈ f ⊂ p1.

Since p1 is prime, we have ai ∈ p1 for at least an i , getting the requiredcontradiction.

27.4 Lasker–Noether Decomposition: Uniqueness

Theorem 27.4.1 (Noether). Let R be a Noetherian ring, f ⊂ R an ideal; let

f =r⋂

i=1

qi =s⋂

j=1

q′j

be two irredundant primary representations of f; for each i, j let pi (respec-tively p′

j ) be the associated prime of qi (respectively q′j ).

Then

• r = s,• for each i, 1 ≤ i ≤ r, there exists j : pi = p′

j ;• for each j, 1 ≤ j ≤ s, there exists i : p′

j = pi .

27.4 Lasker–Noether: Uniqueness 351

Proof. The statement being trivial if f is primary, the proof can be done byinduction on min(r, s) > 1. Let us consider a maximal element among the set

P := pi , 1 ≤ i ≤ r ∪ p′j , 1 ≤ j ≤ s

and wlog let us say it is p1.Let us now quotient by q1 getting

r⋂i=1

(qi : q1) =s⋂

j=1

(q′

j : q1

).

For each i > 1, q1 ⊆ pi , otherwise p1 ⊆ pi , contradicting the maximality ofp1.

If we assume that p1 = p′j , for each j , the same argument proves q1 ⊆ p′

jfor each j . As a consequence we would have

qi : q1 = qi , for each i > 1, q′j : q1 = q′

j , for each j and q1 : q1 = R,

whencer⋂

i=2

qi =r⋂

i=1

(qi : q1) =s⋂

j=1

(q′

j : q1

)=

s⋂j=1

q′j = f ⊆ q1

and a contradiction on the irredundancy of f = ⋂ri=1 qi .

Therefore, we can conclude that every maximal ideal in P occurs on bothrepresentations.

Let us now assume r ≤ s; our aim is to show that r = s and, by a suitablerenumbering, pi = p′

i , for each i .Let us renumber both representations so that p1 = p′

1 and let us quotient byq1q

′1 where, for each i, j > 1,

qi : q1q′1 = qi , q

′j : q1q

′1 = q′

j , q1 : q1q′1 = R, q′

1 : q1q′1 = R,

whencer⋂

i=2

qi =r⋂

i=1

(qi : q1q

′1

) =s⋂

j=1

(q′

j : q1q′1

)=

s⋂j=2

q′j .

By induction assumption, we can assume the results hold for any ideal whichhas an irredundant primary representation as intersection of less than r primaryideals. Therefore r = s and, up to renumbering, pi = p′

i , for each i > 1.

Definition 27.4.2. Let R be a Noetherian ring, f ⊂ R an ideal; let f = ⋂ri=1 qi

be an irredundant primary representation and, for each i , let pi be the associ-ated prime of qi .

The primes pi are called the associated prime ideals of f.A minimal element in pi , 1 ≤ i ≤ r is called an isolated prime ideal of f.The primes which are not isolated are called embedded.A primary qi is called a primary component of f and is called isolated or

embedded, according to whether pi is isolated or embedded.

352 Noether

Theorem 27.4.3 (Noether). Let R be a Noetherian ring, f ⊂ R an ideal; let

f =r⋂

i=1

qi =r⋂

i=1

q′i ,

be two irredundant primary representations of f; for each i, let pi be the asso-ciated prime of both qi and q′

i .If pi is isolated, then qi = q′

i .

Proof. Let us set c := ⋂rj=1j =i

q j and c′ := ⋂rj=1j =i

q′j .

Then by Proposition 27.3.10(1), qi : c = qi and q′i : c = q′

i , so that

qi = f : c = q′i ∩ (

c′ : c)

and qi ⊆ q′i . By symmetry we get q′

i ⊆ qi and qi = q′i .

Example 27.4.4 (Hentzelt). We will present here some examples which willshow that the statements about uniqueness of representation cannot be im-proved.

All the examples are ideals in the polynomial ring Q[X, Y ].

(1) The decomposition

(X2, XY ) = (X) ∩ (X2, XY, Y λ), for each λ ∈ N, λ ≥ 1,

where√

(X2, XY, Y λ) = (X, Y ) ⊃ (X) shows that embedded comp-onents are not unique; however,

(X2, Y ) ⊇ (X2, XY, Y λ), for each λ > 1,

shows that (X2, Y ) is a reduced embedded irreducible component andthat

(X2, XY ) = (X) ∩ (X2, Y )

is a reduced representation.(2) The decompositions

(X2, XY ) = (X) ∩ (X2, Y + aX), for each a ∈ Q,

where√

(X2,Y +aX) = (X,Y ) ⊃ (X), and, clearly, each (X2,Y +aX)

is reduced, show also that reduced representation is not unique; notethat, setting a = 0 we find again the decomposition (X2, XY ) = (X)∩(X2, Y ) found above.

Example 27.4.5. In the same context let us also record the reduced represen-tation

(X2, XY, Y λ) = (X2, Y ) ∩ (X, Y λ)

of the primary ideal (X2, XY, Y λ) into reduced irreducible components.


Also such a decomposition is not unique since we have

(X2, XY, Y λ) = (X2, Y + aX) ∩ (X, Y λ).

Let us also remark that these reduced irreducible components give the irre-dundant primary representations

(X2, XY ) = (X) ∩ (X2, XY, Y λ)

= (X) ∩ (X2, Y + aX) ∩ (X, Y λ)

= (X) ∩ (X2, Y + aX)

in terms of the reduced primary components.

Example 27.4.6 (Noether). In the same context it is worth recording the de-compositions (in Q[X, Y, Z ])

(X2, XY, Y λ) = (X2, XY, Y 2, Y Z) ∩ (X, Y λ),

(X2, XY, Y 2, Y Z) = (X2, Y ) ∩ (X, Y 2, Z),

whence

(X2, XY, Y λ) = (X2, XY, Y 2, Y Z) ∩ (X, Y λ)

= (X2, Y ) ∩ (X, Y λ)

because (X, Y 2, Z) ⊃ (X, Y λ).

We will show in Section 32.3 that in an irredundant primary decompositionof an ideal, for each embedded associated prime p it is possible to determine areduced primary component q associated to it, together with a reduced decom-position of q into irreducible components associated to p.

Remark 27.4.7. In connection with Example 27.4.4 it is worth quoting thecomments by Grobner:5

The fact that an embedded component is not uniquely determined gives the impressionthat the consequences of the Lasker–Noether Theorem, from a geometric point of view,are not very satisfactory, even without geometric meaning. But an accurate interpreta-tion proves that the relevant fact is in perfect agreement with the geometric needs. Infact, as can soon be seen, all polynomials contained in the ideal a = (X2, XY ) have thefixed factor X ; the other factor is an arbitrary polynomial which vanishes at the origin.Therefore the polynomials contained in a represent (reducible) algebraic curves whichcontain the line X = 0 and which have at least a double point in the origin.

The condition of containing the line X = 0 is expressed by the first componentq1 = (X); in order to have also a point which is (at least) double in the origin it issufficient to add the condition that the curve contains a point infinitely near the originin an arbitrary direction (but different from the line X = 0). This condition is expressedby the component q2 = (X2, Y + aX), in particular by q2 = (X2, Y ). Now nothing

5 In W. Grobner, Teoria degli ideali e geometria algebrica, Seminari INDAM 1962–63, p. 7.

354 Noether

changes if we add the further condition such that the curve also passes through npoints successively infinitely near the origin on the line X = 0, because, evidently, thevanishing at such points is already prescribed by the first component q1. This furthercondition is expressed by the component q2 = (X2, XY, Y n) and therefore also thisideal is useful for the same task.

Let us record here also a characterization of the (unique) associated primesof an ideal and of their (unique) isolated primary components:

Theorem 27.4.8. Let R be a Noetherian ring, f ⊂ R an ideal; let f = ⋂ri=1 qi

be an irredundant primary representation of f and, for each i , let pi be theassociated prime of qi .

Then:

(1) A prime ideal p ⊂ R is a prime component of f iff there exists c ∈ Rsuch that c ∈ f,

√(f : c) = p.

(2) For each i , let q′i := x ∈ R : (f : x) ⊂ pi . Then

• q′i ⊆ qi is an ideal;

• if pi is isolated, then q′i = qi .

Proof.

(1) Let us fix i, 1 ≤ i ≤ r ; then, since the representation is irredundant,there exists c ∈ R such that c ∈ ⋂r

j=1j =i

q j and c /∈ qi ; thereforeqi ⊆ (f : c) ⊆ pi .

If xy ∈ (f : c) and x /∈ pi then xyc ∈ f ⊂ qi whence yc ∈ qi sincex /∈ pi ; this allows us to conclude, from c ∈ ⋂r

j=1j =i

q j , that yc ∈ f, that

is y ∈ (f : c). Therefore (f : c) is a primary belonging to pi .Conversely, assume the existence of c ∈ R such that c ∈ f,

√(f : c) = p

for some prime p.Taking the radical of (f : c) = ⋂r

j=1

(q j : c

)we obtain p =⋂r

j=1

√(q j : c

).

The same argument which proved the other implication, applied to

f := q j , allows us to deduce√(

q j : c) = p j unless c ∈ q j in which

case the radical is R.In conclusion p is the intersection of some of the p j s; from Proposi-tion 27.3.10(4), this implies p ⊇ pi for some i , whence p = pi whilep ⊆ p j for j = i.

(2) It is obvious that x ∈ q′i ⇒ yx ∈ q′

i , for each y ∈ R.

Let us now consider x1, x2 ∈ q′i ; therefore there are y1, y2 ∈ R \ pi


such that y j x j ∈ f, j = 1, 2, and y1 y2(x1 − x2) ∈ f and y1 y2 ∈ pi ,proving x1 − x2 ∈ q′

i and the claim that q′i is an ideal.

Moreover, for each x ∈ q′i there exists c ∈ R \ pi such that xc ∈ f ⊂ qi

implying x ∈ qi and q′i ⊆ qi .

Let us now assume pi is isolated; as a consequence, for each j = i,there exists a j ∈ R such that a j /∈ pi , a j ∈ p j and there existsρ j , a

ρ jj ∈ q j for each j = i.

Then, for any x ∈ qi , x∏

j =i aρ jj ∈ f while

∏rj=2 a

ρ jj /∈ pi , implying

x ∈ q′i , whence q′

i = qi .

Let us note here the following result which we will need later.

Definition 27.4.9. For any ideal f its characteristic number is the minimalvalue ρ ∈ N such that (

√f)ρ ⊆ f.

Lemma 27.4.10. If f = ⋂ri=1 qi is an irredundant primary representation of

the ideal f ⊂ R, ρi is the characteristic number of qi , for each i , and ρ is thecharacteristic number of f, then

(1) ρ = maxi ρi ;(2) if p j is maximal, then f + p

ρj = q j .

Proof.

(1) We have√

f = ⋂ri=1 pi and

(√

f)ρ =r⋂

i=1

pρi ⊆

r⋂i=1

pρii ⊆

r⋂i=1

qi = f,

while, for any index i such that ρi = ρ, it is sufficient to take d ∈ pi

such that dρ−1 ∈ qi and any c ∈ R such that

c ∈r⋂

j=1j =i

q j and c /∈ qi ,

to obtain

cd ∈√

f, (cd)ρ−1 ∈(√

f)ρ−1

, (cd)ρ−1 ∈ qi , (cd)ρ−1 ∈ f,

and proving (√

f)ρ−1 ⊆ f.(2) Since p j is maximal, p j + qi = R, for each i = j so that

f + pρj =

r⋂i=1

(qi + p

ρj

)= q j + p

ρj = q j .

356 Noether

27.5 Contraction and Extension

Let us now consider two commutative rings with identity, R and S, and a ho-momorphism φ : R −→ S such that φ(1) = 1 and discuss the behaviour ofideals and ideal decomposition between the two rings.

Remark 27.5.1. The first case to be discussed is projection:6 let R be a Noethe-rian ring, d ⊂ R an ideal, S the residue class ring S := R/d, which also isNoetherian and φ : R −→ S the canonical projection.

Primality and primariety of a ⊂ R depend only on the properties of R/a sothey are preserved by φ as well as radicality –

√φ(a) = φ(

√a), – intersections

– φ(a1 ∩ a2) = φ(a1) ∩ φ(a2) – and inclusion.As a consequence, if we are given an ideal f′ ⊂ S, we set f := φ−1(f′) ⊇ d;

if f = ∩ri=1qi is an irredundant primary representation of f and, for each i , pi

is the associated prime of qi , then f′ = φ(f) = ⋂ri=1 φ(qi ) is an irredundant

primary representation, whose associated primes are φ(pi ); isolated and em-bedded primes are preserved by φ.

Example 27.5.2. In our context we are however mainly interested in the fol-lowing cases:

• R = k[X1, . . . , Xi ][Xi+1, . . . , Xn] and S = Li [Xi+1, . . . , Xn] wherepi ∈ k[X1, . . . , Xi ] is a prime and Li is the field Li := k[X1, . . . , Xi ]/pi

(see Chapter 8),• R = k[X1, . . . , Xn] and S = k(X1, . . . , Xd)[Xd+1, . . . , Xn],• more in general, if we consider a multiplicative system M ⊂ R, that is a set

such that

• m, n ∈ M ⇒ mn ∈ M,

• 1 ∈ M,

• 0 /∈ M

and we further assume that M does not contain zero-divisors, that is

for each r ∈ M, s ∈ R, rs = 0 ⇒ s = 0;then, denoting by ∼ the equivalence relation on R × M defined by

(r, m) ∼ (s, n) ⇐⇒ rn = sm,

the quotient ring

(r, m) : r ∈ R, m ∈ M/ ∼ =: r/m : r ∈ R, m ∈ M =: M−1 R

6 In connection with this remember Proposition 24.7.3.

27.5 Contraction and Extension 357

is a ring under the ‘natural’ extension of the ring structure of R;7 we can thenset S := M−1 R and φ : R −→ S to be the natural immersion φ(r) = r/1.

Definition 27.5.3. For any ideal A ⊂ S the ideal

Ac := φ−1(A) = a ∈ R : φ(a) ∈ A ⊂ R

is called the contraction of A.

For any ideal a ⊂ R the ideal

ae :=∑

i

aiφ(gi ), ai ∈ S, gi ∈ a

⊂ S

is called the extension of a.

Lemma 27.5.4. Let a, b ⊂ R, A,B ⊂ S be ideals. Then

(1) a ⊆ b ⇒ ae ⊆ be; A ⊆ B ⇒ Ac ⊆ Bc;(2) aec ⊇ a; Ace ⊆ A;(3) aece = ae; Acec = Ac;(4) (a + b)e = ae + be; (A + B)c ⊇ Ac + Bc;(5) (a ∩ b)e ⊆ ae ∩ be; (A ∩ B)c = Ac ∩ Bc;(6) (ab)e = aebe; (AB)c ⊇ AcBc;

(7)(√

a)e ⊆ √

ae;(√

A)c = √

Ac;

(8) (a : b)e ⊆ ae : be; (A : B)c ⊆ Ac : Bc;(9) B = be ⇒ (A : B)c = Ac : Bc;

(10) a = Ac, b = Bc ⇒ (ae : be)c = a : b;(11) if φ is a projection and ker(φ) ⊂ a then R/a ∼= S/ae;(12) if φ is a projection and ker(φ) ⊂ a then aec = a.

Proof. Most of the statements are trivial; as regards the others:

(2) If s ∈ Ace, then there are ai ∈ S, gi ∈ Ac such that s = ∑i aiφ(gi );

writing hi := φ(gi) we have hi ∈ A and s =∑i aiφ(gi) =∑

i ai hi ∈A.

(3) Using (2) we obtain both acec = (ace)c ⊆ ac, and acec = (ac)ec ⊇ ac.In the same way we obtain both Acec = (Ac)ec ⊇ Ac, and Acec =(Ace)c ⊆ Ac.

(8) If r ∈ (A : B)c, then for each b ∈ B, φ(r)b ∈ A; therefore for eachr ′ ∈ Bc, φ(r ′) ∈ B so that φ(rr ′) = φ(r)φ(r ′) ∈ A and rr ′ ∈ Ac;this implies r ∈ Ac : Bc.

The other statement follows by (6) and Theorem 26.3.2(15):

(a : b)e be = ((a : b) b)e ⊆ ae.

7 An instance of this construction has already been discussed in Lemma 26.3.10.

358 Noether

(9) Note thatB = be = bece = (

be)ce = Bce.

Therefore(Ac : Bc)e

B = (Ac : Bc)e

Bce = ((Ac : Bc) Bc)e ⊆ (

Ac)e ⊆ A,

that is (Ac : Bc)e ⊆ A : B

whence

Ac : Bc ⊆ (Ac : Bc)ec ⊆ (A : B)c .

Since, by (8), Ac : Bc ⊇ (A : B)c we are through.(10) Setting

B′ := be = Bce ⊆ B and A′ := ae = Ace ⊆ A

and remarking that

B′c = Bcec = Bc and A′c = Acec = Ac,

by the statement above we have(ae : be)c = (

A′ : B′)c = A′c : B′c = Ac : Bc = a : b.

(11) Let us denote by Φ : R −→ S/ae the canonical projection; clearly a ⊂ker(Φ); conversely, if a ∈ R is such that φ(a) ∈ ae, exist s1, . . . , sn ∈S, a1, . . . , an ∈ a such that φ(a) = ∑

i siφ(ai ); also, since φ is aprojection, for each i , there are ri ∈ R such that si = φ(ri ); thereforeb := a − ∑

ri ai ∈ ker(φ) ⊂ a, a ∈ a and ker(Φ) = a.(12) Denoting again by Φ : R −→ S/ae the canonical projection, we have

aec = φ−1(ae) = ker(Φ) = a.

The statement of (1) does not hold if we replace ⊆ with ⊂. In all the otherstatements ⊆ cannot be replaced by equality.

Remark 27.5.5. The statement (3) shows that the strict inclusions of (2)

become equality only for contracted and extended ideals. As a consequence ifwe consider the sets R (respectively S), of all the ideals a ⊂ R (respectivelyA ⊂ S), the maps · e : R −→ S and · c : S −→ R give a duality only on thesubsets

E := ae : a ∈ R ⊂ S and C := Ac : A ∈ S ⊂ R.

The statements above prove also that, while E is closed under sum andmultiplication, C is closed under intersection, radical and quotient; also,intersection, radical and quotient are preserved by · c.


Proposition 27.5.6. Let A ⊂ S be an ideal. Then

(1) if A is prime, so also is Ac;(2) if A is primary, so also is Ac.

Proof. Let x, y ∈ R : y ∈ Ac, xy ∈ Ac; then φ(x)φ(y) ∈ A, φ(y) ∈ A.

If A is primary, then there exists ρ ∈ N such that φ(xρ) = φ(x)ρ ∈ A andxρ ∈ Ac. This proves that Ac is primary.

The same argument, just putting ρ := 1, proves that Ac is prime if A issuch.

As a consequence of the fact that · c preserves radical formation, we canstate a stronger result:

Corollary 27.5.7. Let A ⊂ S be an ideal. Then

(1) if A is primary belonging to the prime B, then Ac is primary belongingto the prime Bc;

(2) if A is radical, so also is Ac.

Proof. If A is primary belonging to the prime B, then

Bc =(√

A)c =

√Ac

and the primary Ac belongs to the prime Bc.

If A is radical, the same argument, that is

Ac =(√

A)c =

√Ac,

proves that Ac also is radical.

The preservation of intersection, radical and quotient by · c allows thepreservation of primary decomposition:

Corollary 27.5.8. Let A ⊂ S be an ideal. Then, if A = ⋂i Qi is an irredun-

dant primary representation, then Ac = ⋂i Qc

i is a (not necessarily irredun-dant) primary representation.

In general, · e does not preserve primality, intersection, radical and quotientformation, thus not preserving primary decomposition.

Our aim now is to restrict ourselves to the case of a quotient ring S :=M−1 R and to prove that in this context · e preserves primality, intersection,radical and quotient formation, thus also preserving primary decomposition.

Let us therefore consider the quotient ring S := M−1 R and the naturalimmersion φ : R −→ S, where M is a multiplicative system containing nozero-divisor. In this setting we have

360 Noether

Theorem 27.5.9. With the notation above, for ideals a ⊂ R and A ⊂ S, wehave

(1) Ac = A ∩ R;(2) ae = a/m : a ∈ a, m ∈ M;(3) aec = r ∈ R : there exists m ∈ M, rm ∈ a;(4) a = aec ⇐⇒ a : m = a, for each m ∈ M;(5) Ace = A.

Proof.

(1) Trivial.(2) If a ∈ a, m ∈ M , then a/m = (1/m)a ∈ ae.

Conversely, if s ∈ ae, there exist ai ∈ a, ri ∈ R, mi ∈ M such thats = ∑

i (ri/mi )ai ; setting

m :=∏

i

mi ∈ M, ni :=∏j =i

m j = m/mi , a :=∑

i

ni ri ai ∈ a,

we obtain s = (∑

i ni ri ai )/m = a/m.

(3) We have

r ∈ aec ⇐⇒ r ∈ ae ∩ R

⇐⇒ there exists a ∈ a, m ∈ M : a = mr

⇐⇒ there exists m ∈ M : mr ∈ a.

(4) We have r ∈ a : m ⇐⇒ mr ∈ a ⇒ r ∈ aec; therefore

a = aec ⇒ a : m = a, for each m ∈ M.

Conversely, from

r ∈ aec ⇒∃ m ∈ M : rm ∈ a ⇐⇒ ∃ m ∈ M : r ∈ (a : m),

we obtain

a : m = a, for each m ∈ M ⇒ a = aec.

(5) We have just to prove A ⊆ Ace.Let s = r/m ∈ A with r ∈ R, m ∈ M ; then r = ms ∈ Ac =: a ands = r/m ∈ ae.

Corollary 27.5.10. If R is Noetherian, so is S.

Proof. For each ideal A ⊂ S we have A = (Ac)e and Ac ⊂ R is finitelygenerated. By definition, a basis of an ideal a ⊂ R is also a basis of ae.

Corollary 27.5.11. Let a ⊂ R; then

ae = (1) ⇐⇒ a ∩ M = ∅.


Proof. As a consequence of Theorem 27.5.9(3) we have

ae = (1) ⇐⇒ 1 ∈ aec ⇐⇒ there exists m ∈ M : m ∈ a.

Continuing the discussion of Remark 27.5.5 and using the same notation,we have


• for each a ∈ R : a ∈ C ⇐⇒ a : m = a, for each m ∈ M;• S = E .

Lemma 27.5.13. Let a, b ⊂ R be ideals. Then

(1) (a ∩ b)e = ae ∩ be;(2)

(√a)e = √

ae;(3) (a : b)e = ae : b = ae : (b)e, for each b ∈ R;(4) (a : b)e = ae : be.

Proof. For each statement, we just need to prove one inclusion.

(1) Let s ∈ S be such that s ∈ ae ∩ be; this implies there are a ∈a, b ∈ b, m, n ∈ M : s = a/m = b/n, na = bm ∈ a ∩ b ands = (na)/(nm) ∈ (a ∩ b)e .

(2) Let s ∈ S be such that s ∈ √ae; this implies there are a ∈ a, m ∈ M,

ρ ∈ N : sρ = a/m and (ms)ρ = mρ−1a ∈ a, ms ∈ √a, s ∈ (√

a)e

.

(3) Let s ∈ S be such that s ∈ ae : (b)e; this implies there are a ∈ a, m ∈M such that

bs = a/m, mbs = a ∈ a, ms ∈ (a : b) , s ∈ (a : b)e .

(4) We recall that · e preserves sum and, by the proof above, intersection.If we consider any basis b1, . . . , bs of b we can deduce

(a : b)e = (a : (b1, . . . , bs))e

=(⋂

i

a : bi

)e

=⋂

i

(a : bi )e

=⋂

i

(ae : (bi )

e)= ae :

((b1)

e + · · · + (bs)e)

= ae : be

362 Noether

Corollary 27.5.14. The set E is closed under intersection, radical and quo-tient.

Moreover, intersection, radical and quotient are preserved by · e.

We must now take into consideration the behaviour of · e with respect toprimariety and primality. The first result we need is to characterize whichprimes/primaries in R are members of C:

Lemma 27.5.15. Let q ∈ R be a primary ideal and let p be its associatedprime. The following conditions are equivalent:

(1) q ∈ C,(2) q = qec,(3) q : m = q, for each m ∈ M,(4) q ∩ M = ∅,(5) p ∈ C,(6) p = pec,(7) p : m = p, for each m ∈ M,(8) p ∩ M = ∅.

Proof.

(1) ⇒ (2) and (5) ⇒ (6): if Q ∈ S is such that q = Qc, then

q = Qc = Qcec = qec.

(2) ⇒ (1) and (6) ⇒ (5) are obvious.(2) ⇐⇒ (3) and (6) ⇐⇒ (7) follow from Theorem 27.5.9(4).(2) ⇒ (4) and (6) ⇒ (8) follow from Corollary 27.5.11, since 1 ∈ q.

(4) ⇒ (3) and (8) ⇒ (7): assume there are m ∈ M and r ∈ R such thatmr ∈ q; since, for each ρ ∈ N, mρ ∈ M , then mρ ∈ q and r ∈ q.

(4) ⇒ (8) Assume r ∈ p ∩ M ; then there exists ρ ∈ N : rρ ∈ q; since Mis also a multiplicative system, rρ ∈ M , contradicting the assumptionq ∩ M = ∅.

(8) ⇒ (4) follows by q ⊂ p.

Corollary 27.5.16. Let Q ⊂ S be a P-primary. Then:

(1) Qc is a Pc-primary;(2) Qc ∩ M = ∅ and Pc ∩ M = ∅;(3) Qce = Q and Pce = P.

Proof.

(1) follows from Corollary 27.5.7.(2) follows by the result above since q := Qc and p := Pc are in C.(3) is an instance of Theorem 27.5.9(5).


Proposition 27.5.17. Let a ⊂ R be an ideal such that a ∩ M = ∅. Then

(1) if a is prime, so also is ae;(2) if a is primary, so also is ae;(3) if a is primary belonging to the prime b, then ae is primary belonging

to the prime be;(4) if a is radical, so also is ae.

Proof. Let x, y ∈ R, m, n ∈ M be such that

y/n ∈ ae, (x/m)(y/n) = (xy)/(mn) ∈ ae;therefore there are z ∈ a, µ ∈ M : xyµ = zmn.

If a is primary, since xyµ ∈ a, y ∈ a, then there exists ρ ∈ N : xρµρ ∈ a;but µρ ∈ M and µρ ∈ a. This further implies that exists σ ∈ N : (xρ)σ ∈ a

and (x/m)ρσ ∈ ae.

This proves that ae is primary if a is such.The same argument, just putting ρ := σ := 1, proves also that ae is prime

if a is such.As a consequence of the fact that · e preserves radical formation, the same

argument as for Corollary 27.5.7 proves the other statements.

Remark 27.5.18. Continuing the discussion of Remark 27.5.5 we have

(1) E = S and C := a ∈ R : a : m = a, for each m ∈ M are closedunder intersection, radical and quotient formation;

(2) the restriction of the maps · e : R −→ S and · c : S −→ R to C andE , that is the maps · e : C −→ E = S and · c : S = E −→ C, givesa duality which preserves intersection, radical and quotient formation,primality, primarity and radicality;

(3) the restriction of · e and · c to the sets

p ∈ C, p is prime = p ∈ R, p is prime and p ∩ M = ∅ ⊂ C ⊂ R

and

P ∈ E, P is prime ⊂ E = S

gives a duality which preserves inclusion;(4) let us now fix a couple of primes P ∈ E = S and p ∈ C ⊂ R – so that

p ∩ M = ∅ and q ∩ M = ∅ for each p-primary q – which are dual toeach other in the sense that Pc = p and pe = P.Then, the restriction of · e and · c to the sets

q ∈ C, q is p-primary and Q ∈ E, Q is P-primary

364 Noether

gives a duality preserving inclusion, intersection and quotient forma-tion.8

This duality allows us to state the converse result to Corollary 27.5.8

Corollary 27.5.19. Let a ⊂ S be an ideal and let a = ⋂ri=1 qi be an irredun-

dant primary representation; assume that

qi ∩ M = ∅ ⇐⇒ i ≤ s ≤ r.

Then

• ae = ⋂si=1 qe

i is an irredundant primary representation;• aec = ⋂s

i=1 qi is an irredundant primary representation.

Proof. We have ae = ⋂ri=1 qe

i = ⋂si=1 qe

i and aec = ⋂si=1 qec

i = ⋂si=1 qi .

So we need only to prove the irredundance; for aec it follows obviouslyfrom the irredundance of the decomposition of a; as regards ae the assumption⋂r

i=1i = j

qei ⊆ qe

j would imply⋂r

i=1i = j

qi = ⋂ri=1i = j

qeci ⊆ qec

j = q j .

27.6 Decomposition of Homogeneous Ideals

Lemma 27.6.1. Let R be a graded ring. Let a, b ⊂ R denote homogeneousideals, q a homogeneous primary ideal. Then:

(1) a + b, a ∩ b, ab, a : b are homogeneous.(2)

√a is homogeneous.

(3) The associated prime of q also is homogeneous.(4) a is prime iff for each homogeneous elements F, G ∈ R we have

F /∈ a, G /∈ a ⇒ FG /∈ a.

(5) a is primary iff for each homogeneous elements F, G ∈ R we have

FG ∈ a, F /∈ a ⇒ G ∈ √a.

Proof.

(1) All statements are obvious except the one regarding a : b which canbe reduced by Theorem 26.3.2(18) to the case in which b = (b) isprincipal.Then if g ∈ (a : b) and g := ∑

i gi , gi being homogeneous of degreei , then bg = ∑

i bgi ∈ a and each of its homogeneous componentsbgi ∈ a so that gi ∈ (a : b).

8 Exceptions are the trivial cases in which

• q1 ⊃ q2 and q1 : q2 = R where, in any case, we have qe1 ⊃ qe

2 and qe1 : qe

2 = S = Re;• Q1 ⊃ Q2 and Q1 : Q2 = S where Qc

1 ⊃ Qc2 and Qc

1 : Qc2 = R = Sc.

27.6 Decomposition of Homogeneous Ideals 365

(2) Let f ∈ √a. Then there exists ρ ∈ N : f ρ ∈ a. Obviously L( f )ρ =

L( f ρ) ∈ a and L( f ) ∈ √a; the argument can then be re-applied to

f − L( f ) ∈ √a.

(3) Directly from the statement above.(4) Let F, G ∈ R be such that F /∈ a, G /∈ a; let F = ∑

i fi , G =∑j g j be their decompositions into homogeneous components and let

fπ , gσ be the highest-degree homogeneous components in F and Grespectively which are not in a.Then if we define

F ′ :=∑i>π

fi , F ′′ :=∑i≤π

fi , G ′ :=∑j>σ

g j , G ′′ :=∑j≤σ

g j ,

we deduce that L(F ′′G ′′) =L(F ′′)L(G ′′) = fπ gσ /∈ a so that F ′′G ′′ /∈a and, since F ′, G ′ ∈ a, FG = F ′G ′ + F ′G ′′ + F ′′G ′ + F ′′G ′′ /∈ a.

(5) Let F, G ∈ R be such that F /∈ a, FG ∈ a; let F = ∑i fi , G = ∑

j g j

be their decompositions into homogeneous components and let fπ bethe highest-degree homogeneous component in F which is not in a,F ′ := ∑

i>π fi , F ′′ := ∑i≤π fi .

Then we have F ′′ /∈ a, F ′ ∈ a, F ′′G = FG − F ′G ∈ a, and either

• fπL(G) = L(F ′′)L(G) = 0 ∈ a or• fπL(G) = L(F ′′)L(G) = L(F ′′G) ∈ a;in either case L(G) ∈ √

a.

This gives us the initial step of a recursive argument: in fact if, for someσ , we set G ′ := ∑

j>σ g j , G ′′ := ∑j≤σ g j and we assume we have

already proved that G ′ ∈ √a and that µ ∈ N is such that G ′µ ∈ a, we

have, for the suitable element H ∈ R

F ′′G ′′µ = F ′′(G − G ′)µ

= F ′′G H + (−1)µF ′′G ′µ ∈ a,

and

fπ gµσ = L(F ′′)L(G ′′)µ = L(F ′′G ′′µ) ∈ a, fπ /∈ a ⇒ gσ ∈ √

a.

Let R be a graded ring; for any ideal a ⊂ R we will denote by a the idealgenerated by all homogeneous elements belonging to a.

Then:

Lemma 27.6.2. Let R be a graded ring; let a ⊂ R be an ideal and f ⊂ R ahomogeneous ideal. Then:

366 Noether

(1) if a is prime such is a;(2) if a is primary such is a;(3) let f = ⋂r

i=1 qi be an irredundant primary representation of f; thenf = ⋂r

i=1 qi is another irredundant primary representation of f.

Proof.

(1) Let f, g ∈ R be homogeneous elements such that f g ∈ a, f /∈ a.Then, by definition, f g ∈ a and f /∈ a so that g ∈ a and g ∈ a.

(2) Again, let f, g ∈ R be homogeneous elements such that f g ∈ a, f /∈a, so that f g ∈ a and f /∈ a and there exists r ∈ N : gr ∈ a andgr ∈ a.

(3) Each qi is primary and we have

⋂ri=1 q

i ⊆ ⋂ri=1 qi = f; also, for each

i , f ⊂ qi , because f ⊂ qi is homogeneous, implying f = ⋂r

i=1 qi .

Corollary 27.6.3. Let R be a Noetherian graded ring R; then each homoge-neous ideal f ⊂ R has an irredundant homogeneous primary representation.

In particular its associated primes and its isolated components are homoge-neous.

Let us now restrict ourselves to the case R := k[X0, . . . , Xn] and werecall that a homogeneous ideal I ⊂ k[X0, . . . , Xn] is called irrelevant if√

I = (X0, . . . , Xn) =: m.

Theorem 27.6.4. Let I be a homogeneous ideal, then there exist a homoge-neous ideal Isat and an irrelevant homogeneous ideal Iirr such that:

(1) I = Isat ∩ Iirr;(2)

√Iirr = (X0, . . . , Xn);

(3) Iirr is maximal, in the sense that for each ideal J

I = Isat ∩ J,√

J = (X0, . . . , Xn), J ⊇ Iirr ⇒ J = Iirr;

(4) Z(Isat) = Z(I);(5) there is s ∈ N such that

f ∈ I homog. , deg( f ) ≥ s = f ∈ Isat homog. , deg( f ) ≥ s;(6) if for some homogeneous ideal J there is s ∈ N such that

f ∈ I homog. , deg( f ) ≥ s = f ∈ J homog. , deg( f ) ≥ s,then J ⊆ Isat;

(7) I = Isat ⇐⇒ Iirr = (X0, . . . , Xn).

The ideal Isat is called the saturation of I and is unique, while the role of Iirr inthis decomposition could be played by different irrelevant ideals.

27.6 Decomposition of Homogeneous Ideals 367

Proof. Either

• m is not an associated prime of I ⊂ m in which case we set Isat := I, Iirr := mand all the statements (except at most (6)) follow obviously, or

• m is an associated prime so that in the homogeneous decomposition

I =r⋂

i=0

qi

one of the primaries, let us say q0, belongs to m, in which case we set Isat :=⋂ri=1 qi , and we choose Iirr among the maximal elements in the set of the

m-primary ideals q0 such that I = q0 ∩ Isat.Therefore (1), (2), (3), (4), (7) hold.

In the second case q0 contains a power ms , which implies

Iirr ⊃ f ∈ I homog. , deg( f ) ≥ sTherefore (5) follows from

f ∈ I homogeneous , deg( f ) ≥ s= f ∈ Isat ∩ Iirr homogeneous , deg( f ) ≥ s= f ∈ Isat homogeneous , deg( f ) ≥ s .

Ad (6): In order to unify both cases let us denote Isat = ⋂ri=1 qi a homoge-

neous decomposition of Isat and let pi be the prime ideal associated to qi . Notethat for each i there is an index ji such that X ji /∈ pi .

Let us consider any homogeneous element F ∈ J and note that, by assump-tion, Xs

jiF ∈ I ⊂ qi , which implies F ∈ qi ; since this holds for each i the

thesis follows.

Example 27.6.5. (See Example 27.4.4) For the ideal (X2, XY ) ⊂ Q[X, Y ] wehave Isat := (X) while the role of Iirr in this decomposition can be played byeach component (X2, Y + aX), a ∈ Q.

Note that Isat can be computed as Isat := I : m∞.

Definition 27.6.6. A homogeneous ideal I ⊂ k[X0, . . . , Xn] is said to besaturated if for any ideal J ⊇ I the existence of s ∈ N such that

f ∈ I homog. , deg( f ) ≥ s = f ∈ J homog. , deg( f ) ≥ simplies J = I.

Please allow me to quench my horror vacui by recalling that the maps

h− : k[X1, . . . , Xn] −→ k[X0, . . . , Xn]

368 Noether

anda− : k[X0, . . . , Xn] −→ k[X1, . . . , Xn]

preserve all ideal operations (sum, product, intersection, colon, radical compu-tation); moreover h− preserves primality and primariety while a− preservesprimality and primariety only for those ideals I such that X0 /∈ I.

As a consequence we have:

• If f=⋂ri=1 qi is an irredundant primary representation of f ⊂ k[X1, . . . , Xn]

then hf = ⋂ri=1

hqi is an irredundant homogeneous primary representationof hf ⊂ k[X0, . . . , Xn].

• If f = ⋂ri=1 qi is an irredundant homogeneous primary representation of f ⊂

k[X0, . . . , Xn] and X0 /∈ qi iff i ≤ s, then af = ⋂si=1

aqi is an irredundantprimary representation of af ⊂ k[X1, . . . , Xn].

27.7 *The Closure of an Ideal at the Origin

Theorem 27.7.1 (Krull). Let R be a Noetherian ring and m ⊂ R be an ideal.Then

⋂d md = (0) iff there is no z ∈ m such that 1 − z is a zero-divisor in

R, that is for each z ∈ m, x ∈ R

x(1 − z) = 0 ⇒ x = 0.

Proof. Let z ∈ m, x ∈ R, be such that x(1 − z) = 0 so that

x = zx = z2x = · · · , zd x = · · · ,and x ∈ md , for each d , so that x = 0.

Conversely, let us assume that in R there is no zero-divisor 1− z, z ∈ m, andlet us write a := ⋂

d md ; in the primary decomposition of a there is at mostone m-primary component a0 and let us denote a1 the intersection of all theother components so that a = a0 ∩ a1.

In the same way, the ideal b := ma can be expressed as b = b0 ∩ b1 whereb0 is its m-primary component and b1 the intersection of all the other ones.

Since (Proposition 27.2.11), for any m-primary ideal c, we have both

a1 : c = a1 and b1 : c = b1,

we have:

a1a0m ⊂ (a0 ∩ a1) m = am = b ⊂ b1 ⇒ a1 ⊂ b1 : a0m = b1, andb1b0 ⊂ b0 ∩ b1 = am ⊂ a ⊂ a1 ⇒ b1 ⊂ a1 : b0 = a1,

whence b1 = a1.

27.7 *Closure of an Ideal at the Origin 369

Since b0 is m-primary, there is δ such that mδ ⊂ b0 so that a = ⋂d md ⊂

mδ ⊂ b0. As a consequence

a ⊂ b0 ∩ a1 = b0 ∩ b1 = b = ma.

Therefore if a1, . . . , as is any basis of a there are elements

mi j ∈ m, 1 ≤ i, j ≤ s

such that, for each i

ai =∑

j

mi j a j ands∑

j=1

(δi j − mi j )ai = 0, where δi j =

1 if i = j ,0 if i = j .

This implies the vanishing of the determinant,

0 = det(δi j − mi j

) ≡ 1 mod m,

and this contradicts the assumption that there is no zero-divisor 1 − z, z ∈ m,

in R.

Let us now consider a primary ideal q ⊂ k[X1, . . . , Xn] =: P , its associatedprime p := √

q and its characteristic number ρ so that pρ ⊂ q, and let m :=(X1, . . . , Xn) denote the maximal ideal at the origin.

Let us denote by R the residue class ring R := P/q and by π the canonicalprojection π : P −→ R; we will also write q := (0) = π(q), p := π(p),m := π(m).

Corollary 27.7.2.⋂

d q + md = q ⇐⇒ 1 /∈ p + m.

Proof. It is sufficient to recall that the set of the zero-divisors of R is p.

Corollary 27.7.3. 1 ∈ ⋂d q + md ⇐⇒ 1 ∈ p + m.

Proof. On the one hand

1 ∈⋂

d

q + md ⇒⋂

d

q + md = q ⇒ 1 ∈ p + m.

Conversely, let z ∈ m, x ∈ p be such that 1 = x + z; this implies that, forsome ρ,

q := (1 − z)ρ = xρ ∈ q.

Then, for the suitable element y ∈ P for which q = 1 − yz and for each d,we have

1 = 1d = (q + yz)d = qp + yd zd ∈ q + md ,

for the suitable p ∈ P .This proves that 1 ∈ ⋂

d q + md .

370 Noether

Lemma 27.7.4. Let q1, q2 be such that

1 ∈ q1 + m and 1 ∈ q2 + m.

Then, for each d ∈ N, 1 ∈ (q1 ∩ q2) + md .

Proof. By assumption there are q1 ∈ q1, q2 ∈ q2, x1, x2 ∈ m such that

q1 + x1 = 1 = q2 + x2;therefore 1 = (q1 + x1)(q2 + x2) = q + x with

q = q1q2 ∈ q1 ∩ q2, x = x1q2 + x2q1 + x1x2 ∈ m,

and, for each d ∈ N,

1 = (q + x)d = qp + xd ∈ (q1 ∩ q2) + md

for the suitable p ∈ P .

Let us now consider an ideal I ⊂ P and its irredundant primary decompositionI = ⋂s

i=1 qi , enumerated so that qi ⊂ m ⇐⇒ i ≤ r and let us write

I0 :=r⋂

i=1

qi , I1 :=s⋂

i=r+1

qi .

Then we have

Proposition 27.7.5.⋂

d I + md = I0.

Proof. We have

I0 ⊆⋂

d

I0 + md =⋂

d

(r⋂

i=1

qi

)+ md ⊆

r⋂i=1

(⋂d

qi + md

)=

r⋂i=1

qi = I0

so that I0 = ⋂d I0 + md .

Also we have P = qi + md , for each i > r and each d ∈ N, so thatP = I1 + md , for each d ∈ N. As a consequence, for each d ∈ N,

I0 = I0P = I0(I1 + md

)= I0I1 + I0md ⊂ I + md ,

and ⋂d

I + md ⊂⋂

d

I0 + md = I0 ⊂⋂

d

I + md

whence the claim.

Corollary 27.7.6 (Lasker). If f1, . . . , fh is a basis of the ideal

I ⊂ m ⊂ k[X1, . . . , Xn]

27.8 Generic System of Coordinates 371

and f ∈ k[X1, . . . , Xn] is such that

f =h∑

i=1

pi fi , p1, . . . , ph ∈ k[[X1, . . . , Xn]],

then there exists g ∈ k[X1, . . . , Xn] \ m : g f ∈ I.

Proof. The assumption implies that f ∈ I + md , for each d ∈ N, so thatf ∈ ⋂

d I + md = I0.Therefore if we consider the primary decomposition I = ⋂s

i=1 qi , enumer-ated so that qi ⊂ m ⇐⇒ i ≤ r and we denote I0 := ⋂r

i=1 qi , I1 :=∩s

i=r+1qi , we have f ∈ ⋂d I + md = I0.

The claim is proved by taking, for each i, r < i ≤ s, an element pi ∈ qi \ mand setting g := ∏

i pi ∈ I1 \ m so that g f ∈ I1I0 ⊂ I.

Definition 27.7.7. With the present notation the ideal⋂

d I + md is called them-closure of I.

An ideal I such that I = ⋂d I + md is called m-closed.

27.8 Generic System of Coordinates

Let

• GL(n, k) be the general linear group, that is the set of all invertible n × nsquare matrices with entries in k,

• B(n, k) ⊂ GL(n, k) be the Borel group of the upper triangular matricesM := (

ci j), that is those such that i > j ⇒ ci j = 0;

• N (n, k) ⊂ B(n, k) be the subgroup of the upper triangular unipotent matri-ces M := (

ci j), that is those such that

i > j ⇒ ci j = 0, and i = j ⇒ ci j = 1.

We will use the shorthand k[Xi j ] and k(Xi j ) to denote, respectively, thepolynomial ring generated over k by the variables


Let us fix any matrix

M := (ci j

) ∈ GL(n, k)

and let us denote (d ji

) = M−1 ∈ GL(n, k),

its inverse.

372 Noether

The matrix M describes the linear transformation

M : k[X1, . . . , Xn] −→ k[X1, . . . , Xn]

defined by

M(Xi ) =∑

jci j X j for each i

whose inverse is the transformation

X j −→∑

id ji Xi for each j

and which satisfies, for each ideal I ⊂ k[X1, . . . , Xn],

Z(I) =(∑

jc1 j b j , . . . ,

∑j

cnj b j

): (b1, . . . , bn) ∈ Z(M(I))

.

Example 27.8.1. The linear transformation (see Section 20.2)

Lc : k[X1, . . . , Xn] −→ k[X1, . . . , Xn]

defined by

Lc(Xi ) :=

Xi + ci Xn if i < n,cn Xn if i = n,

where c := (c1, . . . , cn) ∈ C(n, k), and its inverse

L−1c (Xi ) :=

Xi − ci c−1

n Xn if i < n,c−1

n Xn if i = n

are described by the matrices

ci j = ci if j = n,

1 if i = j < n,0 otherwise

and di j =

⎧⎪⎪⎨⎪⎪⎩

c−1n if i = j = n,

−ci c−1n if i < j = n,

1 if i = j < n,0 otherwise

and we have

Lc( f )(b1, . . . , bn) = 0 ⇐⇒ f (a1, . . . , an) = 0

where

ai :=∑

jci j b j =

bi − ci bn if i < n,cnbn if i = n.

If we also write for each i ,

Yi := M(Xi ) =∑

jci j X j ,

since each homogeneous form in k[X1, . . . , Xn] is uniquely expressed as ahomogeneous form of the same degree in k[Y1, . . . , Yn] and conversely, we

27.8 Generic System of Coordinates 373

obtain a system of coordinates Y1, . . . , Yn and a corresponding change ofcoordinates

k[Y1, . . . , Yn] = k[X1, . . . , Xn],

which we will say is induced by M, and which is defined by

f (X1, . . . , Xn) = f(∑

id1i Yi , . . . ,

∑i

dni Yi

)∈ k[Y1, . . . , Yn],

because X j = ∑i d ji Yi , for each j .

Also, for each polynomial f ∈ k[X1, . . . , Xn], g ∈ k[Y1, . . . , Yn] related by

f (X1, . . . , Xn) = f(∑

id1i Yi , . . . ,

∑i

dni Yi

)= g(Y1, . . . , Yn)

we have

f (a1, . . . , an) = 0 ⇐⇒ g(∑

jc1 j a j , . . . ,

∑j

cnj a j

)= 0,

g(b1, . . . , bn) = 0 ⇐⇒ f(∑

jd1 j b j , . . . ,

∑j

dnj b j

)= 0.

Example 27.8.2. The change of coordinates

k[Y1, . . . , Yn] = k[X1, . . . , Xn]

defined by

Yi :=

X1 + ∑ni=2 ci Xi if i = 1

Xi if i > 1,

and its inverse

Xi :=

Y1 − ∑ni=2 ci Yi if i = 1

Yi if i > 1,

are described by the matrices

ci j = 1 if i = j ,

ci if i = 1, j > 1,0 otherwise

and di j = 1 if i = j ,

−ci if i = 1, j > 1,0 otherwise.

For any ideal I ∈ k[X1, . . . , Xn], setting J := Ik[Y1, . . . , Yn], we have

(a1, . . . , an) ∈ Z(I) ⇐⇒(

a1 +∑n

i=2ci ai , a2, . . . an

)∈ Z(J).

Each linear transformation M ∈ B(n, k) can be uniquely described by

• assigning (see Section 20.3) for each ν, 1 < ν ≤ n, an element cν :=(c1ν, . . . , cνν) ∈ C(ν, k),

374 Noether

• denoting by Lcν both the automorphism

k[X1, . . . , Xν] −→ k[X1, . . . , Xν]

and its polynomial extensions

k[X1, . . . , Xν][Xν+1, . . . , Xn] −→ k[X1, . . . , Xν][Xν+1, . . . , Xn]

defined by

Lcν (Xi ) :=

Xi + ciν Xν if i < ν,cνν Xν if i = ν,

• and setting M := Lc2 · Lc3 · · · Lcn−1 · Lcn .

If we restrict each such linear transformation Lcν to the case in which cνν =1, we obtain the subgroup N (n, k) ⊂ B(n, k).

In both cases, if we assign, for each ν, 1 < ν ≤ n, a polynomialfν(X1, . . . , Xν) and we restrict the transformations M to those such thatfν(c1ν, . . . , cνν) = 0 for each ν, we obtain a non-empty Zariski open set ofB(n, k) and N (n, k) respectively.

It is worth noting that in the applications of Section 20.3 the emphasis wason the fact that such a set of linear transformations M was not empty, thusallowing us to perform successive elimination. In the present context we nowwant also to be sure that the interesting linear change of coordinates can bechosen in a Zariski open set, that is they are ‘generic’.

27.9 Ideals in Noether Position

Let P := k[X1, . . . , Xn] and let p ⊂ P be a prime ideal. Then R := P/p is anintegral domain such that R ⊃ k and if we denote Q its quotient field we haveQ ⊃ R ⊃ k.

With the obvious meaning (following Section 5.3) we will denote

Q := k(x1, . . . , xn) and R := k[x1, . . . , xn].

We recall (Section 9.2) that, given any set A such that Q = k(A) there is asubset B ⊆ A such that

• B is a transcendental basis of Q over k,• A depends algebraically over B,• k ⊆ k(B) ⊆ k(A) = Q,

and, more importantly,

• the cardinality of B is independent of the choice of the set A and is calledthe transcendency degree of Q over k.

27.9 Ideals in Noether Position 375

Moreover, such a transcendental basis can be assumed to consist of ele-ments in R – actually of variables – since it is sufficient to start with A :=x1, . . . , xn ⊂ R; therefore with a slight abuse of notation we will speak oftranscendency degree and transcendental bases of R over k.

Let R = k[x1, . . . , xn] be an integral domain whose transcendency degreeover k is d and let y1, . . . , yd ⊂ R; we recall that R is said to be integralover k[y1, . . . , yd ] ⊂ R if

• for each i ≤ n, there is a monic polynomial fi ∈ k[Y1, . . . , Yd ][T ] such that

fi (y1, . . . , yd , xi ) = 0,

and we remark that, since the transcendency degree of R over k is d, this im-plies that

• y1, . . . , yd are algebraically independent over k,• y1, . . . , yd is a transcendental basis of R over k and• the canonical morphism k[Y1, . . . , Yd ] −→ k[y1, . . . , yd ] is an isomor-

phism.

Theorem 27.9.1 (Noether Normalization Lemma).Let R = k[x1, . . . , xn] be an integral domain and let d be the transcendency

degree of k(x1, . . . , xn) over k.Then for each ‘generic’ change of coordinates 9

M := (ci j

) ∈ GL(n, k) (respectively B(n, k), N (n, k))

defining yi := ∑j ci j x j , for each i , one has that

R is integral over k[y1, . . . , yd ] so thaty1, . . . , yd is a transcendental basis of R over k.

Proof. Let x j1 , . . . , x jd be a transcendental basis of R over k and note thatwe can assume jl ≥ l for each l, so that we can wlog restrict ourselves in theargument to both B(n, k) and N (n, k).

From y1 = ∑j c1 j x j we have

y1 =d∑

l=1

c1 jl x jl + ω, ω =∑

j /∈ j1,... jd c1 j x j ,

where ω is integral over x j1 , . . . , x jd .Therefore, by the Steinitz Lemma (Lemma 9.2.6) we can deduce that for

each M such that c1 j1 = 0 we have

9 This is ‘generic’ in the sense that there is a non-empty Zariski open set N ⊂ GL(n, k) (respec-tively B(n, k), N (n, k)) such that the statement holds for each M ∈ N.

376 Noether

• y1, x j2 , . . . , x jd is a transcendental basis of R over k,• x j1 = c−1

1 j1y1 − ∑

j = j1 c−11 j1

c1 j x j and

• yi = ci j1 x j1 + ∑j = j1 ci j x j = ci j1c−1

1 j1y1 + ∑

j = j1(ci j − c−11 j1

c1 j )x j .

We can therefore assume by induction that there is a polynomial Pδ ∈k[Xlm] such that for each M := (

ci j)

for which Pδ(clm) = 0 we have

• y1, . . . , yδ−1, x jδ , . . . , x jd is a transcendental basis of R over k,• there are polynomials Di j ∈ k[Xlm] such that, setting di j := Di j (clm), one

has for each i ≥ δ

Pδ(clm)yi =δ−1∑j=1

di j y j +d∑

l=δ

di jl x jl +∑

j /∈ j1,... jd di j x j .

From Pδ(clm)yδ = ∑δ−1j=1 dδ j y j + ∑d

l=δ dδ jl x jl + ∑j /∈ j1,... jd dδ j x j , since∑

j /∈ j1,... jd dδ j x j is integral over y1, . . . , yδ−1, x jδ , . . . , x jd , by the SteinitzLemma we can deduce that for each M such that Pδ(clm)Dδ jδ (clm) = 0 wehave

• y1, . . . , yδ, x jδ+1 , . . . , x jd is a transcendental basis of R over k,

• dδ jδ x jδ = Pδ(clm)yδ − ∑δ−1j=1 dδ j y j − ∑d

l=δ+1 dδ jl x jl − ∑j /∈ j1,... jd dδ j x j ,

• and, setting Pδ+1(Xlm) := Pδ(Xlm)Dδ jδ (Xlm) we have, for each i ≥ δ + 1,

Pδ+1(clm)yi = di jδ dδ jδ x jδ +δ−1∑j=1

dδ jδ di j y j +d∑

l=δ+1

dδ jδ di jl x jl

+∑

j /∈ j1,... jd dδ jδ di j x j

=δ−1∑j=1

(dδ jδ di j − di jδ dδ j )y j + Pδ+1(clm)yδ

+d∑

l=δ+1

(dδ jδ di jl − di jδ dδ jl )x jl

+∑

j /∈ j1,... jd (dδ jδ di j − di jδ dδ j )x j

whence the claim by induction.

Let P := k[X1, . . . , Xn] and let Y1, . . . , Yn be a system of coordinates ofP . Let p ⊂ P be a prime and f ⊂ P be an ideal.

Definition 27.9.2 (van der Waerden). The dimension of the prime ideal p ⊂P , denoted by dim(p), is the transcendency degree of P/p over k.

27.9 Ideals in Noether Position 377

The dimension dim(f) of the ideal f ⊂ P is the maximum dimension of theassociated prime ideals of f.

Lemma 27.9.3. For any two prime ideals p ⊂ p′ ⊂ P we have dim(p) >

dim(p′).

Proof. Consider the integral domains R = P/p and R′ = P/p′; the canonicalhomomorphism π : R −→ R′ is surjective.

Therefore, if B′ is a transcendental basis of R′ over k, there is a set B ⊂ Rsuch that

• π(B) = B′,• #(B) = #(B)′,• B is a transcendental set of R.

The Steinitz Lemma (Lemma 9.2.6) allows us to deduce the existence of atranscendental basis C of R such that B C so that dim(p) > dim(p′).

Definition 27.9.4. The ideal p is said to be in Noether position w.r.t.Y1, . . . , Yn – or Y1, . . . , Yn to be a Noether position for p – if P/p is inte-gral over k[y1, . . . , yd ], where d := dim(p).

The ideal f is said to be in Noether position w.r.t. Y1, . . . , Yn – orY1, . . . , Yn to be a Noether position for f – if each associated prime of f

is in Noether position w.r.t. Y1, . . . , Yn.Historical Remark 27.9.5. The reference is not to Emmy Noether but to herfather Max; in fact the Normalization Lemma was stated and proved by him.

As an interesting remark, Max Noether’s Normalization Lemma was a toolin the proof of his Normalization Theorem. Lasker introduced his Decomposi-tion Theorem as a tool for generalization of Noether’s result of which he gave‘the most general and complete expression’.10

Macaulay’s references to the Lasker–Noether Theorem are related to theNormalization Theorem and not to the Decomposition Theorem.

Corollary 27.9.6. The ideal f is in Noether position w.r.t. the ‘generic’ systemof coordinates Y1, . . . , Yn in GL(n, k) (respectively B(n, k), N (n, k)), thatis there is a Zariski open set N ⊂ GL(n, k) (respectively B(n, k), N (n, k))such that for each M := (

ci j) ∈ N, writing

Yi := M(Xi ) =∑

j

ci j X j ,

the ideal f is in Noether position w.r.t. Y1, . . . , Yn.10 F. S. Macaulay, On the Resolution of a given Modular System into Primary Systems Including

Some Properties of Hilbert Numbers, Math. Ann. 74 (1913), p. 67.

378 Noether

27.10 *Chains of Prime Ideals

Note that, if R = k[x1, . . . , xn] is an integral domain and p ⊂ R is a prime,then there is a prime d ⊂ P := k[X1, . . . , Xn] such that

R = P/d and R′ := R/p = P/(d + p);therefore Definition 27.9.2 and Lemma 27.9.3 can be naturally extended to Rby stating

Definition 27.10.1. For an integral domain R and a prime p ⊂ R, the dimen-sion dim(p) of p is the transcendency degree of R/p over k.

Corollary 27.10.2. For any two prime ideals p ⊂ p′ ⊂ R we have dim(p) >

dim(p′).

Proof. Follows directly from Lemma 27.9.3.

Lemma 27.10.3. Let R = k[x1, . . . , xn] be an integral domain over k, s beits transcendency degree over k and p ⊂ R be a minimal prime ideal. Thendim(p) = s − 1.

Proof. Let us first assume that s = n so that R is a polynomial ring in n = svariables, thus being a unique factorization domain so that there is a polyno-mial f ∈ R \ k such that p = ( f ).

Therefore, for some variable, say xn ,

f =t∑

i=0

gi (x1, . . . , xn−1)xin, t ≥ 1,

and each polynomial g f ∈ p is dependent on xn so that p ∩ k[x1, . . . , xn−1] =(0) and x1, . . . , xn−1 are algebraic independent over k and dim(p) = n − 1.

If s < n by the Normalization Lemma (Theorem 27.9.1), we know theexistence of s elements y1, . . . , ys ∈ R such that R is integral over R′ :=k[y1, . . . , ys]; setting p′ := p ∩ R′, p′ is then minimal in R′ so that, by theproof above, dim(p′) = s − 1.

Assume wlog that y1, . . . , ys−1 are transcendental modulo p′. Then forany x ∈ R, there is a polynomial f (Y1, . . . , Ys−1, X) giving an integral depen-dency of x over y1, . . . , ys−1 mod. p′ ⊂ p so that x is integrally dependentover y1, . . . , ys−1 mod. p and dim(p) = s − 1.

Definition 27.10.4. Let R be a commutative ring with unity and let p ⊂ R bea proper prime ideal.11

11 That is R is not allowed, while (0) is allowed, provided it is prime.

27.10 *Chains of Prime Ideals 379

The ideal p is said to have rank r, r(p) = r, if there exists at least one chain

p0 ⊂ p1 ⊂ · · · ⊂ pr−1 ⊂ pr = p

where each pi is a prime ideal, and there is no such chain with more than r +1ideals.

The ideal p is said to have length l, l(p) = l, if there exists at least one chain

R ⊃ p0 ⊃ p1 ⊃ · · · ⊃ pl−1 ⊃ pl = p

where each pi is a prime ideal, and there is no such chain with more than l + 1ideals.

Proposition 27.10.5. Let R = k[x1, . . . , xn] be an integral domain over kand let s be its transcendental degree over k; let p ⊂ R be a prime ideal ofdimension d.

Then r(p) = s − d, l(p) = d.

Proof.

r(p) ≤ s − d We prove this by decreasing induction on d since the statementshold for s = d , that is for p = (0). Let then p = (0) : from

(0) = p0 ⊂ p1 ⊂ · · · ⊂ pr−1 ⊂ pr = p

we deduce

s = dim(p0) > dim(p1) > · · · > dim(pr−1) > dim(pr) = dim(p) = d.

r(p) ≥ s − d In particular the sequence is finite; therefore there exists a primep′ ⊂ p which is maximal for this property.12

This implies that in the integral domain R′ := R/p′ the ideal P suchthat R′/P = R/p is minimal so that

dim(p′) = 1 + dim(P) = 1 + dim(p) = 1 + d.

By inductive argument we can therefore deduce r(p′) ≥ s − d − 1,

whence r(p) = r(p′) + 1 ≥ s − d.

l(p) ≤ d From

R ⊃ p0 ⊃ p1 ⊃ · · · ⊃ pl−1 ⊃ pl = p

we get

0 ≤ dim(p0) < dim(p1) < · · · < dim(pl−1) < dim(pl) = dim(p) = d.

12 That is there is no other prime p′′ such that p′ ⊂ p′′ ⊂ p.

380 Noether

l(p) ≥ d If d = 0, R/p is a field, p is maximal and l(p) = d.We can therefore prove the statement by increasing induction on d,considering the integral domain R′ := R/p, a minimal prime idealp′ ⊂ R′ and the prime ideal P ⊂ R such that P ⊃ p and R′/p′ =R/P.

Then R′ has transcendental degree d and

l(p) − 1 ≥ l(P) = dim(P) = dim(p′) = d − 1.

Corollary 27.10.6. Let p ⊂ P be a prime ideal of dimension d.Then r(p) = n − d, l(p) = d.

Corollary 27.10.7. Let R = k[x1, . . . , xn] be a finite integral domain over kand let s be its transcendental degree over k; let p ⊂ p′ ⊂ R be prime idealsof dimension, respectively d and d ′.

Then there is at least one chain of d − d ′ + 1 prime ideals

p ⊂ p1 ⊂ · · · ⊂ pd−d ′−1 ⊂ p′.

Moreover any chain of q + 1 prime ideals, q < d − d ′

p ⊂ p1 ⊂ · · · ⊂ pq−1 ⊂ p′

can be refined to a chain having the maximal length d − d ′ + 1.

Proof. In the ring S := R/p whose transcendental degree over k is d, the primeP such that S/P = R/p′ whose dimension is d ′ satisfies r(P) = d − d ′,whence the first claim.

The second claim can be obtained by applying the first statement in order torefine each subchain pi−1 ⊂ pi , dim(pi ) > dim(pi−1) + 1, 1 ≤ i ≤ q.

Corollary 27.10.8. Each refined chain of prime ideals

(0) ⊂ p1 ⊂ · · · ⊂ pq ⊂ P

has length n.Each chain in P can be refined to be a chain having the maximal length

n.

27.11 Dimension

Let us begin by noting that in Definition 27.9.2 the dimension of f can beobtained by just taking the maximum dimension of the isolated prime ideals off, as a consequence of Lemma 27.9.3.

27.11 Dimension 381

Theorem 27.11.1 (Grobner). Let P := k[X1, . . . , Xn] and let p ⊂ P be aprime ideal. Then the following conditions are equivalent:

• dim(p) = d;• There exists a subset Xi1 , . . . , Xid of d variables for which we have

p ∩ k[Xi1 , . . . , Xid ] = (0)

while for each subset X j1 , . . . , X jd+1 of d + 1 variables, we have

p ∩ k[X j1 , . . . , X jd+1 ] = (0).

Proof. Let p be such that dim(p) = d; then, by definition, there is a set ofd variables Xi1 , . . . , Xid such that P/p = k[x1, . . . , xn] is algebraic overk[xi1 , . . . , xid ]; therefore p ∩ k[Xi1 , . . . , Xid ] = (0), while for each subsetX j1 , . . . , X jd+1 the set x j1 , . . . , x jd+1 is algebraically dependent, implyingthe existence of a polynomial f (X j1 , . . . , X jd+1) ∈ p.

Conversely, p∩ k[Xi1 , . . . , Xid ] = (0) implies that in P/p = k[x1, . . . , xn],xi1 , . . . , xid are algebraically independent.

On the other side, each set x j1 , . . . , x jd+1 of d + 1 generators satisfies analgebraic relation f (x j1 , . . . , x jd+1) = 0 because there is a polynomial f ∈p ∩ k[X j1 , . . . , X jd+1 ].

Corollary 27.11.2 (Grobner). Let P := k[X1, . . . , Xn] and let q ⊂ P be aprimary ideal. Then the following conditions are equivalent:

• dim(q) = d;• there exists a subset Xi1 , . . . , Xid of d variables for which we have

q ∩ k[Xi1 , . . . , Xid ] = (0)


q ∩ k[X j1 , . . . , X jd+1 ] = (0).

Proof. Let p be the associated prime of q.

Since there exists ρ ∈ N such that pρ ⊂ q ⊂ p we have, for each subsetXi1 , . . . , Xiδ of δ variables,

q ∩ k[Xi1 , . . . , Xiδ ] = (0) ⇐⇒ p ∩ k[Xi1 , . . . , Xiδ ] = (0).

Corollary 27.11.3 (Grobner). Let P := k[X1, . . . , Xn] and let f ⊂ P be anideal. Then the following conditions are equivalent:

382 Noether

• dim(f) = d;• there exists a subset Xi1 , . . . , Xid of d variables for which we have

f ∩ k[Xi1 , . . . , Xid ] = (0)


f ∩ k[X j1 , . . . , X jd+1 ] = (0).

Proof. Let f = ⋂ri=1 qi , be an irredundant primary representation of f and, for

each i, pi the associated prime of qi .Since d = dim(f) ≥ dim(qi ), for each i, 1 ≤ i ≤ r, and each subset

X j1 , . . . , X jd+1 of d + 1 variables, there exist

fi (X j1 , . . . , X jd+1) ∈ qi ∩ k[X j1 , . . . , X jd+1 ], fi = 0,

so that

f (X j1 , . . . , X jd+1) =∏

i

fi ∈ f ∩ k[X j1 , . . . , X jd+1 ].

On the other hand, let qi be a component such that d = dim(qi ). By defini-tion there is a subset Xi1 , . . . , Xid of d variables for which we have

f ∩ k[Xi1 , . . . , Xid ] ⊂ qi ∩ k[Xi1 , . . . , Xid ] = (0).

On the basis of this result, let us introduce

Definition 27.11.4. Let P := k[X1, . . . , Xn] and let f ⊂ P be an ideal.A subset Xi1 , . . . , Xid of d variables for which we have

f ∩ k[Xi1 , . . . , Xid ] = (0)

is called a set of independent variables for f.

If, for each j /∈ i1, . . . , id, we have

f ∩ k[Xi1 , . . . , Xid , X j ] = (0)

Xi1 , . . . , Xid is called a maximal set of independent variables,

and let us reformulate the notion of Noether position in terms of this definition:

Corollary 27.11.5. Let P := k[X1, . . . , Xn], let Y1, . . . , Yn be a system ofcoordinates of P and f ⊂ P be an ideal.

Let f = ⋂ri=1 qi , be an irredundant primary representation of f and, for

each i, let pi be the associated prime of qi and di := dim(pi ).

27.11 Dimension 383


• Y1, . . . , Yn is a Noether position for f,• for each i, Y1, . . . , Ydi is a maximal set of independent variables for pi ,• for each i, P/pi is integral over k[Y1, . . . , Ydi ].

We state here a stronger characterization of dimension, which will be provedlater:

Fact 27.11.6. Let P := k[X1, . . . , Xn] and let p ⊂ P be a prime ideal. Thenthe following conditions are equivalent:

• dim(p) = d;• there exists a subset Xi1 , . . . , Xid of d variables for which we have

p ∩ k[Xi1 , . . . , Xid ] = (0)

while, for each subset X j1 , . . . , X jd+1 of d + 1 variables, we have

p ∩ k[X j1 , . . . , X jd+1 ] = (0);

• p has rank n − d, r(p) = n − d;• p has length d, l(p) = d;• the Hilbert polynomial Hp(T ) of p has degree d.

Proof. Compare Theorem 27.11.1, Proposition 27.10.5 and Corollary 36.2.9.

Corollary 27.11.7. Let P := k[X1, . . . , Xn] and let f ⊂ P be an ideal. Thenthe following conditions are equivalent:

• dim(f) = d;• there exists a subset Xi1 , . . . , Xid of d variables for which we have

f ∩ k[Xi1 , . . . , Xid ] = (0)

while, for each subset X j1 , . . . , X jd+1 of d + 1 variables, we have

f ∩ k[X j1 , . . . , X jd+1 ] = (0);

• the Hilbert polynomial Hf(T ) of f has degree d.

Proof. Compare Corollary 27.11.3 and Corollary 36.2.9.

Macaulay’s result (Corollary 23.3.2) which reduced the computation of theHilbert function of an ideal I to that of the monomial ideal T<(I) and, in

384 Noether

general, Macaulay’s paradigm, according to which problems on I can be re–duced to the combinatorial ones on T<(I), have a direct illustration in Kredeland Weispfenning’s algorithm for the computation of a maximal independentset of variables for an ideal f ⊂ k[X1, . . . , Xn]:

Lemma 27.11.8 (Kredel–Weispfenning). Let

f ⊂ k[X1, . . . , Xn]

be an ideal, < be any term ordering and T<(f) the corresponding monomialideal.

If Xi1 , . . . , Xid is a set of variables such that T<(f)∩k[Xi1 , . . . , Xid ] = ∅then f ∩ k[Xi1 , . . . , Xid ] = (0).

Proof. If there exists f ∈ f ∩ k[Xi1 , . . . , Xid ], f = 0, then T<( f ) ∈ T<(f) ∩k[Xi1 , . . . , Xid ].

Corollary 27.11.9 (Kredel–Weispfenning). Let f ⊂ k[X1, . . . , Xn] be anideal, < be any degree-compatible term ordering 13 and T<(f) the correspond-ing monomial ideal.

Let Xi1 , . . . , Xid be a maximal set of independent variables for√

T<(f);then

• dim(f) = d,• Xi1 , . . . , Xid is a maximal set of independent variables for f.

Proof. One has dim(√

T<(f)) = dim(T<(f)) and Xi1 , . . . , Xid is a maximalset of independent variables for

√T<(f) iff it is a maximal set of independent

variables for T<(f).Then, by the lemma above, Xi1 , . . . , Xid is a set of independent variables

for f, and is also maximal because dim(T<(f)) = dim(f) since they have thesame Hilbert polynomial.

27.12 Zero-dimensional Ideals and Multiplicity

Lemma 27.12.1. Let k ⊂ K be a field extension.Let f1, . . . , fr ∈ k[X1, . . . , Xn]. Let

I := ( f1, . . . , fr ) ⊂ k[X1, . . . , Xn],

J := ( f1, . . . , fr )K [X1, . . . , Xn] ⊂ K [X1, . . . , Xn].

13 The result holds also without the restriction that < be degree-compatible. One has just to extendthe characterization and the relevant results of the notion of Hilbert function to a graded ring,where Macaulay’s result already holds. Then the same argument can be repeated verbatim.

27.12 Zero-dimensional Ideals and Multiplicity 385

Then

(1) for each f ∈ k[X1, . . . , Xn], f ∈ I ⇐⇒ f ∈ J;(2) for any subset Xi1 , . . . , Xid of d variables,

I ∩ k[Xi1 , . . . , Xid ] = (0) ⇐⇒ J ∩ K [Xi1 , . . . , Xid ] = (0).

Proof. In both cases, the implication ⇒ is trivial, so we limit ourselves toproving the converse.

If f ∈ J, then there exist gi ∈ K [X1, . . . , Xn] such that f = ∑i gi fi .

The coefficients of the gi s, being finite, can be expressed linearly as ak-combination of a finite set of k-linearly independent elements α1 =1, α2, . . . , αt in K , so that one has

gi =∑

j

gi jα j , gi j ∈ k[X1, . . . , Xn],

whence

f =∑

i

(∑j

gi jα j

)fi =

∑j

(∑i

gi j fi

)α j .

Therefore,

(1) since f ∈ k[X1, . . . , Xn], f = ∑i gi1 fi and

∑i gi j fi = 0 for j > 1;

(2) since f ∈ K [Xi1 , . . . , Xid ],∑

i gi j fi ∈ k[Xi1 , . . . , Xid ] for each j .

Let us record this interesting converse:

Remark 27.12.2 (Traverso). Let k ⊂ K be a separable normal field extensionand let J ⊂ K [X1, . . . , Xn] be an ideal which is invariant for the Galois groupG(K/k) – that is σ(J) = J for each σ ∈ G(K/k). Then J ⊂ k[X1, . . . , Xn]because its Grobner basis F is also invariant and therefore consists of elementsin k[X1, . . . , Xn].

By way of the lemma above, the characterization of the dimension in termsof maximal sets of independent variables allows us to give the following char-acterization of zero-dimensional ideals:

Theorem 27.12.3. Let I ⊂ k[X1, . . . , Xn] be a non-trivial ideal. Then thefollowing conditions are equivalent:

(1) Z(I) is finite;(2) for each i there exists pi ∈ I ∩ k[Xi ];(3) k[X1, . . . , Xn]/I is a finite-dimensional k-vectorspace;

386 Noether

(4) I is zero-dimensional;(5) for each i there exists 14 di ∈ N such that Xdi

i ∈ T(I).

Proof.

(1) ⇒ (2) Let k be the algebraic closure of k and

Z(I) =: a1, . . . , as ⊂ kn, ai := (ai1, . . . , ain).

Let

qi (Xi ) :=s∏

j=1

(Xi − a ji ) ∈ k[Xi ].

Then qi ∈ √J, and qρi

i ∈ J ∩ k[Xi ] for some ρi ∈ N; therefore, bythe lemma above, there exists pi ∈ I ∩ k[Xi ].

(2) ⇒ (1) If (a1, . . . , an) ∈ Z(I), then pi (ai ) = 0 for each i , which leavesonly finitely many possibilities.

(2) ⇐⇒ (4) Obvious.(2) ⇒ (5) T(pi ) ∈ T(I).(5) ⇒ (3) N(I) ⊂ Xa1

1 . . . Xann : ai < di for each i.

(3) ⇒ (2) There is a linear dependence modI between the powers of Xi .

Remark 27.12.4. We are now able to discriminate between three differentcases for the ideal I ⊂ k[X1, . . . , Xn] and to do that by means of a Grobnerbasis G of I w.r.t. any ordering:

• Z(I) = ∅ ⇐⇒ 1 ∈ I ⇐⇒ 1 ∈ G;• Z(I) is finite iff k[X1, . . . , Xn]/I is a finite dimensional k-vectorspace iff for

each i there exists di ∈ N : Xdii ∈ T(G) ⊂ T(I);

• Z(I) is infinite iff k[X1, . . . , Xn]/I is an infinite dimensional k-vectorspaceiff there exists i such that for each d ∈ N : Xd

i /∈ T(G) = T(I).

Let us now discuss the structure of the zero-dimensional ideal I ⊂k[X1, . . . , Xn] and its relation with its roots, where k is a field and k denotesits algebraic closure.

We begin with the assumption that k = k is an algebraic closure. Let usconsider a zero-dimensional ideal J ⊂ k[X1, . . . , Xn].

Thus, if J is maximal, then k[X1, . . . , Xn]/J ⊃ k is a field and an algebraicextension; therefore, since k is an algebraic closure, we necessarily have

• k[X1, . . . , Xn]/J = k,• #Z(J) = 1, say Z(J) = (a1, . . . , an),• J = (X1 − a1, . . . , Xn − an).

14 This statement holds for each ordering; the value di of course is not stable under the change ofordering.


If we perform the change of coordinates

L : k[X1, . . . , Xn] −→ k[X1, . . . , Xn]

defined by

L( f ) = f (X1+a1, . . . , Xn +an), for each f (X1, . . . , Xn) ∈ k[X1, . . . , Xn],

we have L(J) = (X1, . . . , Xn).As a consequence, if we also make use of Corollary 23.3.2, we easily have

Proposition 27.12.5. Let k be an algebraic closure, let J ⊂ k[X1, . . . , Xn] bean ideal such that M := √

J is a maximal ideal. Then, there are a1, . . . , an ∈ ksuch that, denoting by L : k[X1, . . . , Xn] −→ k[X1, . . . , Xn] the change ofcoordinates defined by

L( f ) = f (X1+a1, . . . , Xn+an), for each f (X1, . . . , Xn) ∈ k[X1, . . . , Xn],

we have

• M = (X1 − a1, . . . , Xn − an),• Z(J) = (a1, . . . , an),• #N(J) = #N(L(J)) = HJ(T ) = k0(J).

Let us now assume that J ⊂ k[X1, . . . , Xn] is just a zero-dimensional idealand let us consider its irredundant primary representation J = ⋂r

i=1 qi , anddenote, for each i by mi the associated (maximal) prime of qi .

Denote Q := k[X1, . . . , Xn], π : Q −→ Q/J and πi : Q −→ Q/qi thecanonical projections and Φ : Q −→ ⊕r

i=1Q/qi the morphism defined byΦ( f ) = (π1( f ), . . . , πr ( f )), for each f ∈ Q. Then:15

Lemma 27.12.6. With the notation above, we have:

• ker(Φ) = J and• Φ is surjective, so that• Q/J ∼= ⊕r

i=1 Q/qi .

Proof. One has, for each f ∈ Q,

Φ( f ) = 0 ⇐⇒ πi ( f ) = 0, ∀i ⇐⇒ f ∈ qi , ∀i ⇐⇒ f ∈r⋂

i=1

qi = J.

In order to prove that Φ is surjective, we must consider, for each i , anyelement fi ∈ Q, and show the existence of an element f ∈ Q such thatπi ( f ) = πi ( fi ), for each i .

The proof will be done by induction: we will assume that we have an elementg such that πi (g) = πi ( fi ) for each i < j and we will produce an element f

15 Note that this is nothing more than a multivariate reformulation of the Chinese RemainderTheorem.

388 Noether

such that πi ( f ) = πi ( fi ) for each i ≤ j , the induction being guaranteed whenj = 2 by setting g := f1.

Applying Proposition 27.2.18 with q = q j and m = ⋂ j−1i=1 qi , we know that

there are c1, c2 ∈ Q, m1 ∈ q j , m2 ∈ m : c1m1 + c2m2 = 1; therefore setting

u := c1(g − f j ), f := f j + um1 and v := c2(g − f j ),

since

g − f j = (g − f j )(c1m1 + c2m2) = um1 + vm2,

we have π j ( f ) = π j ( f j + um1) = π j ( f j ) and, for each i < j ,

πi ( f ) = πi ( f + vm2) = πi ( f j + um1 + vm2) = πi (g) = πi ( fi ).

In the decomposition J = ⋂ri=1 qi , each associate prime mi is maximal and

there is a root ai := (ai1, . . . , ain) ∈ kn such that mi = (X1 − ai1, . . . , Xn −ain) and writing

Li : k[X1, . . . , Xn] −→ k[X1, . . . , Xn]

the isomorphism defined by

Li ( f ) = f (X1+ai1, . . . , Xn+ain), for each f (X1, . . . , Xn) ∈ k[X1, . . . , Xn],

and

Ni := N(Li (qi )) = T \ T(Li (qi )) and µi := #(Ni ) = Hqi (T ) = k0(qi )

one has

Corollary 27.12.7. With the notation above, we have:

• mi = (X1 − ai1, . . . , Xn − ain), for each i;• Z(J) = a1, . . . , ar ;• HJ(T ) = k0(J) = #(N(J)) = ∑r

i=1 µi , (w.r.t. any ordering);• if J is radical, then HJ(T ) = k0(J) = #(N(J)) = #Z(I).

Proof. The equality #(N(J)) = ∑ri=1 µi is a consequence of Lemma 27.12.6

since

#(N(J)) = dimk(P/J) =r∑

i=1

dimk(P/qi ) =r∑

i=1

#(Ni ).

If we now relax the assumption that k = k is an algebraic closure, givena zero-dimensional ideal I ⊂ k[X1, . . . , Xn], we can consider its extensionJ := Ik[X1, . . . , Xn]; then, using the same notation as above, we have



• Z(I) = Z(J) = a1, . . . , ar ;• HI(T ) = k0(I) = #(N(I)) = #(N(J)) = ∑r

i=1 µi , (w.r.t. any ordering);• if I is radical, then HI(T ) = k0(I) = #(N(I)) = #Z(I).

Proof. The equality Z(I) = Z(J) is trivial.For any term ordering, one has N(I) = N(J) because T(I) = T(J) as a

consequence of Lemma 27.12.1.

Definition 27.12.9. The degree or multiplicity of the zero-dimensional ideal Iis

deg(I) := #(N(I)).

The multiplicity in I ⊂ k[X1, . . . , Xn] both of the root ai ∈ Z(I) ⊂ kn andof the primary component qi ⊂ k[X1, . . . , Xn] is

µi =: mult(ai , I).

From Corollary 27.12.8 we directly obtain


• deg(I) = deg(J) = ∑ri=1 mult(ai , I) = ∑r

i=1 deg(qi ) and• deg(I) = deg(J) = #Z(I) if I is radical.

If we now consider an irredundant primary representation I = ⋂si=1 qi in

k[X1, . . . , Xn] = P , where the associated primes mi := √qi are maximal,

each mi corresponds to a set of k-conjugate zeros of I, whose coordinates livein the finite algebraic extension Ki := P/mi of k, k ⊂ Ki ⊂ k.

If mi is linear, mi = (X1−a1, . . . , Xn−an), (a1, . . . , an) ∈ kn , the structureof qi is described in Proposition 27.12.5.

If mi is not linear, we can consider the irredundant primary representationsqi = ⋂ri

j=1 qi j and mi = ⋂rij=1 mi j in Ki [X1, . . . , Xn , which satisfy

the mi j s are k-conjugate,each mi j is linear and defines a root bi j ∈ K n

i ,the bi j s are k-conjugate,mi = mi j ∩ P ,up to a renumbering,

√qi j = mi j ,

the qi j s are k-conjugate, andqi = qi j ∩ P ,for each j, l, 1 ≤ j, l ≤ ri ,

mult(bi j , I) = deg(qi j ) = deg(qil) = mult(bil , I),

390 Noether

ri = deg(mi ) = [Ki : k],deg(qi ) = ∑ri

j=1 deg(qi j ),for each j, 1 ≤ j ≤ ri , deg(qi ) = deg(mi ) deg(qi j ).

Moreover, sincer⋂

i=1

qi = J =s⋂

i=1

ri⋂j=1

qi j

are both irredundant primary representation, we have also

r =s∑

i=1ri .

In the context above, and with the same notation, it also holds:

Lemma 27.12.11. Let

I ⊂ k[X1, . . . , Xn] =: P be a zero-dimensional ideal,m ⊂ k[Z1, . . . , Zn] a maximal ideal,K := k[Z1, . . . , Zn]/m = k[α1, . . . , αn],b ∈ K n a root of I, b ∈ Z(I),q ⊂ K [X1, . . . , Xn] = k[α1, . . . , αn][X1, . . . , Xn] the primary component

of I in K [X1, . . . , Xn] whose root is b.

If m is generated by fi (Z1, . . . , Zn), 1 ≤ i ≤ m and q by

g j (α1, . . . , αn, X1, . . . , Xn), 1 ≤ j ≤ µ ∈ K [X1, . . . , Xn],

then, denoting by Q ⊂ k[Z1, . . . , Zn, X1, . . . , Xn] the ideal generated by

fi (Z1, . . . , Zn), 1 ≤ i ≤ m ∪ g j (Z1, . . . , Zn, X1, . . . , Xn), 1 ≤ j ≤ µ,Q ∩ k[X1, . . . , Xn] is the primary component q of I in k[X1, . . . , Xn] whoseroot is b.

Proof. In fact if

ψ : k[Z1, . . . , Zn, X1, . . . , Xn] → K [X1, . . . , Xn]

is the morphism defined by ψ(Zi ) = αi , then Q = ψ−1(q), ψ(Q) = q so thatq = q ∩ k[X1, . . . , Xn] = Q ∩ k[X1, . . . , Xn].

27.13 Unmixed Ideals

Let P := k[X1, . . . , Xn], let Y1, . . . , Yn be a system of coordinates of P andf ⊂ P be an ideal.

Let f = ⋂ri=1 qi , be an irredundant primary representation of f and, for each

i, let pi be the associated prime of qi and di := dim(pi ).

27.13 Unmixed Ideals 391

Definition 27.13.1. The ideal f is said to be unmixed if for each i, di = dim(f).

Let d := max(di ) = dim(f) and, for each j, u j := ⋂i :di = j qi .

Lemma 27.13.2. With the notation above, the following holds:

(1) f := ⋂dj=1 u j ;

(2) for each j either

• u j = (1), or• u j is unmixed and dim(u j ) = j ;

(3) for all j such that u j = (1), u j ⊇ ⋂i = j ui .

Definition 27.13.3. An irredundant equidimensional representation of f is arepresentation f := ⋂d

i=1 ui which satisfies the conditions of the lemma above.The top-dimensional component of f is

Top(f) := ud :=⋂

i :δ(i)=d

qi .

Remark 27.13.4. The non-uniqueness of the embedded primary componentsimplies the non-uniqueness of equidimensional decomposition. The best resultis as follows. Let

f =d⋂

i=1

ui =δ⋂

i=1

vi

be two equidimensional decompositions. Then

• d = δ,• ud = vd ,

• ui = (1) ⇐⇒ vi = (1),

• √ui = √

vi , for each i .

In particular, the top-dimensional component is unique.

Remark 27.13.5. If one is interested only in the topological structure of the setZ(f) of the roots of f, then multiplicity and even embedded components areirrelevant and one could be interested in the decomposition√

f =⋂

i∈Mpi , where M = i : pi is isolated.

Let us assume, wlog, that the primaries are ordered so that, for a suitablevalue 1 ≤ s ≤ r,

392 Noether

• X1, . . . , Xd is a maximal set of independent variables for qi ⇐⇒ i ≤ s.

If we therefore consider the ring k(X1, . . . , Xd)[Xd+1, . . . , Xn], whichis the quotient ring of k[X1, . . . , Xn] w.r.t. the multiplicative systemk[X1, . . . , Xd ] \ 0 and the canonical homomorphism

φ : R := k[X1, . . . , Xd ][Xd+1, . . . , Xn]

−→ k(X1, . . . , Xd)[Xd+1, . . . , Xn] =: S,

all the notations and results of Section 27.5 are available. In particular, fromCorollary 27.5.19 we obtain


• fec = fk(X1, . . . , Xd)[Xd+1, . . . , Xn] ∩ k[X1, . . . , Xn];• fe = ⋂s

i=1 qei is an irredundant primary representation;

• fec = ⋂si=1 qi is an irredundant primary representation;

• fe is zero-dimensional;• fec is unmixed.

If, moreover, X1, . . . , Xn is a Noether position for f, then

• Top(f) = fec = fk(X1, . . . , Xd)[Xd+1, . . . , Xn] ∩ k[X1, . . . , Xn].

Proof. We have qi ∩ k[X1, . . . , Xd ] \ 0 = ∅ iff dim(qi ) ≥ d, andX1, . . . , Xd is contained in a maximal set of independent variables for qi .

Since dim(qi ) ≤ dim(f) = d we have

qi ∩ k[X1, . . . , Xd ] \ 0 = ∅ ⇐⇒ i ≤ s.

Definition 27.13.7. For an unmixed ideal f of rank r = n − d in Noetherposition w.r.t. X1, . . . , Xn and whose irredundant primary representation isf = ⋂s

i=1 qi ,

the degree or multiplicity of f is the degree of the zero-dimensional ideal

fe = fk(X1, . . . , Xd)[Xd+1, . . . , Xn];the multiplicity in f of the primary component qi is the multiplicity in fe of

qei = qi k(X1, . . . , Xd)[Xd+1, . . . , Xn].

28

Moller I

Part four is devoted to discussing the linear algebra tools which allowus to describe and compute the k-vectorspace structure of an ideal I ⊂k[X1, . . . , Xn] := P.

The task is indicated by Hilbert’s notion of characteristic function, whichrequires us to describe the linear equations satisfied by the coefficients of apolynomial f ∈ P in order to be a member of I, thus indicating that we haveto consider the P-module P∗ := Homk(P, k) of all k-linear functionals.

In Section 28.1 I recall the properties of the duality between finite-k-dimensional P-modules L ⊂ P∗ and zero-dimensional ideals I ⊂ P.

In Section 28.2 I introduce the computational tool needed in order, given aP-module L ⊂ P∗, to compute the corresponding dual ideal

I := g ∈ P : (g) = 0, for each ∈ L;such a tool is the algorithm introduced by Moller which essentially consists ofa multivariate version of Newton interpolation which takes good advantage ofthe properties of the Grobner basis of I.

28.1 Duality

Let us fix the polynomial ring P := k[X1, . . . , Xn] and let us denote by

P∗ := Homk(P, k)

the k-vectorspace of all k-linear functionals : P → k.Each k-linear functional : P → k is characterized by its value on any basis

B of P; in fact, each f ∈ P can be uniquely expressed as f = ∑β∈B c( f, β)β,

with c( f, β) ∈ k, and, by k-linearity, we have

( f ) =∑β∈B

c( f, β)(β).

393

394 Moller I

Remark 28.1.1. If we use, as a basis of P , the canonical basis

T := Xa11 · · · Xan

n : (a1, . . . , an) ∈ Nn,

each k-linear functional ∈ P∗ can be encoded by means of the series∑t∈T

(t)t ∈ k[[X1, . . . , Xn]]

in such a way that to each such series∑

t∈T γ (t)t ∈ k[[X1, . . . , Xn]] is as-sociated the k-linear functional ∈ P∗ defined, on each polynomial f =∑

t∈T c( f, t)t , by

( f ) :=∑t∈T

c( f, t)γ (t).

The module P∗ has a natural structure as P-module, which is obtained bydefining, for each ∈ P∗ and f ∈ P , ( · f ) ∈ P∗ as

( · f )(g) := ( f g), for each g ∈ P.

Definition 28.1.2. Let L = 1, . . . , r ⊂ P∗ and q = q1, . . . , qs ⊂ P .The sets L and q are said to be

• triangular if r = s and i (q j ) = 0, for each i < j ,

• biorthogonal if r = s and i (q j ) = δi j =

1 if i = j ,0 if i = j.

From a triangular set q = q1, . . . , qs ⊂ P of L = 1, . . . , r ⊂ P∗, abiorthogonal set

q′ = q ′1, . . . , q ′

s ⊂ P

is easily obtained by defining

q ′s := s(qs)

−1qs andq ′

j := j (q j )−1q j − ∑

i> j i (q j )−1q ′

i for j := s − 1..1.

Lemma 28.1.3.

(1) Given L = 1, . . . , s ⊂ P∗, the following conditions are equivalent:

(a) L is linearly independent;(b) there exists q = q1, . . . , qs ⊂ P biorthogonal to L;(c) there exists q = q1, . . . , qs ⊂ P triangular to L.

(2) Given q = q1, . . . , qs ⊂ P , the following conditions are equivalent:

(a) q is linearly independent;

28.1 Duality 395

(b) there exists L = 1, . . . , s ⊂ P∗ biorthogonal to q;(c) there exists L = 1, . . . , s ⊂ P∗ triangular to q.

Proof. Let us consider a set L = 1, . . . , s ⊂ P∗.Assume first that L is not linearly independent.Then there are c1, . . . , cs ∈ k not all zero, say cs = 0, such that

∑i cii = 0

and assume that q = q1, . . . , qs ⊂ P is biorthogonal to L. Then

0 =∑

i

cii (qs) = css(qs) = cs = 0,

giving a contradiction.Assume now that L is linearly independent, and let us prove the existence

of q = q1, . . . , qs ⊂ P biorthogonal to L, arguing by induction on s.If s = 1, the linear independence of 1 means 1 = 0 and so the existence

of g1 ∈ P such that 1(g1) = 0.So let us assume the existence of q1, . . . , qs−1 ⊂ P which is biorthogonal

to 1, . . . , s−1. Since s ∈ Spank (1, . . . , s−1) then

:= s − s(q1)1 − · · · − s(qs−1)s−1 = 0

and there is g ∈ P such that (g) = 0. Setting

g′ := g − 1(g)q1 − · · · − s−1(g)qs−1

we have

s(g′) = s(g) − 1(g)s(q1) − · · · − s−1(g)s(qs−1) = (g) = 0,

while for i < s

i (g′) = i (g)−1(g)i (q1)−· · ·−s−1(g)i (qs−1) = i (g)−i (g)i (qi ) = 0,

so that

q := q1, . . . , qs−1, s(g′)−1g′ ⊂ P

is triangular to L, from which we obtain the biorthogonal set

q′ := q ′1, . . . , q ′

s ⊂ P

by setting

q ′s := s(g′)−1g′ and

q ′j := q j − s(q j )s(g′)−1g′ for j < s.

The statement related to q = q1, . . . , qs ⊂ P is proved dually.

396 Moller I

For each k-vectorsubspace L ⊂ P∗, let

P(L) := g ∈ P : (g) = 0, for each ∈ Land for each k-vectorsubspace P ⊂ P , let

L(P) := ∈ P∗ : (g) = 0, for each g ∈ P.Lemma 28.1.4. For each k-vector subspace P ⊂ P and each k-vectorsubspace L ⊂ P∗ the following holds

• P is an ideal iff L(P) is a P-module.• L is a P-module iff P(L) is an ideal.

Proof. Since

( f )(g) = ( f g), for each g ∈ P, f ∈ P, ∈ P∗,

the three statements

• P is an ideal,• ( f )(g) = ( f g) = 0, for each g ∈ P, f ∈ P, ∈ L(P),• L(P) is a P-module

are trivially equivalent.Dually also the statements

• L is a P-module,• ( f )(g) = ( f g) = 0, for each ∈ L , f ∈ P, g ∈ P(L),• P(L) is an ideal

are equivalent.

Lemma 28.1.5. For all k-vectorsubspaces P1, P2 ⊂ P and all k-vectorsubspaces L1, L2 ⊂ P∗ we have

(1) P1 ⊂ P2 ⇒ L(P1) ⊃ L(P2);(2) L1 ⊂ L2 ⇒ P(L1) ⊃ P(L2);(3) L(P1 ∩ P2) ⊃ L(P1) + L(P2);(4) P(L1 ∩ L2) ⊃ P(L1) + P(L2);(5) L(P1 + P2) = L(P1) ∩ L(P2);(6) P(L1 + L2) = P(L1) ∩ P(L2).

Proof.

(1) and (2) are trivial.(3) and (4) The inclusions follow directly from (1) and (2).

28.1 Duality 397

(5) The inclusion L(P1 + P2) ⊂ L(P1) ∩ L(P2) follows directly from (1).Conversely, for each ∈ L(P1) ∩ L(P2), g1 ∈ P1, g2 ∈ P2,

(g1 + g2) = (g1) + (g2) = 0 and ∈ L(P1 + P2).

(6) The inclusion P(L1 + L2) ⊂ P(L1) ∩ P(L2) follows directly from (2).Conversely, for each g ∈ P(L1) ∩ P(L2), 1 ∈ L1, 2 ∈ L2,

(1 + 2)(g) = 1(g) + 2(g) = 0 and g ∈ P(L1 + L2).

Proposition 28.1.6. For each k-vectorsubspace P ⊂ P and each k-vectorsubspace L ⊂ P∗, we have

• L ⊂ LP(L);• P ⊂ PL(P).

Proof. We have, by definition of P( · ),

(g) = 0, for each ∈ L , g ∈ P(L),

so that, by definition of L( · ) we have ∈ L(P(L)) for each ∈ L .

Dualling the same argument we have

(g) = 0, for each g ∈ P, ∈ L(P)

so that g ∈ P(L(P)) for each g ∈ P.

Lemma 28.1.7. For each k-vector subspace P ⊂ P and each g ∈ P we have

(g) = 0, for each ∈ L(P) ⇒ g ∈ P.

Proof. For any g /∈ P we need to exhibit an element ∈ L(P) such that(g) = 0. So let us consider a k-basis B of P and a set B′ such that B∪g∪B′

is a k-basis of P , and let us define ∈ P∗ to be the unique linear functionalsuch that

(β) =⎧⎨⎩

0 iff β ∈ B,1 iff β = g,0 iff β ∈ B′.

Then ∈ L(P) and (g) = 0 as required.

Corollary 28.1.8. For each k-vector subspace P ⊂ P we have

P = PL(P).

398 Moller I

Example 28.1.9. In general, for a k-vectorsubspace L ⊂ P∗ it does not neces-sarily hold that L = LP(L).

Let us consider P = k[X ] and let us denote, for each i ∈ N, by λi ∈ P∗ thelinear functional such that

λi (X j ) = 1 if i = j,

0 otherwise.

Then (see Remark 28.1.1) for L := Spankλi , i ∈ N ⊂ P∗, we have

P(L) = 0 and LP(L) = P∗ = L

since L consists only of the functionals encoded by polynomials in k[[X ]] ∼=P∗ while functionals encoded as series – like the linear functional λ defined asλ(X j ) := 1, for each j ∈ N – are not members of L .

Example 28.1.10. Also if we assume L ⊂ P∗ to be a P-module, L = LP(L)

does not necessarily hold for the same reason.Let us for instance consider P = k[X1, X2] and let us denote, for each

(i, j) ∈ N2, λi j ∈ P∗ the linear functionals such that

λi j (Xk1 Xl

2) =

1 if (i, j) = (k, l),0 otherwise.

Then

L := Spank L ⊂ P∗, L := λi0, i ∈ N ∪ λ0 j , j ∈ N,is clearly a P-module, since

X1λ00 = 0, X2λ00 = 0,

X1λi0 = λi−1 0, X2λi0 = 0, i > 0,

X1λ0 j = 0, X2λ0 j = λ0 j−1, j > 0.

We have P(L) = (X1 X2) and, using the encoding introduced in Re-mark 28.1.1,

LP(L) =⎧⎨⎩

∑i∈N

(Xi1)Xi

1 +∑

j∈N\0(X j

2)X j2

⎫⎬⎭ ⊂ k[[X1, X2]].

Lemma 28.1.11. For each finite-dimensional k-vectorsubspace L ⊂ P∗ andeach ∈ P∗ we have

(g) = 0, for each g ∈ P(L) ⇒ ∈ L .

Proof. For any /∈ L we need to exhibit an element g ∈ P(L) such that(g) = 0. Let L = 1, . . . , s ⊂ P∗ be a k-basis of L and let s+1 := ∈ Lso that L∪s+1 is linearly independent and there is a set q1, . . . , qs, qs+1 ⊂P biorthogonal to L ∪ s+1.

28.1 Duality 399

In particular s+1(qs+1) = 1 while i (qs+1) = 0, for each i ≤ s, so thatqs+1 ∈ P(L).

Corollary 28.1.12. For each finite dimensional k-vector subspace L ⊂ P∗ wehave L = LP(L).

Lemma 28.1.13. Let L = 1, . . . , s ⊂ P∗ and q = q1, . . . , qs ⊂ P betwo biorthogonal sets.

Writing L := Spank(L) and Q := Spank(q), we have:

(1) P ∼= Q ⊕ P(L), P/P(L) ∼= Q;(2) P∗ ∼= L ⊕ L(Q), P∗/L(Q) ∼= L .

Proof. If q ∈ Q ∩ P(L) then q ∈ Q ⇒ q = ∑j c j q j and

q ∈ P(L) ⇒ ci =∑

j

c ji (q j ) = i (q) = 0, for each i,

so that Q ∩ P(L) = 0.Let q ∈ P and let q(1) := ∑

i i (q)qi , q(2) := q − q(1) so that q(1) ∈ Q,i (q(2)) = 0 for each i, q(2) ∈ P(L) and P = Q ⊕ P(L).

Corollary 28.1.14. For each finite-k-dimensional P-module L ⊂ P∗, P(L)

is a zero-dimensional ideal and dimk(P) = deg(P(L)).

For each zero-dimensional ideal P ⊂ P , the P-module L(P) is finite-k-dimensional and deg(P) = dimk(L(P)).

Theorem 28.1.15. The mutually inverse maps L( · ) and P( · ) give a bi-univocal, inclusion reversing, correspondence between the set of the zero-dimensional ideals P ⊂ P and the set of the finite-k-dimensional P-modulesL ⊂ P∗.

Moreover, for any P ⊂ P we have deg(L) = dimk(L(P)) and, for anyfinite-k-dimensional P-module L ⊂ P∗ we have dimk(P) = deg(P(L)).

Corollary 28.1.16. For each zero-dimensional ideal P1, P2 ⊂ P and eachfinite-k-dimensional P-module L1, L2 ⊂ P∗ we have:

• L(P1 ∩ P2) = L(P1) + L(P2);• P(L1 ∩ L2) = P(L1) + P(L2).

Proof. Remarking that, under the assumptions, L(P1) + L(P2) is a finite-dimensional k-vectorspace, we have

L(P1∩P2) = L(PL(P1)∩PL(P2)) = LP(L(P1)+L(P2)) = L(P1)+L(P2),

400 Moller I

and

P(L1 ∩ L2)

= P(LP(L1) ∩ LP(L2)) = PL(P(L1) + P(L2)) = P(L1) + P(L2).

Theorem 28.1.17. Let L = 1, . . . , s ⊂ P∗ be a linearly independent set,let q = q1, . . . , qs ⊂ P be biorthogonal to L and L := Spank(L).

Then, for each (c1, . . . , cs) ∈ ks and each g ∈ P we have

i (g) = ci , for each i ⇐⇒ there exists h ∈ P(L) : g = h +∑

j

c j q j .

Proof. For g = h + ∑j c j q j , h ∈ P(L), we have, for each i ,

i (g) = i (h) +∑

j

c ji (q j ) = ci .

If i (g) = ci , for each i , then for h := g − ∑j c j q j we have, for each i ,

i (h) = i (g) −∑

j

c ji (q j ) = ci − ci = 0,

so that h ∈ P(L).

Theorem 28.1.18 (Vandermonde Criterion). Let L = 1, . . . , s ⊂ P∗

and p = p1, . . . , ps ⊂ P be two linearly independent sets. Writing

L := Spank(L), P := Spank(p),

the following conditions are equivalent:

(1) P = P(L);(2) det(i (pl)) = 0;(3) L = L(P).

Proof.

(1) ⇒ (2) Let q = q1, . . . , qs be biorthogonal to L. Therefore

P = P(L) ⇐⇒ Spank(p) = Spank(q).

Denoting (c jl) the invertible matrix such that pl =∑

j c jlq j , we have

i (pl) =∑

j

c jli (q j ) = cil , for each i, l,

and det(i (pl)) = det(cil) = 0.

(2) ⇒ (1) Let (al j ) be the inverse of the matrix (cil), cil := i (pl), so that∑l i (pl)al j = δi j , and let q j := ∑

l al j pl , for each j ; then we have

i (q j ) =∑

l

al ji (pl) = δi j

28.2 Moller Algorithm 401

and q = q1, . . . , qs is biorthogonal to L so that P = Spank(q) =P(L).

(1) ⇒ (3) L = LP(L) = L(P).(3) ⇒ (1) P = PL(P) = P(L).

28.2 Moller Algorithm

Let P := k[X1, . . . , Xn], T := Xa11 . . . Xan

n : (a1, . . . , an) ∈ Nn and < be

any term ordering. Let

L = 1, . . . , s ⊂ P∗

be a (not necessarily linearly independent) set of k-linear functionals such thatL := Spank(L) is a P-module, and let us write, for each f ∈ P ,

v( f, L) := (1( f ), . . . , s( f )) ∈ ks .

Since L is a finite-k-dimensional P-module, I := P(L) is a zero-dimen-sional ideal and the order ideal N(I) := N<(I) = T \ T<(I) satisfies

#(N(I)) = deg(I) = dimk(L) =: r ≤ s.

Let us therefore write N(I) = t1, . . . , tr , and let us consider the s × r ma-trix i (t j ) whose columns are the vectors v(t j , L) and are linearly independent,since any relation

∑j c jv(t j , L) = 0 would imply

i

(∑j

c j t j

)=

∑j

c ji (t j ) = 0 and∑

j

c j t j ∈ P(L) = I

contradicting the definition of N(I).The matrix i (t j ) has rank r ≤ s and it is possible to extract an ordered

subset

Λ := λ1, . . . , λr ⊂ L, SpankΛ = SpankLand to renumber the terms in N(I) in such a way that each principal minorλi (t j ), 1 ≤ i, j ≤ σ ≤ r is invertible.

Therefore, if we consider a set

q := q1, . . . , qr ⊂ P

which is triangular w.r.t. Λ, and (ai j ) denotes the invertible matrix such that,for each i ≤ r , qi = ∑r

j=1 ai j t j , then

• q1, . . . , qσ and λ1, . . . , λσ are triangular, for each σ ≤ r ;• Spankt1, . . . , tσ = Spankq1, . . . , qσ , for each σ ≤ r ;• (ai j ) is lower triangular.

402 Moller I

If we now further assume that1

dimk(L) = r = s andeach subvectorspace Lσ := Spank(1, . . . , σ ) is a P-module so thateach Iσ = P(Lσ ) is a zero-dimensional ideal andthere is a chain I1 ⊃ I2 ⊃ · · · ⊃ Is = I,

then, for each σ ≤ r

• λσ = σ

• N(Iσ ) = t1, . . . , tσ is an order ideal,• Iσ ⊕ Spankq1, . . . , qσ = P ,• T(qσ ) = tσ .

We can summarize these remarks in the following

Theorem 28.2.1 (Moller). Let P := k[X1, . . . , Xn], and < be any term or-dering. Let L = 1, . . . , s ⊂ P∗ be a set of k-linear functionals such thatP(Spank(L)) is a zero-dimensional ideal.

Then there are

• an integer r ∈ N,• an order ideal N := t1, . . . , tr ⊂ T ,• an ordered subset Λ := λ1, . . . , λr ⊂ L,• an ordered set q := q1, . . . , qr ⊂ P ,

such that, writing L := Spank(L) and I := P(L), we have:

• r = deg(I) = dimk(L),• N(I) = N,

1 There are instances in which this assumption is natural.For instance if each functional i consists of the polynomial evaluation at the point ai :=

(ai1, . . . , ain) ∈ kn so that

i ( f ) = f (ai1, . . . , ain) for each f (X1, . . . , Xn) ∈ P,

then any permutation π(1), . . . , π(s) has this property since each

Iσ = P(Spank (π(1), . . . , π(σ )) = f ∈ P : f (aπ( j)1, . . . , aπ( j)n), 1 ≤ j ≤ σ

is a zero-dimensional ideal.We will see further (see Corollary 32.3.3) that (at least if k is algebraically closed) any

zero-dimensional ideal I ⊂ P has a specific set of functionals L = 1, . . . , s such thatI = P(Spank (L) and has this property.

Let us explicitly remark that such property depends on a specific good enumeration of the setL and can be easily lost under a permutation.


• Spank(Λ) = Spank(L),• Spankt1, . . . , tσ = Spankq1, . . . , qσ , for each σ ≤ r ,• q1, . . . , qσ and λ1, . . . , λσ are triangular, for each σ ≤ r .

If, moreover, dimk(L) = r = s and Lσ := Spank(1, . . . , σ ) is a P-module, for each σ ≤ r , then it further holds that

• λσ = σ ,• N(Iσ ) = t1, . . . , tσ is an order ideal,• Iσ ⊕ Spankq1, . . . , qσ = P ,• T(qσ ) = tσ ,

for each σ ≤ r , where Iσ = P(Lσ ),

and give a more precise formulation of Theorem 28.1.17.

Corollary 28.2.2 (Lagrange Interpolation Formula). Let P := k[X1, . . . ,

Xn], and < be any term ordering. Let

L = 1, . . . , s ⊂ P∗

be a set of linearly dependent k-linear functionals such that I := P(Spank(L))

is a zero-dimensional ideal.There exists a set q = q1, . . . , qs ⊂ P such that

(1) qi = Can(qi , I) ∈ Spank(N(I)),(2) L and q are triangular,(3) P/I ∼= Spank(q).

There exists a set q′ = q ′1, . . . , q ′

s ⊂ P such that

(1) q ′i = Can(q ′

i , I) ∈ Spank(N(I)),(2) L and q′ are biorthogonal,(3) P/I ∼= Spank(q

′).

Let c1, . . . , cs ∈ k and let q := ∑i ci q ′

i ∈ P . Then, if g1, . . . , gt denotesa Grobner basis of I, one has

(1) q is the unique polynomial in Spank(N(I)) such that i (q) = ci , foreach i ,

(2) for each p ∈ P it is equivalent

(a) i (p) = ci , for each i ,(b) q = Can(p, I),

404 Moller I

(c) there exist h j ∈ P such that

p = q +t∑

j=1

h j g j , T(h j )T(g j ) ≤ T(p − q).

Lemma 28.2.3. Let P := k[X1, . . . , Xn], and < be any term ordering. LetL = 1, . . . , r ⊂ P∗ be a set of linearly independent k-linear functionalssuch that I := P(Spank(L)) is a zero-dimensional ideal and let

N := t1, . . . , tr ⊂ T ,q := q1, . . . , qr ⊂ P ,G := g1, . . . , gt ⊂ P

be such that

• N is an order ideal,• Spankt1, . . . , tr = Spankq1, . . . , qr ,• q1, . . . , qr and 1, . . . , r are triangular,• (g) = 0 for each g ∈ G and each ∈ L ,• N T<(G) = T ,• for each g ∈ G, g − lc(g)T<(g) ∈ Spank (N),

then G is a reduced Grobner basis of P(Spank(L)) w.r.t. <.

Proof. Even if G := g1, . . . , gt were not necessarily a Grobner basis, thecondition

N T<(G) = T

is sufficient to imply that, for any polynomial f ∈ P , CanonicalForm( f, G)

(Figure 22.2) returns polynomials h j ∈ P such that

f −t∑

j=1

h j g j ∈ Spank (N)

so that there are constants ci such that f = ∑ri=1 ci qi + ∑t

j=1 h j g j .

Therefore the condition

i (g j ) = 0, 1 ≤ i ≤ r, 1 ≤ j ≤ t,

allows us to deduce that:

G := g1, . . . , gt is the Grobner basis of the ideal J it generates: otherwisethere is f ∈ Spank (N) ∩ J, f = 0 and there are polynomials h j ∈ P


and constants ci such that f = ∑ri=1 ci qi = ∑t

j=1 h j g j , whence,for each l,

cl =r∑

i=1

cil(qi ) =t∑

j=1

l(h j )l(g j ) = 0

and f = 0;and such an ideal is P(Spank(L)), since G ⊂ P(Spank(L)) and, foreach

f =σ∑

i=1

ci qi +t∑

j=1

h j g j ∈ P(Spank(L))

f = ∑tj=1 h j g j ∈ J because, for each l,

cl =σ∑

i=1

cil(qi ) +t∑

j=1

l(h j )l(g j ) = l( f ) = 0.

Finally G is reduced since for each g ∈ G,

g − lc(g)T<(g) ∈ Spank (N) = N(P(Spank(L))).

Definition 28.2.4. We will say that a set L = 1, . . . , s ⊂ P∗ of k-linearfunctionals is given if it is possible to compute i ( f ), for each f ∈ P and eachi, 1 ≤ i ≤ s.

If we are given a set L ⊂ P∗ of k-linear functionals, there is not in generalan algorithm verifying whether Spank(L) is a P-module, so that P(Spank(L))

is an ideal, nor one verifying the linear independence of L.While the second algorithm that we are going to describe (Algorithm 28.2.7)

does not need the linear independence of L and actually extracts from it alinear basis Λ of Spank(L), the correctness of the algorithms depends on theassumption that P(Spank(L)) is an ideal. As we will see, in all the applications,this assumption can be easily deduced theoretically.

Algorithm 28.2.5 (Moller). Let us assume we are given a set

L = 1, . . . , s ⊂ P∗

of k-linear functionals such that I := P(Spank(L)) is an ideal.In this algorithm we will further assume that dimk(L) = s and Lσ :=

Spank(1, . . . , σ ) is a P-module, for each σ ≤ s, so that each Iσ = P(Lσ )

is an ideal and all the results of Theorem 28.2.1 hold.

406 Moller I

Fig. 28.1. Moller Dual Algorithm

(G1, . . . , Gs , N, q) := G-basis(L, <)where

P := k[X1, . . . , Xn],T := Xa1

1 . . . Xann : (a1, . . . , an) ∈ N

n,L = 1, . . . , s ⊂ P∗ is a linearly independent set such that Lσ :=Spank(1, . . . , σ ) is a P-module, for each σ ≤ s,< is a term ordering on P ,Iσ = P(Lσ ), for each σ ≤ s,Gσ ⊂ Iσ is the reduced Grobner basis of Iσ w.r.t. <, for each σ ≤ s,N := t1, . . . , ts is an order ideal,q := q1, . . . , qs ⊂ P is a set triangular to L,Nσ := t1, . . . , tσ = N(Iσ ), for each σ ≤ s,qσ ∈ SpankNσ , and T(qσ ) = tσ , for each σ ≤ s,Spankt1, . . . , tσ = Spankq1, . . . , qσ , for each σ ≤ s,q1, . . . , qσ and 1, . . . , σ are triangular for each σ ≤ s,

σ := 1, t1 := 1, N := t1,q1 := 1(1)−1t1, q := q1,G1 := Xh − 1(Xh), 1 ≤ h ≤ n,%% Nσ T(Gσ ) = T , j ( f ) = 0 for all f ∈ Gσ , 1 ≤ j ≤ σ .For σ := 2..s do

t := minT( f ) : f ∈ Gσ−1, σ ( f ) = 0,Let f ∈ Gσ−1 : T( f ) = t ,tσ := t, qσ := −1

σ ( f ) f ,N := N ∪ tσ , q := q ∪ qσ ,Gσ := f − σ ( f )qσ : f ∈ Gσ−1,For each h = 1..n such that Xht /∈ T(Gσ ) do

p := Xht ,For i = 1..σ do p := p − i (p)qi ,Gσ := Gσ ∪ p,

%% Nσ T(Gσ ) = T , j ( f ) = 0 for all f ∈ Gσ , 1 ≤ j ≤ σ .G1, . . . , Gs , N, q

Under these assumptions we present here an algorithm (see Figure 28.1)which allows us to compute

• the reduced Grobner basis Gσ of Iσ w.r.t. <, for each σ ≤ s,• an order ideal N := t1, . . . , ts ⊂ T and• an ordered set q := q1, . . . , qs ⊂ P ,

which satisfy the conditions of Theorem 28.2.1, thus making effective the La-grange Interpolation Formula.

The algorithm performs iteration on σ and, in each step, it will produce

• a set Gσ ⊂ I,• a term tσ ∈ T ,• a polynomial qσ ∈ P ,


which satisfy

• Nσ := t1, . . . , tσ ⊂ N(I) is an order ideal,• Nσ T(Gσ ) = T ,• qσ ∈ Spankt1, . . . , tσ , and T(qσ ) = tσ ,

• Spankt1, . . . , tσ = Spankq1, . . . , qσ ,• q1, . . . , qσ and 1, . . . , σ are triangular,• for each g ∈ Gσ , g − lc(g)T(g) ∈ Spank (Nσ ),• i (g) = 0 for each g ∈ Gσ and each i ≤ σ .

Therefore, for each σ , Lemma 28.2.3 gives that Gσ is the reduced Grobnerbasis of Iσ .

In particular, at termination, it is sufficient to set

N := Ns, and q := q1, . . . , qs.We begin the iteration by setting

t1 := 1, q1 := λ1(1)−1t1, G1 := Xh − λ1(Xh), 1 ≤ h ≤ n,which trivially satisfy the required conditions.

Then iteratively,

• we select t ∈ T(Gσ ) and f ∈ Gσ such that 2

t := T( f ) = minT(g) : g ∈ Gσ , σ+1(g) = 0;• and we set tσ+1 := t, qσ+1 := −1

σ+1( f ) f ;• then we modify all elements in g ∈ Gσ in order that they also satisfy

σ+1(g) = 0 by replacing them with g − σ+1(g)qσ+1;• and we enlarge the resulting set

G := g − σ+1(g)qσ+1 : g ∈ Gσ , g = f to a set Gσ+1 satisfying Nσ+1 T(Gσ+1) = T by including, for each τ /∈Nσ+1 T(G), the single polynomial g := τ − ∑σ+1

i=1 c(g, ti )ti such thati (g) = 0 for each i ≤ σ + 1.

Example 28.2.6. Let us compute the Grobner basis w.r.t. the lex ordering in-duced by X1 < X2 of the (X1, X2)-primary ideal

I := f1, f2, f3, f4 ⊂ k[X1, X2],

f1 := X32 − X1 X2

2, f2 := X21 X2, f3 := X3

1 − X22 + X1 X2, f4 := X4

2,

2 Their existence is given by the linear independency of L.

408 Moller I

which, as we will prove in Example 32.7.3, satisfies

I = P(Spank(L)), L = 1, . . . , 7

where each i is encoded as elements in k[[X1, X2]] by

1 := 1, 2 := X1,

3 := X2, 4 := X21,

5 := X22 + X1 X2, 6 := X3

1 − X1 X2,

7 := X41 + X3

2 + X1 X22

and (see Example 32.7.3) L is ordered so that each Spank(1, . . . , i is aP-module.

We have

t1 := 1, q1 := 1, G1 := X1, X2;t2 := X1, q2 := X1, G2 := X2

1, X2;t3 := X2, q3 := X2, G3 := X2

1, X1 X2, X22;

t4 := X21, q4 := X2

1, G4 := X31, X1 X2, X2

2;t5 := X1 X2, q5 := X1 X2, G5 := X3

1, X21 X2, X2

2 − X1 X2;t6 := X3

1, q6 := X31, G6 := X4

1, X21 X2, X2

2 − X1 X2 − X31;

t7 := X41, q7 := X4

1, G7 := X51, X2

1 X2, X22 − X1 X2 − X3

1.The reader can easily check not only that is G7 the lex Grobner basis of I butthat each basis Gi , 1 ≤ i ≤ 7, is the lex Grobner basis of the ideal

Ii = P(Spank(1, . . . , i ))

it generates.

Algorithm 28.2.7 (Moller). Algorithm 28.2.5 iterates on the ordered set of thefunctionals and, as a consequence, produces a set of terms N(I) whose orderinghas no relation to the term ordering <.

This is a price which is worth paying under the assumption that each sectionLσ defines an ideal Iσ = P(Lσ ), since the algorithm describes each such ideal.But in a general case, it could be preferable to preserve the < ordering on N(I),the more so if, as in many applications, the computation is connected withGrobner basis structure and normal form computation.

We present here a variation of Algorithm 28.2.5 which iterates on the <-ordered set N(I), thus preserving this ordering while reshuffling L.

We now assume that we are given a set L = 1, . . . , s ⊂ P∗ of k-linearfunctionals whose only property is that I := P(Spank(L)) is an ideal and wepresent here an algorithm (see Figure 28.2) which computes


Fig. 28.2. Moller Algorithm

(G, r, N, Λ, q) := G-basis(L, <)where

P := k[X1, . . . , Xn],T := Xa1

1 . . . Xann : (a1, . . . , an) ∈ N

n,L = 1, . . . , s ⊂ P∗ is a set such that I := P(Spank(L)) is a zero-dimensional ideal;< is a term ordering on P ,G ⊂ I is the reduced Grobner basis of I w.r.t. <,r = deg(I) = dimk(Spank(L)),N := t1, . . . , tr = N(I),1 = t1 < t2 < · · · < ti < ti+1 < · · · < tr ,Λ := λ1, . . . , λr ⊂ L, is a linearly independent basis of Spank(L),q := q1, . . . , qr ⊂ P is a set triangular to Λ,qi ∈ Spankt1, . . . , ti , T(qi ) = ti , for each i ≤ r ,Spankt1, . . . , ti = Spankq1, . . . , qi , for each i ≤ r ,q1, . . . , qi and λ1, . . . , λi are triangular, for each i ≤ r .

G := ∅, r := 1, t1 := 1, N := t1,v := (1(t1), . . . , s(t1)),µ := min j : j (1) = 0,λ1 := µ, Λ := λ1,q1 := λ1(1)−1t1, q := q1, vect(1) := λ1(1)−1v,%% vect(1) = (1(q1), . . . , s(q1)),While N T(G) = T do

t := min<τ ∈ T , τ /∈ N T(G),q := t, v := (1(q), . . . , s(q)),For j = 1..r do

v := v − λ j (q) vect( j), q := q − λ j (q)q j ,%% v = (1(q), . . . , s(q)).

If v = 0 thenG := G ∪ q,

elser := r + 1,tr := t, N := N ∪ tr ,µ := min j : j (q) = 0,λr := µ, Λ := Λ ∪ λr ,qr := λr (q)−1q, q := q ∪ qr , vect(r) := λr (q)−1v,%% vect(i) = (1(qi ), . . . , s(qi )) for each i, 1 ≤ i ≤ r ,

G, r, N, Λ, q

• the reduced Grobner basis G of I w.r.t. <,• the integer r ∈ N, such that deg(I) = r ≤ s,• an order ideal N := t1, . . . , tr ⊂ T , 1 = t1 < · · · < ti < ti+1 < · · · < tr ,• an ordered subset Λ := λ1, . . . , λr ⊂ L and• an ordered set q := q1, . . . , qr ⊂ P ,

which satisfy the conditions of Theorem 28.2.1.

410 Moller I

The algorithm performs by iteration on σ and, in each step, it produces

• a set Gσ ⊂ I,• a term tσ ∈ T ,• a functional λσ ∈ L,• a polynomial qσ ∈ P ,

which satisfy

• Nσ := t1, . . . , tσ ⊂ N(I) is an order ideal,• 1 = t1 < · · · < ti < ti+1 < · · · < tσ ,• Nσ ∩ T(Gσ ) = ∅,• Λσ := λ1, . . . , λσ ⊂ L is a linearly independent set,• qσ ∈ Spankt1, . . . , tσ , and T(qσ ) = tσ ,

• Spankt1, . . . , tσ = Spankq1, . . . , qσ ,• q1, . . . , qi and λ1, . . . , λi are triangular, for each i ≤ σ ,• v(q1, L), . . . , v(qσ , L) ⊂ ks is a linearly independent set,• for each g ∈ Gσ , g − lc(g)T<(g) ∈ Spank (N),• (g) = 0 for each g ∈ Gσ and each ∈ L

until Nσ T<(Gσ ) = T so that all the conditions of Lemma 28.2.3 are satisfiedand Gσ is the reduced Grobner basis of P(Spank(Λσ )) = P(Spank(L)) = I.

At termination, we can therefore set

G := Gσ , r := σ, N := Nσ , Λ := Λσ , q := qσ .

We begin the iteration by setting

σ := 1, G1 := ∅, t1 := 1,λ1 := µ where3 µ := min j : j (t1) = 0,q1 := λ1(1)−1t1,

which trivially satisfy the required conditions.Iteratively, if Nσ T(Gσ ) = T we set

t := min<

τ ∈ T , τ /∈ Nσ T(Gσ )and we check, by means of Gaussian reduction, whether v(t, L) is linearlyindependent w.r.t. v(q1, L), . . . , v(qσ , L), computing

g1 := t, v1 := v(t, L),

g j+1 := g j − λ j (g j )q j , v j+1 := v j − λ j (g j )v(q j , L), 1 ≤ j < σ,

g := gσ − λσ (gσ )qσ , v := vσ − λσ (gσ )v(qσ , L)

3 The existence of such a µ is a consequence of the assumption that I = P(Spank (L)) is an ideal:in fact, otherwise we obtain the contradiction 1 ∈ I, I = P and 0 = L(I) ⊇ L.


so that, for each i, 1 ≤ i ≤ σ , vi = v(gi , L), g = gi − ∑σj=i λ j (g j )q j , and

λi (g) = λi (gi ) −σ∑

j=i

λ j (g j )λi (q j ) = λi (gi ) − λi (gi )λi (qi ) = 0.

Then

• if v = v(g, L) = 0, that is g ∈ I, we set Gσ := Gσ ∪ lc(g)−1g;• otherwise, we set

• Gσ+1 := Gσ ,• tσ+1 := t ,• λσ+1 := µ where 4 µ := min j : j (g) = 0,• qσ+1 := λσ+1(g)−1g,• σ := σ + 1,

which trivially satisfy the required conditions.

Termination of the algorithm is granted, because each loop is itemized bychecking whether

Nσ T(Gσ ) = T

and in each such loop either

Gσ and T(Gσ ) are enlarged, and this can be performed, by Noetherianity,only a finite number of times, or

a new element is inserted in Nσ and this too can be performed only a finitenumber of times, because #(Nσ ) ≤ N(I) ≤ s.

Example 28.2.8. Let us apply this version of the Moller Algorithm in the sameproblem as in Example 28.2.6. We have

1: t := 1, g := 1, σ := 1, G1 := ∅,τ1 := 1, λ1 := 1, q1 := 1;

X1: t := X1, g := X1, σ := 2, G2 := ∅,τ2 := X1, λ2 := 2, q2 := X1

X21: t := X2

1, g := X21, σ := 3, G3 := ∅,

τ3 := X21, λ3 := 4, q3 := X2

1X3

1: t := X31, g := X3

1, σ := 4, G4 := ∅,τ4 := X3

1, λ4 := 6, q4 := X31

X41: t := X4

1, g := X41, σ := 5, G5 := ∅,

τ5 := X41, λ5 := 7, q5 := X4

1X5

1: t := X51, g := X5

1, G5 := X51

4 Such µ exists because v(g, L) = 0; moreover µ /∈ Spank (Λσ ) since λ j (g) = 0 for each j ≤ σ.

412 Moller I

X2: t := X2, g := X2, σ := 6, G6 := X51,

τ6 := X2, λ6 := 3, q6 := X2,X1 X2: t := X1 X2, g := X1 X2 + X3

1, σ := 7, G7 := X51,

τ7 := X1 X2, λ7 := 5, q7 := X1 X2 + X31,

X21 X2: t := X2

1 X2, g := X21 X2, G7 := X5

1, X21 X2

X22: t := X2

2, g := t − λ7(t)q7 = X22 − X1 X2 − X3

1,G7 := X5

1, X21 X2, X2

2 − X1 X2 − X31.

Remark 28.2.9 (Lazard). The efficiency of this algorithm strongly depends onthe efficiency of the procedure to select the next term

t := min<

τ ∈ T , τ /∈ Nσ T(Gσ )to be treated.

Let us note that if, at each step, we denote

Bσ := Xhtl : 1 ≤ h ≤ n, 1 ≤ l ≤ σ \ Nσ

then

min<

τ ∈ T , τ /∈ Nσ T(Gσ ) = min<

τ ∈ Bσ , τ /∈ T(Gσ ).Therefore the algorithm can be adapted by creating a list of terms, ordered

by <, whose minimal element is the next term to be treated by the algorithmin each loop; such a list initially consists of the set

Xh, 1 ≤ h ≤ n = Xht1, 1 ≤ h ≤ nand is enlarged, any time a new term tσ+1 ∈ N(G) is produced, by appendingto it the set Xhtσ+1, 1 ≤ h ≤ n.

An efficient way to encode any such term Xhtl is to store the triple

(Xhtl , h, l).

Remark 28.2.10. In our description of the algorithm, we assume that in eachloop a new term Xhtl ∈ T(Gσ ) is treated, but the reader can easily realize thatthe same algorithm would work perfectly well even if the computation wereapplied to the polynomial Xhql .5

In this case, the reduction loop would compute the polynomial

g := Xhql −σ∑

j=1

λ j (Xhql)q j ;

therefore if

5 Note that T(Xhql ) = Xhtl .


• g ∈ I we have

Xhql ≡σ∑

j=1

λ j (Xhql)q j mod I,

while, if• g ∈ I, the algorithm inserts it in q setting qσ+1 := g, so that

Xhql ≡σ∑

j=1

λ j (Xhql)q j + qσ+1 mod I.

Thus, if, also after the termination granted by the test Nσ T(Gσ ) = T , weperformed the reduction loop on each remaining element (Xhql , h, l) of thelist, we would obtain the values ah

l j , 1 ≤ l ≤ r, 1 ≤ j ≤ r, 1 ≤ h ≤ n, suchthat

Xhql ≡r∑

j=1

ahl j q j mod I, for each h, 1 ≤ h ≤ n, l, 1 ≤ l ≤ r,

that is the ring structure of P/I.As we will see later (see Corollary 31.6.12), this approach will give a further

bonus: in many instances, it is often less time consuming to deduce v(Xhql , L)

from the knowledge of v(ql , L) than directly computing v(Xhtl , L); moreoverv(Xhql , L) is often sparser than v(Xhtl , L), especially if L is properly ordered,thus simplifying the Gaussian computation.

29

Lazard

Knowing the Grobner basis of a zero-dimensional ideal

I ⊂ k[X1, . . . , Xn] := P

w.r.t. the lexicographical ordering is a powerful tool for solving, mainly asa consequence of Corollary 26.2.4 (but see also Section 34.6); unfortunately,experimental considerations indicate that computation of the Grobner basis ofan ideal I w.r.t. the lexicographical ordering is often not feasible and, in general,at least time–space consuming in comparison with other term orderings, thebest candidate of which is the degrevlex ordering.

This prompted the FGLM problem: to produce an efficient algorithm whichallows us to deduce the Grobner basis of I w.r.t. the lexicographical ordering,from knowing that w.r.t. the degrevlex ordering (Section 29.1).

The original solution, the FGLM algorithm (Section 29.2) of this problem isessentially an independent, and stronger, discovery of a version of the Molleralgorithm.

This led to a deeper analysis of the linear-algebra description of the vec-torspace k[N<(I)] = Spank(N<(I)) ∼= P/I suggesting formalization of the no-tions of border bases, Grobner representation, linear representation, Grobnerdescription (Section 29.3), and allowed the complexity of the Moller algo-rithm – O(n2s3) where n is the number of variables and s the degree of I – tobe evaluated and the production of a new and improved version of the Molleralgorithm, which, within such expected complexity, produces all the data pre-senting the linear-algebra description of k[N<(I)] (Section 29.4).

In Section 29.5, I present the two other efficient alternatives to FGLM al-gorithm for solving the FGLM problem, the Hilbert Driven Algorithm and theGrobner Walk.

Finally, in Section 29.6, I show how by dualling the Moller algorithm oneobtains an algorithm by Macaulay, which, given a finite set L ⊂ P∗ of linearly

414

29.1 The FGLM Problem 415

independent k-linear functionals generating a P-module, allows us to computethe P-module structure of Spank(L).

29.1 The FGLM Problem

A natural paradigm for solving polynomial systems is provided by elimina-tion: it, given an ideal I ⊂ k[X1, . . . , Xn], produces an ideal J ⊂ k[X1, . . . ,

Xn−1] and, for each b := (b1, . . . , bn−1) ∈ Z(J), a polynomial hb ∈k[X1, . . . , Xn−1][Xn], which satisfies

(b1, . . . , bn−1, b) ∈ Z(I) ⇐⇒ h(b) = 0,

where h(Xn) := hb(b1, . . . , bn−1, Xn) ∈ k[Xn].This approach was first proposed by Kronecker (Section 20.4), according

to whose proposal J was obtained by computation of the resultant betweena fixed basis element of I and the ‘generic’ linear combination of the otherbasis elements, and was then re-formulated by Grobner (Section 20.3) whosuggested that J := I ∩ k[X1, . . . , Xn−1] should be used.

The introduction of Grobner bases made this approach computational; asnoted by Spear (Section 26.2), given a Grobner basis G of I w.r.t. the lexico-graphical ordering < induced by X1 < · · · < Xn , each G ∩ k[X1, . . . , Xi ] is aGrobner basis of the elimination ideal J := I ∩ k[X1, . . . , Xi ].

Moreover, in the zero-dimensional case, setting J := I ∩ k[X1, . . . , Xn−1],for each (b1, . . . , bn−1) ∈ Z(J) we have that the principal ideal

g(b1, . . . , bn−1, Xn) : g(X1, . . . , Xn) ∈ I ⊂ k[Xn]

is generated by

h(Xn) := gcd(g(b1, . . . , bn−1, Xn) : g(X1, . . . , Xn) ∈ G)

and (Trink’s Algorithm)

(b1, . . . , bn−1, b) ∈ Z(I) ⇐⇒ h(b) = 0.

However, the role of the lexicographical ordering as a tool for solving isflawed by the same original sin that Macaulay saw in Kronecker’s solver:

Konig’s treatise might be regarded as in some measure complete if it were admittedthat a problem is finished with when its solution has been reduced to a finite number offeasible operations. If however the operations are too numerous or too involved to becarried out in practice the solution is only a theoretical one.F. S. Macaulay, The Algebraic Theory of Modular Systems.

416 Lazard

Macaulay’s bound (Section 23.9) proved that performing elimination ofa ‘generic’ trivial ideal would lead to a doubly exponential complexity.Macaulay’s result must however be put in the proper perspective: his criticismwas directed towards the iterative application of the resultant as a solving tool,it being responsible for the double exponentiality; in the same example, hehimself pointed out that I ∩ k[X1] = (Xdn

1 ), corresponding to an unavoidablesingle exponential bound.

And, in the zero-dimensional case, the double exponentiality pointed out byMacaulay is just an effect of the solving method, while the degree bound forany Grobner basis and H-bases is single exponential, an unavoidable bound;therefore double exponentiality does not haunt zero-dimensional ideals.

In any case, even in the zero-dimensional single-exponential-bounded case,we know, both theoretically (Section 38.4) and experimentally, that theGrobner basis computation is much more efficient for a degrevlex orderingthan for a lexicographical one, which very often is computationally untreat-able while a degrevlex one is time consuming but still manageable. On theother hand, as we have already said, the lexicographical ordering is in generalmore powerful as a solving tool than the degrevlex one.

These remarks prompted the following:

Problem 29.1.1 (FGLM problem). Given a term ordering < on the polyno-mial ring P := k[X1, . . . , Xn], a zero-dimensional ideal I ⊂ P and its re-duced Grobner basis G≺ w.r.t. the term ordering ≺, deduce the Grobner basisG< of I w.r.t. <.

An efficient, both theoretically and practically, solution of this problemwould make the scenario feasable in which, given a zero-dimensional idealI ⊂ P , one first applies Buchberger’s algorithm in order to obtain the easy-to-compute reduced Grobner basis G≺ w.r.t. the degrevlex term ordering ≺ andthen uses the solution of the FGLM problem to efficiently deduce the hard-to-compute reduced Grobner basis G< of I w.r.t. a lexicographical ordering <, inorder to apply it as a solving tool.

In this context ‘efficient, both theoretically and practically’ refers to ‘poly-nomial complexity’.

The original solution of this problem consisted of an independent rediscov-ery of Moller’s Algorithm 28.2.7; the different problem which prompted thisrediscovery gave a new perspective on the algorithm, suggested some impor-tant improvements and allowed a complexity analysis to be deduced. In ourdiscussion we will begin by showing how Moller’s Algorithm 28.2.7 solvedthe problem and then we will improve it to make it a sharp tool for solving theFGLM problem.

29.1 The FGLM Problem 417

In order to apply Moller’s Algorithm 28.2.7 to the solution of the FGLMproblem all we need do is describe a set L of functionals such that I =P(Spank(L)) because then (G, r, N, Λ, q) := G-basis(L, <) would give therequired Grobner basis G< := G.

The solution is obvious: since we know the Grobner basis of I w.r.t. ≺, wealso deduce the order ideal

N≺(I) := τ1, . . . , τs = T \ T≺(I)

and for each g ∈ P we can explicitly compute the unique vector

Rep(g, N≺(I)) := (γ (g, τ1, ≺), . . . , γ (g, τs, ≺))

such that

Can(g, I, ≺) =s∑

j=1

γ (g, τ j , ≺)τ j .

The maps i : P → k, 1 ≤ i ≤ s, defined, for each g ∈ P , by

i (g) := γ (g, τi , ≺), for each i,

that is, equivalently,

Can(g, I, ≺) =s∑

j=1

j (g)τ j

are k-linear functionals and L := 1, . . . , s satisfies

g ∈ P(Spank(L)) ⇐⇒ i (g) = 0 for each i

⇐⇒ Can(g, I, ≺) = 0

⇐⇒ g ∈ I,

so that I = P(Spank(L)).1

Applying Moller’s Algorithm 28.2.7 in order to solve the FGLM problemwith the crucial requirement of preserving polynomial complexity requires apriori the ability to solve two problems:

(1) how to compute

v := (1(t), . . . , s(t)) = (γ (t, τ1, ≺), . . . , γ (t, τs, ≺)) = Rep(t, N≺(I))

for all terms t dealt with in the While-loop; and

1 We record here a formula which allows us to compute the P-module structure of Spank (L): forf = ∑

ω∈T c( f, ω)ω, τi ∈ N≺(I), we have

f · i =s∑

j=1

∑ω∈T

c( f, ω)i (ωτ j ) j .

418 Lazard

(2) how to perform the crucial procedure of testing whether

N T(G) = Tand, in the positive case, computing the term

t := min<

τ ∈ T , τ /∈ N T(G),which is dealt with in the While-loop.

The obvious solutions are not feasible; in fact:

(1) of course we cannot contemplate performing Buchberger’s reductionof t via the Grobner basis G≺ because we cannot rule out the possi-bility that the leading terms of the intermediate reductions run over thewhole set of all monomials ω such that ω < t ; this would require us toperform 2

(δ+n

n

) ≈ δn, δ = deg(t), reduction steps for each element t ,thus obtaining the undesired exponential complexity,

(2) Moller’s original proposal for managing the While-loop was just totreat each term one after the other in increasing ordering w.r.t. <; thetermination condition to be tested became the achievement of a degreed in which Td ⊂ T(G), thus trivially granting N T(G) = T . In ourcontext this is again unacceptable, since it would mean performing theWhile-loop

(G(I)+nn

) ≈ G(I)n times, where G(I) := maxdeg(g) : g ∈G<, re-introducing the exponentiality that we are trying to avoid.

This means that we must find a polynomial solution for both problems; as wewill see in the next section it is sufficient to characterize precisely the set ofmonomials on which the While-loop must be performed.

29.2 The FGLM Algorithm

Let us begin by recalling (see Remark 28.2.9), using freely the same notationas in Algorithm 28.2.7, that, in the σ th While-loop of the algorithm, if wewrite

Bσ := Xhtl : 1 ≤ h ≤ n, 1 ≤ l ≤ σ \ Nσ

= XhT<(ql) : 1 ≤ h ≤ n, 1 ≤ l ≤ σ \ Nσ ,

the set Bσ consists of all potential candidates for the next term to be treated,and we have

min<

τ ∈ T, τ /∈ Nσ T<(Gσ ) = min<

τ ∈ Bσ , τ /∈ T<(Gσ ).

2 This assumes ≺ is degree-compatible; but we can obtain a similar bound for any ordering ≺ bymeans of Bayer’s results, see Proposition 24.9.7 and Corollary 24.10.4.

29.2 The FGLM Algorithm 419

The remark immediately gives some advantages:

• we can now precisely describe the set of all the terms which can be poten-tially treated within a While-loop of the algorithm, which is

Xht : 1 ≤ h ≤ n, t ∈ N = XhT<(ql) : 1 ≤ h ≤ n, 1 ≤ l ≤ s;• as a consequence we can give a good upper bound on the number of the

While-loops performed by the algorithm, that is ns.• the algorithm can now be adapted so as to perform the choice of the next

term to be treated in an efficient way: it is sufficient to create a list B ofterms, ordered by <, whose minimal element is the next term to be treatedby the algorithm in each loop; such a list initially consists of the set

B1 := Xh, 1 ≤ h ≤ n = XhT<(q1), 1 ≤ h ≤ nand, whenever a new term tσ+1 ∈ N<(G) is produced and inserted in N, itis enlarged by setting

Bσ+1 := Bσ ∪ XhT<(qσ+1), 1 ≤ h ≤ n.An efficient way to encode each element XhT<(ql) of B is to store the triple

(XhT<(ql), h, l); we moreover assume that B is ordered by any ordering satisfying

(ω1, h1, l1) (ω2, h2, l2) ⇒ ω1 ≤ ω2,

so that

τ := min<

τ ∈ T , τ /∈ Nσ T<(Gσ ), (ω, h, l) := min (Bσ ) ⇒ τ = ω.

Having thus solved the problem of how to manage the choice of the ele-ments to be dealt with in the While-loop, let us now discuss how to producean efficient way to evaluate each vector Rep(Xhtl , N≺(I)); for technical rea-sons, we intend to discuss the more general case in which we want to computeeach vector Rep(Xhql , N≺(I)): when the While-loop is treating a term Xhql ,we previously managed the polynomial ql – since otherwise (XhT<(ql), h, l)would not have been included in the list B – so that we computed

Rep(ql , N≺(I)) = (γ (ql , τ1, ≺), . . . , γ (ql , τs, ≺)),

which satisfies the relation

ql −s∑

j=1

γ (ql , τ j , ≺)τ j = ql − Can(ql , I, ≺) ∈ I,

420 Lazard

so that Xhql − ∑sj=1 γ (ql , τ j , ≺)Xhτ j ∈ I, and

Can(Xhql , I, ≺) =s∑

j=1

γ (ql , τ j , ≺) Can(Xhτ j , I, ≺)

=s∑

i=1

(s∑

j=1

γ (ql , τ j , ≺)γ (Xhτ j , τi , ≺)

)τi .

As a consequence, if we begin with a preprocessing in which we compute

Rep(ω, N≺(I)), for each ω ∈ Xhτl : 1 ≤ h ≤ n, 1 ≤ l ≤ s,the computation of each

γ (Xhql , τi , ≺) =s∑

j=1

γ (ql , τ j , ≺)γ (Xhτ j , τi , ≺)

is reduced to performing linear combinations.

Algorithm 29.2.1. (see Figure 29.1) The computation of the whole set

Rep(ω, N≺(I)), for each ω ∈ Xhτl : 1 ≤ h ≤ n, 1 ≤ l ≤ s,can be essentially performed by adapting the same scheme as the improvedversion of Moller’s Algorithm 28.2.7 we are discussing here.

We begin by considering ω := 1 and setting

N := 1 ⊂ N≺(I) and B := Xh : 1 ≤ h ≤ n.Then, iteratively, until B = ∅, we pick

ω := Xhτl := min≺ (B)

and:

• if ω ∈ T≺(I) then ω ∈ N≺(I), so that we add ω to N and ωXh : 1 ≤ h ≤ nto B;

• if there are g ∈ G≺ such that T≺(g) = ω and g = ω −∑τ∈N≺(I)γ (ω, τ,≺) τ,

since the procedure iterates on ≺-increasing values of ω, we have

γ (ω, τ,≺) = 0 ⇒ τ ≺ ω ⇒ τ ∈ N;• if there are H, 1 ≤ H ≤ n, τ ∈ T≺(I) such that ω = X H τ , since we have

also X H τ = ω = Xhτl with τl ∈ N, we can deduce that

X H | τl , there exists ι ≤ s : τι := τl

X H∈ N≺(I), and τ = Xhτι;


Fig. 29.1. Linear Representaion Algorithm

(N≺,M) := FGLM-Matrix(G≺)where

P := k[X1, . . . , Xn],T := Xa1

1 · · · Xann : (a1, . . . , an) ∈ N

n,≺ is a term ordering on P ,I ⊂ P is a zero-dimensional ideal,G≺ ⊂ I is the reduced Grobner basis of I w.r.t. ≺;s = deg(I),N≺ := τ1, . . . , τs = N≺(I),1 = τ1 ≺ τ2 ≺ · · · ≺ τ j ≺ τ j+1 ≺ · · · ≺ τs ,

M = M(N≺) =(

a(h)l j

)∈ ks2

, 1 ≤ h ≤ n

is the set of the square matrices

defined by the equalities Xhτl = ∑j a(h)

l j τ j in P/I = Spank(N≺);r := 1, τ1 := 1, N≺ := τ1, B := (Xh , h, 1) : 1 ≤ h ≤ n,While B = ∅ do

(ω, h, l) := min(B),B := B \ (ω, h, l),If ω /∈ T≺(I) then

r := r + 1,τr := ω, N≺ := N≺ ∪ τr ,B := B ∪ (Xhτr , h, r) : 1 ≤ h ≤ n,a(h)

lr := 1;elseif exists g := T≺(g)−∑r

j=1 γ (ω, τ j , ≺)τ j ∈ G≺ : T≺(g) = ω = Xhτl then

For j = 1..r do a(h)l j := γ (ω, τ j , ≺)

elseLet H, ι : 1 ≤ H ≤ n, 1 ≤ ι ≤ r : Xhτι ∈ T≺(G≺), τl = X H τι;For i = 1..r do a(h)

li := ∑rj=1 a(h)

ιj a(H)j i

For each (τ, H, i) ∈ B : τ = ω doB := B \ (τ, H, i),For j = 1..r do a(H)

i j := a(h)l j ;

N≺,M

therefore τ < ω has already been treated so that we have obtained a repre-sentation Can(τ, I, ≺) = ∑s

j=1 γ (τ, ≺, τ j )τ j ; since in such a representationwe have

γ (τ, ≺, τ j ) = 0 ⇒ τ j ≺ τ ⇒ τ j ∈ N, X H τ j < X H τ = ω,

we also have the representation

Can(X H τ, I, ≺) =s∑

j=1

γ (τ, ≺, τ j ) Can(X H τ j , I, ≺)

422 Lazard

and we can use the same formula as above to derive

γ (Xhτl , τi , ≺) = γ (X H τ, τi , ≺) =s∑

j=1

γ (τ, τ j , ≺)γ (X H τ j , τi , ≺)

=s∑

j=1

γ (Xhτι, τ j , ≺)γ (X H τ j , τi , ≺).

Example 29.2.2. If we apply the algorithm to the ideal I of Example 28.2.6whose Grobner basis w.r.t. the degrevlex ordering induced by X1 < X2 is

X32 − X1 X2

2, X21 X2, X3

1 − X22 + X1 X2

we obtain N≺ = 1, X1, X2, X21, X1 X2, X2

2, X1 X22 and the multiplication ta-

bles are⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 −1 1 00 0 0 0 0 0 00 0 0 0 0 0 10 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

and

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 00 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

where the non-trivial results come from

X1 · X21 = −X1 X2 + X2

2X1 · X1 X2 = X2 · X2

1 = 0X2 · X2

2 = X1 X22

X1 · X1 X22 = X2 · X2

1 X2 = X2 · 0 = 0X2 · X1 X2

2 = X1 · X32 = X1 · X1 X2

2 = 0

Algorithm 29.2.3. This discussion leads directly to the adapted version ofMoller’s Algorithm 28.2.7 as a solution of the FGLM problem presented inFigure 29.2.

About this presentation, we must note a few points:

• the algorithm works exactly in the same way if each While-loop treats theterms Xhtl instead of the polynomial Xhql ;

• the central test is to verify whether Xhql (respectively Xhtl ) is linearly de-pendent on q (resp. N); in our presentation, this test is performed by meansof Gaussian reduction as in Figure 28.2; any linear-algebra approach withoptimal complexity could be freely used here;


Fig. 29.2. FGLM Algorithm

(G, N, q) := FGLM(G≺, <)where

P := k[X1, . . . , Xn],T := Xa1

1 . . . Xann : (a1, . . . , an) ∈ N

n,< and ≺ are term orderings on P ,I ⊂ P is a zero-dimensional ideal,G≺ ⊂ I is the reduced Grobner basis of I w.r.t. ≺;s = deg(I),N≺ := τ1, . . . , τs = N≺(I),1 = τ1 ≺ τ2 ≺ · · · ≺ τ j ≺ τ j+1 ≺ · · · ≺ τs ,

M = M(N≺) =(

a(h)l j

)∈ ks2

, 1 ≤ h ≤ n


defined by the equalities Xhτl = ∑j a(h)

l j τ j in P/I = Spank(N≺);B ⊂ (τ, h, l) : τ ∈ T , 1 ≤ h ≤ n, 1 ≤ l ≤ s is a set ordered by so that

(ω1, h1, l1) (ω2, h2, l2) ⇒ ω1 ≤ ω2;G ⊂ I is the reduced Grobner basis of I w.r.t. <,N := t1, . . . , ts = N<(I),1 = t1 < t2 < · · · < t j < t j+1 < · · · < ts ,µ : 1, . . . , s → 1, . . . , s is a permutation,q := q1, . . . , qs⊂P is a set triangular to γ (· , τµ(1), ≺), . . . , γ (· , τµ(s), ≺),qi ∈ Spankt1, . . . , ti , T<(qi ) = ti , for each i ≤ s,q1, . . . , qi and γ (· , τµ(1), ≺), . . . , γ (· , τµ(i), ≺) are triangular for all i ≤ s.

(N≺,M) := FGLM-Matrix(G≺)G := ∅, r := 1, t1 := 1, N := t1, q1 := 1, q := q1,vect(1) := (1, 0, . . . , 0), µ(1) := 1,%% vect(1) = Rep(q1, N≺), µ(1) = min j : γ (q1, τ j , ≺) = 0.Let B := (Xh , h, 1), 1 ≤ h ≤ n.While B = ∅ do

(t, h, l) := min(B),%% t = Xhtl = XhT<(ql ).B := B \ (t, h, l),If t /∈ T<(G) then

q := Xhql

For i = 1..s do vi := ∑sj=1 γ (ql , τ j , ≺)a(h)

j i ;v := (v1, . . . , vs),%% v = Rep(q, N≺),For j = 1..r do

v := v − γ (q, τµ( j),≺) vect( j), q := q − γ (q, τµ( j), ≺)q j ,%% v = Rep(q, N≺)

If v = 0 thenG := G ∪ q,

elser := r + 1,tr := t, N := N ∪ tr ,µ(r) := min j : γ (q, τ j , ≺) = 0,qr := γ (q, τµ(r), ≺)−1q, vect(r) := γ (q, τµ(r), ≺)−1v,%%vect(i) = Rep(qi , N≺), for each i, 1 ≤ i ≤ r ,q := q ∪ qr ,B := B ∪ (Xhtr , h, r), 1 ≤ h ≤ n,

G, N, q

424 Lazard

• there is another improvement which allows us to reduce the complexity fromO(n3s3) to O(n2s3): each element t = Xhtl is inserted in B when tl is in-serted in N, not being a leading term of an element of G. Since the terms aretreated by <-increasing ordering, when t is treated each factor tl = t/Xh –and there are as many such factors as there are variables dividing t – hasalready been treated; therefore if t has been inserted fewer times than thenumber of variables dividing it, this means that at least one of the factorstl = t/Xh has not been inserted in N, being the multiple of the leading termof an element of G and, therefore, that t ∈ T<(G).

It is therefore sufficient to count both the number of variables factoringa term t when it is inserted in B and the number of times in which it isinserted there and to compare the two numbers in order to decide whethert /∈ T<(G).

Example 29.2.4. We can now apply this algorithm to compute the Grobnerbasis w.r.t. the lex ordering < induced by X1 < X2 of the ideal I of Exam-ple 28.2.6 using the structural information obtained from its Grobner basisw.r.t. the degrevlex ordering ≺ induced by X1 ≺ X2 which has been obtainedin Example 29.2.2:

1: t1 := 1, q1 := 1, vect(1) := (1, 0, 0, 0, 0, 0, 0), µ := 1,

N< = 1, B = X1, X2;X1: q := X1 · 1, v := (0, 1, 0, 0, 0, 0, 0),

t2 := X1, q2 := X1, vect(2) := (0, 1, 0, 0, 0, 0, 0), µ := 2,

N< = 1, X1, B = X21, X2, X1 X2;

X21: q := X1 · X1, v := (0, 0, 0, 1, 0, 0, 0),

t3 := X21, q3 := X2

1, vect(3) := (0, 0, 0, 1, 0, 0, 0), µ := 4,

N< = 1, X1, X21, B = X3

1, X2, X1 X2, X21 X2;

X31: q := X1 · X2

1, v := (0, 0, 0, 0, −1, 1, 0),

t4 := X31, q4 := −X3

1, vect(4) := (0, 0, 0, 0, 1, −1, 0), µ := 5,

N< = 1, X1, X21, X3

1, B = X41, X2, X1 X2, X2

1 X2, X31 X2;

X41: q := X1 · X3

1, v := (0, 0, 0, 0, 0, 0, 1),

t5 := X41, q5 := X4

1, vect(5) := (0, 0, 0, 0, 0, 0, 1), µ := 7,

N< = 1, X1, X21, X3

1, X41, B = X5

1, X2, X1 X2, X21 X2, X3

1 X2, X41 X2;

X51: q := X1 · X4

1, v := (0, 0, 0, 0, 0, 0, 0),

q := X51, v := (0, 0, 0, 0, 0, 0, 0), G := X5

1,N< = 1, X1, X2

1, X31, X4

1, B = X2, X1 X2, X21 X2, X3

1 X2, X41 X2;

X2: q := X2 · 1, v := (0, 0, 1, 0, 0, 0, 0),

t6 := X2, q6 := X2, vect(6) := (0, 0, 1, 0, 0, 0, 0), µ := 3,

N< = 1, X1, X21, X3

1, X41, X2, B = X1 X2, X2

1 X2, X31 X2, X4

1 X2, X22;

X1 X2: q := X1 · X2, v := (0, 0, 0, 0, 1, 0, 0),


q := q − q4 = X1 X2 + X31, v := (0, 0, 0, 0, 0, 1, 0),

t7 := X1 X2, q7 := X1 X2 + X31, vect(7) := (0, 0, 0, 0, 0, 1, 0), µ := 6,

N< = 1, X1, X21, X3

1, X41, X2, X1 X2,

B = X21 X2, X3

1 X2, X41 X2, X2

2, X1 X22;

X21 X2: q := X1 · X1 X2, v := (0, 0, 0, 0, 0, 0, 0),

q := X21 X2, v := (0, 0, 0, 0, 0, 0, 0), G := X5

1, X21 X2,

N< = 1, X1, X21, X3

1, X41, X2, X1 X2, B = X3

1 X2, X41 X2, X2

2, X1 X22;

X31 X2: X3

1 X2 ∈ T<(G): in fact X1 · X21 X2 has not been inserted in B;

X41 X2: X4

1 X2 ∈ T<(G) since X1 · X31 X2 has not been inserted in B;

X22: q := X2 · X2, v := (0, 0, 0, 0, 0, 1, 0),

q := q − q7 = X22 − X1 X2 − X3

1, v := (0, 0, 0, 0, 0, 0, 0),

G := X51, X2

1 X2, X22 − X1 X2 − X3

1,N< = 1, X1, X2

1, X31, X4

1, X2, X1 X2, B = X1 X22;

X1 X22: X1 X2

2 ∈ T<(G) since X1 · X22 has not been inserted in B.

Remark 29.2.5. We can now compute the complexity 3 of the FGLM algo-rithm:

• as we already remarked the algorithm performs at most ns While-loops;• therefore over the whole algorithm we must

• check if each term t has been inserted as many times as variables divideit, in order to decide whether it is a member t ∈ T<(G),4 with a total costof O(ns) operations;

• evaluate all vectors v by performing O(s2) operations, for a total costO(ns3);

• perform Gaussian reduction on each v via vect(1), . . . , vect(r) and qvia q, for a total cost, again, of O(ns3) operations;

• moreover, in the s times in which N is enlarged, one must also merge andre-order the ordered lists B – which have at most n(s − 1) elements – and(Xhtr , h, r), 1 ≤ h ≤ n which requires at most n(s − 1) + n term-comparisons, thus costing O(n2s2) operations;

• note also that, for each i , qi is the combination of exactly i terms and thatvect(i) has at most s − i non-zero entries, so each step of Gaussian reductiondeals at most with s entries;

3 In our evaluation we will assume that the size of the elements of the field k is 1 and that eacharithmetical operation will cost 1.

4 Without this trick, we should pay O(ns × ns × n) ≈ O(n3s2) in order to compare each of thesn terms with the at most sn leading terms of elements in G.

426 Lazard

• finally, for FGLM – Matrix(G≺) we obtain the same complexity by thesame arguments.

In conclusion FGLM – Matrix(G≺) and FGLM(G≺, <) cost O(n2s3),where

• the evaluation of the canonical forms of the treated polynomials costsO(ns3),

• the linear-algebra operations cost O(ns3), and• the management of the list B costs O(n2s2).

The size of the information to be stored has a similar complexity, since oneneeds to store

• the s terms in N≺(I) each having size n,• the at most ns − s terms G≺(I) := T≺G≺ and, for each of them, their

canonical form with size n + s.

Remark 29.2.6 (Sweedler–Taylor). The FGLM algorithm (Figure 29.2) can bedirectly adapted to a more general setting: let P := k[X1, . . . , Xn] and let <

be a term ordering on it; let M be a P-module such that dimk(M) is finiteand on which it is possible to determine k-linear dependencies. Then, given aP-module morphism π : P → M , the FGLM algorithm can be adapted tocompute the Grobner basis of ker(π) w.r.t. <.

29.3 Border Bases and Grobner Representation

In Section 22.1 we imposed on T the decomposition T = T≺(I) N≺(I)depending on the assignment of an ideal I ⊂ P and a term ordering ≺ on T ;the FGLM algorithm implicitly introduced a finer decomposition which playsa central role within the linear-algebra description of polynomial ideals.

We can in fact remark that there is a decomposition

T = I≺(I) B≺(I) N≺(I)

where

B≺(I) := Xhτ : 1 ≤ h ≤ n, τ ∈ N≺(I) \ N≺(I),

I≺(I) := T≺(I) \ B≺(I).

We also denote by G≺(I) ⊂ B≺(I) the unique minimal basis of T≺(I) andwe introduce the set

C≺(I) := τ ∈ N≺(I) : Xhτ ∈ T≺(I), for each h.

29.3 Border Bases and Grobner Representation 427

Definition 29.3.1. The set B≺(I) is called the border set of I w.r.t. ≺.The set C≺(I) is called the corner set of I w.r.t. ≺.For any term τ ∈ T and each h : 1 ≤ h ≤ n for which Xh | τ , the term

τ/Xh is called the hth-predecessor of τ.

For any term τ ∈ T and each h : 1 ≤ h ≤ n, the term Xhτ is called thehth-successor of τ.

Lemma 29.3.2. With this notation we have

• T≺(I) = τ ∈ T for which there exists g ∈ I : T≺(g) = τ ;• I≺(I) = τ ∈ T≺(I) such that all its predecessors ω ∈ T≺(I);• B≺(I) = τ ∈ T≺(I) such that at least one of its predecessors ω ∈ N≺(I);• G≺(I) = τ ∈ T≺(I) such that all its predecessors ω ∈ N≺(I);• C≺(I) = τ ∈ N≺(I) such that all its successors ω ∈ B≺(I);• N≺(I) = τ ∈ T for which no g ∈ I satisfies T≺(g) = τ ;• τ ∈ I≺(I) iff all its predecessors are in T≺(I);• τ ∈ B≺(I) \ G≺(I) iff there exist h, H : τ/Xh ∈ N≺(I), τ/X H ∈ B≺(I) ⊂

T≺(I);• if τ ∈ B≺(I) \ G≺(I) then all its predecessors are in N≺(I) ∪ B≺(I);• τ ∈ N≺(I) ∪ G≺(I) iff all its predecessors are in N≺(I);• τ ∈ T≺(I) ∪ C≺(I) iff all its successors are in T≺(I);• τ ∈ N≺(I) \ C≺(I iff there exists h : Xhτ ∈ N≺(I);• C≺(I) ∪ T≺(I) is a monomial ideal;• N≺(I) ∪ G≺(I) and N≺(I) ∪ B≺(I) are order ideals.

The result computed by the algorithm of Figure 29.1 can be encoded in twodifferent, but equivalent, ways:

• by giving the set

M(N≺) :=(

a(h)l j

)∈ ks2

, 1 ≤ h ≤ n

of the square matrices describing the effect

Xhτl =s∑

j=1

a(h)l j τ j

of the multiplication by each variable Xh on the linear basis N≺(I) =τ1, . . . , τs of the algebra P/I = Spank(N≺);

• by giving, for each τ ∈ N≺(I) ∪ B≺(I), the value of

Can(τ, I, ≺) =s∑

j=1

γ (τ, τ j , ≺)τ j =s∑

j=1

γ (τ, τ j , N<(I))τ j .

428 Lazard

Moreover both Moller’s Algorithm 28.2.7 and our presentation of the FGLMalgorithm implicitly produce also the matrices describing the effect of the vari-able multiplication over the linear basis q.

Definition 29.3.3. The border basis of I w.r.t. ≺ is the set

τ − Can(τ, I, ≺) : τ ∈ B≺(I).

A Grobner representation of I is the assignment of

• a linearly independent set q = q1, . . . , qs, q1 = 1, such that P/I =Spank(q),

• the set M = M(q) :=(

a(h)l j

)∈ ks2

, 1 ≤ h ≤ n

of the square matrices(a(h)

l j

)defined by the equalities

Xhql =∑

j

a(h)l j q j , for each l, j, h, 1 ≤ l, j ≤ s, 1 ≤ h ≤ n,

in P/I = Spank(q).

For each f ∈ P the Grobner description of f in terms of a Grobner repre-sentation (q,M) is the unique vector

Rep( f, q) := (γ ( f, q1, q), . . . , γ ( f, qs, q)) ∈ ks

such that f − ∑j γ ( f, q j , q)q j ∈ I.

The linear representation of I w.r.t. the term ordering ≺ is the Grobner rep-resentation (N≺(I),M(N≺(I))) where q = N≺(I).

Historical Remark 29.3.4. Grobner (see Example 24.0.1) introduced the sem-inal idea which led to Buchberger’s algorithm and to the bases named afterhimself, in the effort of producing, for a zero-dimensional ideal I ⊂ P , a lin-early independent set q = q1, . . . , qs, q1 = 1, consisting of monomials, suchthat P/I = Spank(q), and the set of values q(k)

i j defined by qhql = ∑j q(h)

l j q j .The ideas discussed in this section are essentially the best solution to his

problem.

With these definitions, if ≺ is a term ordering and N≺(I) = τ1, . . . , τs, theGrobner description

Rep( f, N≺(I)) := (γ ( f, τ1, N≺(I)), . . . , γ ( f, τs, N≺(I)))

of f in terms of the linear representation of I w.r.t. ≺ is a convoluted synonym


of the notion of the canonical form

Can( f, I, ≺) =s∑

j=1

γ ( f, τ j , ≺)τ j =s∑

j=1

γ ( f, τ j , N≺(I))τ j

of f in terms of ≺.However it is encoded, this information contains the algebraic structure of

the P-module P/I and allows us to efficiently compute the Grobner description(and the canonical form) of a polynomial f ∈ P modulo I.

If f = ∑µi=1 c(ti , f )ti and, for each i , m(i)

l ∈ T , Xv(i,l), l ≤ deg(ti ) := di ,denote the terms and variables such that

m(i)0 = 1, m(i)

l = Xv(i,l)m(i)l−1, m(i)

di= ti

we have

γ ( f, q j , q) =µ∑

i=1

c(ti , f )γ (ti , q j , q),

γ (m(i)l , q j , q) =

s∑ι=1

γ (m(i)l−1, qι, q)γ (Xv(i,l)qι, q j , q).

Clearly the complexity of computing the Grobner description is O(µds2),

d = deg( f ).

We obtain however a better complexity if we use a more efficient represen-tation of f ∈ P , which is in any case the most common representation used ingood computer algebra software.

Definition 29.3.5. A recursive Horner representation of a polynomial

f ∈ P := k[X1, . . . , Xn]

is inductively defined as the assignment of

• a constant a0 ∈ k and the recursive Horner representation of a polynomialg ∈ k[X1], deg1(g) = d − 1, if

f ∈ k[X1], deg1( f ) = d, and f = a0 + X1g;

• the recursive Horner representations of a polynomial f0 ∈ k[X1, . . . , Xν−1]and of a polynomial g ∈ k[X1, . . . , Xν], degν(g) = d − 1, if

f ∈ k[X1, . . . , Xν], degν( f ) = d, and f = f0 + Xνg.

The Horner complexity of f , Hor( f ), is the number of + operations re-quired by the recursive Horner representation of f .

430 Lazard

Example 29.3.6. For instance the polynomial X22 − X1 X2 − X3

1 has the recur-sive Horner representation(

0 + X1

(0 + X1

(0 + X1(−1)

)))+ X2

((0 + X1

(−1)) + X2

(1))

and the Horner complexity is 6.

Remark 29.3.7 (Traverso). Clearly for a polynomial f := ∑µi=1 c(ti , f )ti ,

d = deg( f ), we have Hor( f ) ≤ µd so that the recursive computation

γ ( f, q j , q) = γ ( f0, q j , q) +s∑

ι=1

γ (g, qι, q)γ (Xνqι, q j , q),

whose complexity is O(Hor( f )s2), is more efficient.

Algorithm 29.3.8 (Traverso). Once a zero-dimensional ideal I ⊂ P is given bymeans of a Grobner representation

q = q1, . . . , qs, q1 = 1,M :=(

a(h)l j

), 1 ≤ h ≤ n

,

for any finite set of elements F := g1, . . . , gr ⊂ P , given via their Grobnerdescriptions c(i) = (c(i)

1 , . . . , c(i)s ), c(i)

j = γ (gi , q j , q), for each i, j, 1 ≤ i ≤ r,

1 ≤ j ≤ s, so that gi − ∑sj=1 c(i)

j q j ∈ I, the algorithm of Figure 29.3 allowsus to compute with good complexity the Grobner representation of the idealJ := I + (F).

The basic idea is the following: if we consider an element g ∈ F , having theGrobner description

g −ι∑

j=1

c j q j ∈ I, cι = 0,

and we enlarge I by adding g to it, then we obtain the relation

qι ≡ −ι−1∑j=1

c−1ι c j q j mod I ∪ g;

the decomposition P = I Spank(q) of P into disjoint k-vectorspaces is thentransformed into

P = (I ∪ g) Spank(q \ qι),and we have to replace, in each Grobner description

∑sj=1 d j q j of the poly-

nomials gi and Xhql – which are respectively encoded in the vectors c(i) and

in the rows(

a(h)l1 , . . . , a(h)

ls

)of the matrices of M – the instances of qι with

− ∑ι−1j=1 c−1

ι c j q j thus getting∑

j (d j − c−1ι c j dι)q j .


Fig. 29.3. Extending a Grobner representation

(q′,M′) := FGLM(q,M, c(i) : 1 ≤ i ≤ r)where

P := k[X1, . . . , Xn],I ⊂ P is a zero-dimensional ideal,q := q1, . . . , qs ⊂ P,

M = M(q) :=(

a(h)l j

)∈ ks2

, 1 ≤ h ≤ n

,

(q,M) is a Grobner representation of I,c(i) ∈ ks , for each i, 1 ≤ i ≤ r,gi := ∑

j cij q j , for each i, 1 ≤ i ≤ r,

J := I + (g1, . . . , gr ),σ := deg(J);q′ := q ′

1, . . . , q ′σ ⊂ P,

M′ = M(q′) :=(

d(h)l j

)∈ ks2

, 1 ≤ h ≤ n

,(

q′,M′) is a Grobner representation of J,

B := c(1), . . . , c(r), I := 1, . . . , s,While B = ∅ do

Choose c = (c1, . . . , cs) ∈ BB := B \ cB := B ∪ cM : M ∈ MLet ι := max j ∈ I : c j = 0I := I \ ιFor all j ∈ I do

q j := q j − c−1ι c j qι

For all l = 1..s, h = 1..n do a(h)l j := a(h)

l j − c−1ι c j a(h)

lιB′ := B, B := ∅For all (d1, . . . , ds) ∈ B′ do

For j := 1..s do d j := d j − c−1ι c j dι

If (d1, . . . , ds) = (0, . . . , 0) do B := B ∪ (d1, . . . , ds)q′ := qi , i ∈ I , M′ :=

(a(h)

l j

)l, j∈I

Since J is an ideal, the inclusion in it of g implies that J contains also thepolynomials Xh g,5 which are inserted in the list F in order to be treated in thesame way.

5 If the current Grobner representation is (q′,M′), q′ := q ′1, . . . , q ′

σ , M′ = M(q′) :=(d(h)

l j

)and g = ∑s

l=1 cl q′l then

Xh g =s∑

l=1

cl Xhq ′l =

s∑j=1

(s∑

l=1

cl d(h)l j

)q j .

432 Lazard

At termination, if I ⊂ 1, . . . , n denotes the set of indices of the elementsq j which have not been removed from q in this procedure, then J is describedby the Grobner representation

q′ = q j , i ∈ I ,M′ =(

a(h)l j

), l, j ∈ I, 1 ≤ h ≤ n

.

Example 29.3.9. Let us consider the linear representation of the ideal I ofExample 28.2.6 which has been computed in Examples 29.2.2 and let us com-pute the linear representation of I + (X2

1).We have

B = (0, 0, 0, 1, 0, 0, 0), I := 1, 2, 3, 4, 5, 6, 7,c := (0, 0, 0, 1, 0, 0, 0), B = (0, 0, 0, 0, −1, 1, 0), I := 1, 2, 3, 5, 6, 7,c := (0, 0, 0, 0, −1, 1, 0), B = (0, 0, 0, 0, 0, 0, 1), I := 1, 2, 3, 5, 7, 6

c := (0, 0, 0, 0, 0, 0, 1), B = ∅, I := 1, 2, 3, 5,which gives N≺ = 1, X1, X2, X1 X2 and the multiplication tables⎛

⎜⎜⎝0 1 0 00 0 0 00 0 0 10 0 0 0

⎞⎟⎟⎠ and

⎛⎜⎜⎝

0 0 1 00 0 0 10 0 0 00 0 0 0

⎞⎟⎟⎠ .

In fact the degrevlex Grobner basis of I + (X21) is X2

1, X22 − X1 X2.

29.4 Improving Moller’s Algorithm

One could remark that in the algorithms of Figure 28.2 and Figure 29.2 theestimate of the number of terms to be treated and of the While-loops to beperformed, that is ns, is just an upper bound since in the set

XhT<(ql) : 1 ≤ h ≤ n, 1 ≤ l ≤ s• some monomials XhT<(ql), while they are in T<(I), are not minimal gen-

erators of it, that is they are in B<(I) \ G<(I),• and others are represented in more than a single way:

XhT<(ql) = X H T<(qL), h = H.

While this is true, one can counter that it could be better, instead of trying toget an improved assessment, to evaluate how much information is obtainedwithin that complexity by performing the While-loop of the algorithm ofFigure 28.2 not just on the terms Xhtl belonging to the set N<(I) ∪ G<(I)but on the whole set N<(I) ∪ B<(I); in doing so we will again use freely thesame notation as in Algorithm 28.2.7.

6 Note that X2 · τ3 = X2 · X2 = τ6 now has the representation X2 · τ3 = τ5.

29.4 Improving Moller’s Algorithm 433

The While-loop which manages the term t := XhT<(ql) needs to knowthe vector v(Xhtl , L) = 1(Xhtl), . . . , s(Xhtl), and produces both a poly-nomial g = t − ∑σ

j=1 c j t j and the corresponding vector v(g, L); at thesame time, keeping track of the computation we obtain also the representa-tion of g − t = ∑σ

j=i λ j (g j )q j , in terms of the k-vectorspace basis qσ =q1, . . . , qσ .

Moreover:

• if v(g, L) = 0, then g ∈ I, t ∈ T<(I) and such information gives us, fort = Xhtl , its Grobner description in terms of the basis qσ and its canonicalform/Grobner description w.r.t. <:

Can(Xhtl , I, <) =σ∑

j=1

c j t j ,

γ (Xhtl , t j , <) = γ (Xhtl , t j , N<(I)) =

c j if j ≤ σ,

0 if j > σ,

Can(Xhtl , I, <) = t − g = −σ∑

j=1

λ j (g j )q j ,

• while, if v(g, L) = 0, then qσ+1 = λ−1σ+1(g)g ∈ qσ+1, t = Xhtl = tσ+1 ∈

N<(I) and such information still gives us, for t = Xhtl , its Grobner de-scription in terms of the basis qσ+1 and its (trivial) canonical form/Grobnerdescription w.r.t. <:

Can(t, I, <) = Can(Xhtl , I, <) = tσ+1,

γ (Xhtl , t j , <) = γ (Xhtl , t j , N<(I)) =

1 if j = σ + 1,

0 otherwise,

Can(Xhtl , I, <) = t = −σ∑

j=1

λ j (g j )q j + λσ+1(g)qσ+1,

qσ+1 = λ−1σ+1(g)g = λ−1

σ+1(g)tσ+1 −σ∑

j=1

λ−1σ+1(g)c j t j .

Therefore, if we perform the algorithm on the whole set N<(I) ∪ B<(I),we obtain within the same complexity, not only the Grobner basis G and thetriangular set q, but also the border basis, the Grobner representation by q, thelinear representation by N and the Grobner description of each qi in terms ofthe linear representation.7

It is interesting now to consider what modifications are needed if we con-sider (as we already did in our presentation of the FGLM algorithm) a variant

7 Which is nothing more than the change of basis between N and q.

434 Lazard

of the Moller algorithm in which the computations performed by the While-loop commanded by t = Xhtl are performed not on q := t but on q := Xhql .This choice has essentially no effect in the algorithm of Figure 29.2, but, in themore general case, such an improved version of the algorithm of Figure 28.2gives some potential benefits, at least whenever – as often can be easily done(see Corollary 32.3.3) – the set L is effectively ordered so that, for each i ,Ii = P(Spank(1, . . . , i )) is an ideal:

• the algorithm provides the whole structure not just for I but also for eachideal Ii in the chain;

• the required evaluation of the vector

v(Xhql , L) = 1(Xhql), . . . , s(Xhql)is simplified since we have j (Xhql) = 0, for each j ≤ l; and

• the loop in which v and q are reduced respectively by vect( j) and q j runs onthe indices j, l < j ≤ r , improving the computation by avoiding the indicesj, 1 ≤ j ≤ l.

On the other hand, the result of the While-loop which manages the polyno-mial Xhql , producing the polynomial g = Xhql −

∑σj=1 c j t j while also giving

the Grobner description g − Xhql = ∑σj=1 λ j (g j )q j , so that

• if v(g, L) = 0 we have Xhql = − ∑σj=1 λ j (g j )q j in P/I,

• and if v(g, L) = 0, the algorithm inserts it in q setting qσ+1 := λσ+1(g)−1gso that Xhql = − ∑σ

j=1 λ j (g j )q j + λσ+1(g)qσ+1 in P/I,

does not give, instead, a Grobner description of Xhql (respectively qσ+1) interms of the linear representation Nσ (respectively Nσ+1) since

g = Xhql −σ∑

j=1

c j t j ∈ Spank(Nσ ∪ Xhtl) ⇐⇒ Xh(ql − tl) ∈ Spank(Nσ )

but we can only claim that Xh(ql − tl) ∈ Spank(Bσ ); this implies that, withoutany modification,

• v(g, L) = 0 ⇒ ∑σj=1 c j t j = Can(Xhql , I, <) and

• v(g, L) = 0, ⇒ qσ+1 = λσ+1(g)−1g ∈ Spank(Nσ+1).

The required modification simply applies the While-loop not to Xhql but toXhtl −Can(Xh(tl −ql), I, <). Such modification can be done, because we havetl − ql = ∑l−1

j=1 γ (tl − ql , t j , <)t j which implies

Can(Xh(tl − ql), I, <) =l−1∑j=1

γ (tl − ql , t j , <) Can(Xht j , I, <)


and we have already obtained each Can(Xht j , I, <) because Xht j < Xhtl andwhen the While-loop manages Xhtl it has already managed all terms ω ∈B<(I) ∪ N<(I) such that ω < Xhtl .

Definition 29.4.1. Let P := k[X1, . . . , Xn], and < be any term ordering.Let L = 1, . . . , s ⊂ P∗ be an ordered set of k-linear functionals suchthat I := P(Spank(L)) is a zero-dimensional ideal and let r = deg(I) =dimk(Spank(L)).

The structural description of the ideal I in terms of L and < is the assignmentof the set

G, N, Λ, q,B,N ,Q, B, Bwhere

N := t1, . . . , tr ⊂ T is an order ideal,Λ := λ1, . . . , λr ⊂ L is an ordered subset,q := q1, . . . , qr ⊂ P is an ordered set,

which satisfy the conditions of Theorem 28.2.1 and

G ⊂ I is the reduced Grobner basis of I w.r.t. <,B = (

bl j) ∈ GL(r, k) is the invertible matrix defined by ql = ∑

j bl j t j ,

N = M(N) =(

a(h)l j

)∈ kr2

, 1 ≤ h ≤ n


defined by the equalities Xhtl = ∑j a(h)

l j t j in P/I = Spank(N),

Q = M(q) =(

q(h)l j

)∈ kr2

, 1 ≤ h ≤ n


defined by the equalities Xhql = ∑j q(h)

l j q j ,

B ⊂ I is the border basis of I w.r.t. <,B := B<(I).

Algorithm 29.4.2. All the comments above can be summarized in the algo-rithm of Figure 29.4 which produces the structural description of an ideal interms of a given set L of k-linear functionals and a term ordering <. Such analgorithm is obtained by merging into the algorithm of Figure 28.2 the ideasintroduced by the algorithm of Figure 29.2 and the ones discussed here, mainlythe application of the While-loop to the polynomials Xhql instead of the termsXhtl , and the explicit extraction of all the free information provided by thecomputation.

It is clear that the analysis we performed for the FGLM algorithm can berepeated verbatim, so that the stored information has size at most O(ns2+n2s)and the complexity depends on three different kinds of computation:

• the linear algebra operations which cost O(ns3);• the management of the list B which costs O(n2s2);

436 Lazard

Fig. 29.4. Enhanced Moller Algorithm

(r, G, N, Λ, q,B,N ,Q, B, B) := structure(L, <)where

P := k[X1, . . . , Xn],T := Xa1

1 . . . Xann : (a1, . . . , an) ∈ N

n,L = 1, . . . , s ⊂ P∗ is a set such that I := P(Spank(L)) is a zero-

dimensional ideal;< is a term ordering on PB ⊂ (τ, h, l) : τ ∈ T , 1 ≤ h ≤ n, 1 ≤ l ≤ s is a set ordered by so that

(ω1, h1, l1) (ω2, h2, l2) ⇒ ω1 ≤ ω2;r = deg(I) = dimk(Spank(L))G, N, Λ, q,B,N ,Q, B, B is the structural description of the ideal I in terms

of L and < (here presented with the same notation as in Definition 29.4.1)1 = t1 < t2 < . . . < t j < t j+1 < . . . < tr ,

G := ∅, r := 1, t1 := 1, N := t1, B := ∅, B := ∅,v := (1(1), . . . , s(1)),µ := min j : j (1) = 0, λ1 := µ, Λ := λ1,b11 := λ1(t1)−1, q1 := b11t1, q := q1, vect(1) := b11v,Let B := (Xh , h, 1), 1 ≤ h ≤ nWhile B = ∅ do

(t, h, l) := min(B), B := B \ (t, h, l),%% t = Xhtl = XhT<(ql )

%% Xhql = Xhtl + ∑l−1j=1 bl j Xht j

q := Xhtl + ∑ri=1

(∑l−1j=1 bl j a(h)

j i

)ti

For i = 1..r do a(h)li := −∑l−1

j=1 bl j a(h)j i

If t ∈ T<(G) then B := B ∪ q, B := B ∪ t,else

v := (1(q), . . . , s(q)),For j = 1..r do

q(h)l j := λ j (q), v := v − q(h)

l j vect( j), q := q − q(h)l j q j

For i = 1..r do a(h)li := a(h)

li + q(h)l j a(h)

j iIf v = 0 then G := G ∪ q, B := B ∪ q, B := B ∪ t,else

r := r + 1 tr := t, N := N ∪ tr ,µ := min j : j (q) = 0, λr := µ, Λ := Λ ∪ λr ,qr := λr (q)−1q, q := q ∪ qr , vect(r) := λr (q)−1v

For j = 1..r − 1 do br j := −λr (q)−1a(h)l j , q(h)

l j := λr (q)−1q(h)l j

a(h)lr := 1, q(h)

lr := λr (q)

B := B ∪ (Xhtr , h, r), 1 ≤ h ≤ n,For each (τ, κ, ι) ∈ B : τ = t do

B := B \ (τ, κ, ι),For j = 1..r do a(κ)

ιj := a(k)l j

q := Xκqι, v := (1(q), . . . , s(q)),For j = 1..r do

q(κ)ιj := λ j (q), v := v − q(κ)

ιj vect( j), q := q − q(κ)ιj q j

r, G, N, Λ, q,B,N ,Q, B, B


• the evaluation of each s functional on each ns polynomial. We will see soonthat for the most common functionals on which Figure 29.4 has been ap-plied, the total cost is at most O(ns3); the only exceptions are computationsinvolving changes of coordinates, which cost O(n2s3).

Let us now discuss the main applications proposed for the algorithm andestimate the cost of the evaluation of the functionals at a polynomial:

canonical forms: This is the original application of Figure 29.2 and we havealready seen that the evaluation of s functionals at the ns polynomialsq := Xhql costs O(ns3).

Other than solving the FGLM Problem, the same functionals havebeen applied by Lakshman as a tool to compute a Grobner basis ofa zero-dimensional ideal I := ⋂

qi , knowing the Grobner bases (notnecessarily w.r.t. the same ordering) of the primary components qi ;the complexity becomes O(ns

∑i µ(i)2) where µ(i) = deg(qi ), for

each i , so that∑

i µ(i) = s.change of coordinates: As the FGLM problem aims to apply the powerful

properties of the Grobner basis w.r.t. the lexicographical orderingwithout paying the cost of direct computation, the same approach canbe applied to avoid the cost of performing generic changes of coordi-nates.

If we are given a basis F ⊂ k[X1, . . . , Xn] of an ideal I and weperform a change of coordinates

k[Y1, . . . , Yn] = k[X1, . . . , Xn]

by fixing an invertible matrix M = (ci j

) ∈ GL(n, k) and its inverse(di j

) = M−1 ∈ GL(n, k) and setting Yi := ∑j ci j X j , for each i ,

so that X j = ∑i d ji Yi , for each j , then the knowledge of a Grobner

basis of the ideal

J := Ik[Y1, . . . , Yn]

:=

f(∑

id1i Yi , . . . ,

∑i

d1i Yi

), f ∈ I

⊂ k[Y1, . . . , Yn],

gives an advantage not dissimilar to the one offered by the lexico-graphical ordering (e.g. in the computation of a primary decomposi-tion).

However, the cost to be paid in order to perform such a change ofcoordinates is definitely not affordable: the problem is not necessarilythe cost of performing Buchberger’s algorithm on the basis

F ′ :=

f(∑

id1i Yi , . . . ,

∑i

d1i Yi

), f ∈ F

⊂ k[Y1, . . . , Yn],

438 Lazard

of J, but it is simply that of storing F ′; in fact, for any polynomial f ∈P, deg( f ) = d , f (

∑i d1i Yi , . . . ,

∑i d1i Yi ) is a linear combination

of(d+n

n

) ≈ dn terms.On the basis of the obvious equality

Can(gYi , I, <) =∑

j

ci j Can(gX j , I, <),

it is trivial to modify Figure 29.2 in order to obtain the Grobner basisof J in k[Y1, . . . , Yn] w.r.t. < at the cost of O(ns3) operations in thefield k. Direct applications of this approach to decomposition algo-rithms will be discussed in Section 35.7.

point evaluation: This is the original application of Figure 28.2. We can as-sume that we are given a set of points ai := (ai1, . . . , ain) ∈ kn and,for each such point, a set of functionals (i)

1 , . . . , (i)µ(i) defining an

mi -primary

qi := P(Spank(i)1 , . . . ,

(i)µ(i),

where mi = (X1 − ai1, . . . , Xn − ai1) and the aim is to compute theideal I := ⋂

qi . There are of course different cases to be considered:

simple rational point evaluation: for all i , we have ai ∈ kn andµ(i) = mult(qi ) = 1, that is qi = mi ; in this case we have(i)1 := evai where, for each f ∈ P , evai ( f ) = f (ai1, . . . ,

ain), and each evaluation on each polynomial q := Xhql hascost 1, since

evai (q) = aihevai (ql);therefore evaluating s such functionals at the ns polynomialsq := Xhql costs O(ns2);

simple algebraic point evaluation: here we can wlog assume thatwe are given just a single point ai for each conjugate class.

It is natural to assume that we are using the Kronecker–Duval Model and the Grobner technology; therefore wlogthe ring Ki := P/Ii is given as a quotient of P by a zero-dimensional ideal Ii whose roots are ai and its conjugatesand of which we have the border basis.

If σi = deg(ai ) = [Ki : k] the evaluation at all σi

points in the conjugate class requires the single evaluationevai (Xhql) = aihevai (ql) which costs σ 2

i operations. There-fore the complexity becomes

O(

ns∑

i

σ 2i

)≤ O(ns3);


multiple rational point evaluation: in Chapter 31 we will discussadvanced techniques on how to represent mi -primaries; upto now it is sufficient to note that a possible (and an effi-cient, conservative) solution is to give each qi by means of aGrobner representation. Thus we obtain the complexity

O(ns∑

i

µ(i)2) ≤ O(ns3).

We will see later that, in general, different representations donot improve such complexity;

multiple algebraic point evaluation: the evaluation of the ideal

qi ∈ Ki [X1, . . . , Xn], deg(qi ) = µ(i), [Ki : k] = σi

returns an ideal I ⊂ P such that deg(I) = µ(i)σi and itis performed by evaluating a la FGLM the canonical formsCan( · , qi , ≺) modulo qi w.r.t. any suitable term ordering atthe cost of O(nsµ(i)2) operations on the ring Ki , each suchoperation costing O(σ 2

i ) computations in k. The total cost istherefore

O(

ns∑

i

µ(i)2σ 2i

)≤ O

⎛⎝ns

(∑i

µ(i)σi

)2⎞⎠ = O(ns3).

Algorithm 29.4.3 (Moller). We present here an algorithm which applies thesame improvements on Algorithm 28.2.5 (see Figure 28.1) which iterates onan ordered set of k-linear functionals

L = 1, . . . , s ⊂ P∗,

which satisfies dimk(L) = s and Lσ := Spank(1, . . . , σ ) is a P-module,for each σ ≤ s, so that each Iσ = P(Lσ ) is an ideal and all the results ofTheorem 28.2.1 hold.

This version has some further advantages:

• For each ideal Ir some of the informations can be extracted directly:

the Grobner representation is q1, . . . , qr ,the linear one is t1, . . . , tr ,the matrix encoding the change of coordinates between them is the r th

principal minor of the matrix (bl j ).

• It also explicitly provides the border set and the border bases of each Ir , andthe multiplication structure N of t1, . . . , tr modulo Ir . It is not obvioushow to extract this directly from the algorithm of Figure 29.4, although it isimplicitly present there.

440 Lazard

• The computation of each multiplication structure Q of q1, . . . , qr mod-ulo Ir requires a shorter computation since in the representation Xhql =∑r

j=1 q(h)l j q j we have q(h)

li = 0, 1 ≤ i < l, because, for each i < l,

Xhql ∈ Ii ⇒ 0 = i (Xhql) =r∑

i=1

q(h)l j i (q j ) = q(h)

li .

• It is quite natural in many applications that more new points to be evaluatedare taken into consideration later; so one could require an algorithm, likethe one in Figure 29.5, in which the structural description of the ideal Ii :=P(Spank(1, . . . , i )) and the functional i+1 are given and the structureof Ii+1 is required.

• More importantly, this algorithm can be applied as a technical tool in orderto derive important theoretical results on configuration of points (see Chap-ter 33).

• The only negative aspect of this algorithm w.r.t. the one of Figure 29.5 isthat the management of F costs O(n3s2) because the improvement appliedin the FGLM algorithm cannot be applied here.

29.5 Hilbert Driven and Grobner Walk

Let us now discuss two interesting alternatives to the FGLM algorithm as asolution of the FGLM problem.

Algorithm 29.5.1 (Traverso; Hilbert Driven Algorithm). The first solution as-sumes knowledge of the Hilbert function H(T ; I), information which can beextracted from knowledge of the Grobner basis w.r.t. any degree-compatibleordering; it requires the assumption that the ideal is homogeneous but this as-sumption is easily by-passed.8

The algorithm consists of

• applying Buchberger’s algorithm with a selection strategy which managesthe S-pairs by increasing value of their degree;

8 In fact if we are given an (affine) ideal I ⊂ k[X1, . . . , Xn ] and its Hilbert function H(T ; I)and we want to obtain its Grobner basis w.r.t. any term ordering < we have just to consider thehomogeneous ideal h I ⊂ k[X0, X1, . . . , Xn ], whose Hilbert function hH(T ; h I) = H(T ; I) weknow, and deduce from this algorithm the Grobner basis G of h I w.r.t. the term ordering <hdefined by

τ1 <h τ2 ⇐⇒ deg(τ1) < deg(τ2) or deg(τ1) = deg(τ2) and aτ1 < aτ2;

the required Grobner basis is then simply ag : g ∈ G.

29.5 Hilbert Driven and Grobner Walk 441

Fig. 29.5. Enhanced Moller Dual Algorithm

(r, G, N, Λ, q,B,N ,Q, B, B) := structure(L, <, )where

P := k[X1, . . . , Xn],T := Xa1

1 . . . Xann : (a1, . . . , an) ∈ N

n,L = 1, . . . , s ⊂ P∗ is a set such that J := P(Spank(L)) is a zero-

dimensional ideal;< is a term ordering on P ∈ P∗ \ Spank(L) is a functional such that

L′ := L ∪ ⊃ Spank(L) is a P-module, so that

I := P(Spank(L ∪ )) is a zero-dimensional ideal;r = deg(I) = dimk(Spank(L′) = dimk(Spank(L) + 1 = deg(J) + 1G, N, Λ, q,B,N ,Q, B, B is the structural description of the ideal I in terms

of L′ and < (here presented with the same notation as in Definition 29.4.1)

(r, G, N, Λ, q,B,N ,Q, B, B) := structure (L, <)t := minT( f ) : f ∈ G, ( f ) = 0Let f ∈ G, : T( f ) = t ,Let h, l, 1 ≤ h ≤ n, 1 ≤ l ≤ r : t = XhtlLet c1, . . . , cr ∈ k : f = lc( f )t + ∑r

j=1 c j t jG := G \ f , B := B \ f , B := B \ t,r := r + 1,tr := t, qr := ( f )−1 f, N := N ∪ tr , q := q ∪ qr For j = 1..r − 1 do br j := ( f )−1c j ,

brr := lc( f )( f )−1, a(h)lr := 1,

Bold := B, B := ∅,For each f ∈ Bold do

f := f − ( f )qr ,For each h, l, 1 ≤ h ≤ n, 1 ≤ l ≤ r : T( f ) = Xhtl do

For j = 1..r do a(h)l j := a(h)

l j − ( f )br jB := B ∪ f

For h = 1..n doIf Xhtr ∈ B then

Let κ, ι : Xhtr = Xκ tιFor j = 1..r do a(h)

r j := a(κ)ιj

elset := Xhtr , f := Xhqr ,%% f ∈ J, λ( f ) = 0 for all λ ∈ L

f := f − ( f )qrLet c1, . . . , cr ∈ k : f = ∑r

j=1 c j t j

For j = 1..r do a(h)r j := c j

B := B ∪ f , B := B ∪ tG := f ∈ B : T( f ) ∈ T(I), Λ := Λ ∪ For l = 1..r , h = 1..n do

%% Xhql = ∑si=1 bli Xhti = ∑r

j=1∑s

i=1 bli a(h)i j t j

f := ∑rj=1

(∑si=1 bli a(h)

i j

)t j ,

For j = l..r , do q(h)l j := r ( f ), f := f − r ( f )qr

(r, G, N, Λ, q,B,N ,Q, B, B)

442 Lazard

• upgrading the value of hH(T ; h I),9 any time a new element g is inserted inthe current basis G;

• finding the first value δ ∈ N such that hH(δ; (G)) > hH(δ; I),10 and• discarding as a useless pair, from the set B of all S-pairs to be treated, each

pair i, j ∈ B such that deg(S(gi , g j )) < δ.

The rationale is that, since we are dealing with a homogeneous ideal, the re-duction of such a pair S(gi , g j ) is a polynomial g,

g ∈ I, g /∈ (G), deg(g) = deg(S(gi , g j )) < δ;the equality h H(deg(g); (G)) = h H(deg(g); I) then implies g = 0, that is theuselessness of i, j.

Traverso’s Hilbert Driven Algorithm can also be successfully applied to res-olution computation: in fact almost all known algorithms for computing reso-lutions apply Schreier’s result (Proposition 23.7.4) which states that if G is aGrobner basis of a module, the lifting of its S-pairs generates a Grobner basisof the module of its syzygies; since knowledge of the Hilbert function of amodule also gives freely the Hilbert function of its syzygies, the lifting of theS-pairs can therefore be controlled by means of the Hilbert Driven Algorithmin order to avoid useless liftings.

Algorithm 29.5.2 (Collart–Kalkbrener–Mall; Grobner Walk). Macaulay’s re-sults (Lemma 23.2.4 and Corollary 24.5.6) state that, for any weight functionw and any weight-compatible ordering on k[X1, . . . , Xn]

if G is a Grobner basis of I ⊂ k[X1, . . . , Xn] w.r.t. then G is a standardbasis of I and Lw(G) is a Grobner basis of Lw(I) w.r.t. and, con-versely;

if G ′ is a standard basis of I such that Lw(G ′) is a Grobner bases of Lw(I)w.r.t. then G ′ is a Grobner basis of I w.r.t. .

Let < and ≺ be two term orderings on P := k[X1, . . . , Xn] and let I ⊂ Pbe any ideal.

The results on the state polytope (Section 24.10) of an ideal inform us thatin Z

n there are weight vectors δ(<) and δ(≺) such that < (respectively ≺) isthe refinement of δ(<) (respectively δ(≺)), T<(I) = Lδ(<)(I) and T≺(I) =Lδ(≺)(I).

9 In this setting we have a monomial ideal J := T(G), whose Hilbert function is known, and anew term τ := T(g) and we need to compute the Hilbert function of

T(G ∪ g) = J′ = J + τ ,which can be obtained, via Equation (23.5), by the computation of that of (J : τ ).

10 Since (G) ⊂ I necessarily we have h H(t; (G)) ≥ h H(t; I), for each t ∈ N.

29.5 Hilbert Driven and Grobner Walk 443

If we consider the line segment in Qn defined by

(1 − t)d(<) + td(≺), 0 ≤ t ≤ 1it is possible to compute the maximal value T ≤ 1 for which

(1 − t)d(<) + td(≺), 0 ≤ t < T ⊂ C(I, d(<));then, for the weight vector w := (1 − T )d(<) + T d(≺), if we denote by <′and ≺′ the refinements of w by < and ≺, we know that:

if G := g1, . . . , gr is the Grobner basis of I w.r.t. <, then

G is also the Grobner basis of I w.r.t. <′ andLw(G) is the Grobner basis of Lw(I) w.r.t. <′;

if h1, . . . , hs ⊂ Lw(I), where

hi :=∑

j

pi jLw(g j ), w(hi ) = w(pi j ) + w(g j ),

is a Grobner basis of Lw(I) w.r.t. ≺′, then, setting Hi := ∑j pi j g j ,

H1, . . . , Hs ⊂ I is a Grobner basis of I w.r.t. ≺′.

Therefore, by iteration, it is possible to compute weight vectors δ(<) :=w1, . . . , wt =: δ(≺) such that, denoting by <i and ≺i the refinement of wi by< and ≺, then

the Grobner basis G1 of I w.r.t. < = <1 is such that

G1 is also the Grobner basis of I w.r.t. <2 andLw2(G1) is the Grobner basis of Lw2(I) w.r.t. <2

· · ·if Gl := g1, . . . , gr denotes the Grobner basis of I w.r.t. <l and

h1, . . . , hs ⊂ Lwl (I), hi :=∑

j

pi jLwl (g j ), wl(hi )

= wl(pi j ) + wl(g j )

is a Grobner basis of Lwl (I) w.r.t. ≺l , then, setting Hi := ∑j pi j g j ,

Gl+1 := H1, . . . , Hs ⊂ I is a Grobner basis of I w.r.t. ≺l and <l+1

· · ·if Gt := g′

1, . . . , g′R denotes the Grobner basis of I w.r.t. <t and

h′1, . . . , h′

S ⊂ Lwl (I), h′i :=

∑j

p′i jLwt (g

′j ), wt (h

′i )

= wt (p′i j ) + wt (g

′j )

is a Grobner basis of Lwt (I) w.r.t. ≺t then, setting H ′i := ∑

j p′i j g

′j ,

Gt+1 := H ′1, . . . , H ′

S ⊂ I is a Grobner basis of I w.r.t. ≺t = ≺.

444 Lazard

Therefore a sequence of computations of the Grobner basis of the Leitide-alen Lwl (I) allows us to obtain the Grobner basis Gt+1 of I w.r.t. ≺ from theGrobner basis G1 of I w.r.t. <.

According to experimental analyses, a good implementation of the FGLMalgorithm is better than the Hilbert Driven Algorithm which is better by farthan the Grobner Walk.

It is more important to note that both the Hilbert Driven and the GrobnerWalk Algorithms have the advantage, over the FGLM one, of being also ap-plicable in the higher-dimensional case. While, in principle, FGLM can alsobe adapted to that case 11 nobody in his right mind would implement such anadaptation because of its obvious complexity.

While checking the proofs, Sala communicated me a new approach to theFGLM problem which seems very promising.

Let

P := k[X1, . . . , Xn],T := Xa1

1 · · · Xann : (a1, . . . , an) ∈ N

n,< be any termordering,P ′ := k[X1, . . . , Xn−1],T ′ := T [1, n − 1] = T ∩ P ′,<′ the restriction of < to T ′,w := (w1, . . . , wn) the weight under which < is weight compatible, I ⊂ Pan ideal,G a minimal Grobner basis of I wrt <.

Definition 29.5.3. < is called a pseudo-lex ordering for I if, for each g ∈ G

T < (g) ∈ T ′ ⇐⇒ g ∈ P ′, for each g ∈ G

Lemma 29.5.4 (Sala). With the present notation, the following holds:

(1) If < is a pseudo-lex ordering for I, then G ∩ P ′ is the Grobner basiswrt <′ of I ∩P ′

(2) For any I, if wn >> wi for i = n then < is a pseudo-lex ordering for I.

Proof.

(1) Let f ∈ I ∩ P ′; then T( f ) ∈ T ′ and there are τ ∈ T ′, g ∈ G suchthat T( f ) = τT(g); therefore T(g) ∈ T ′ and hence g ∈ G ∩ P ′.

11 It is ‘sufficient’ to perform the algorithm over all the monomials in N<(I) whose degree isbounded by the highest degree of the elements in G<(I), exporting the termination conditionby the Hilbert Driven Algorithm, that is comparing the Hilbert function of I with that of themonomial ideal under construction.

29.6 *The Structure of the Canonical Module 445

In other words, for any f ∈ I ∩ P ′, there is g ∈ G ∩ P ′ suchthat T(g) | T( f ).

(2) Obvious.

Algorithm 29.5.5 (Sala-Zanoni). Sala and Zanoni proposed an algorithm thatexploits Sala’s Criterion as follows:

• for i = n..2, repeatedly– apply Lemma 29.5.4(2) to choose12 a term ordering <i on T [1, i] and– compute the Grobner basis Gi of I ∩ k[X1, . . . , Xi ]until <i is a pseudo-les ordering for I ∩ k[X1, . . . , Xi ], thus obtaining theGrobner basis Gi−1 := Gi ∩ k[X1, . . . , Xi−1] wrt <′

i of I∩ k[X1, . . . , Xi−1];• in particular, for i = 1. H1 := G1 = g is the generator of the principal

ideal I ∩ k[X1];• for i = 2..n :

– apply Buchberger’s algorithm to the basis Hi−1 ∪ Gi in order to computethe Grobner basis Hi of the ideal I ∩ k[X1, . . . , Xi ] wrt the lex orderinginduced by X1 < · · · Xi .

The implicit assumption13 of this approach is that this sequence of ‘con-trolled’ lex Grobner basis computation could compare with the FGLM Algo-rithm. Up to now, no deep experimention has been performed, but the first testsare promising. Also this algorithm can be applied in the higher-dimensionalcase.

29.6 *The Structure of the Canonical Module

Let

P := k[X1, . . . , Xn],T := Xa1

1 . . . Xann : (a1, . . . , an) ∈ N

n,< be any term ordering,L := 1, . . . , r ⊂ P∗ be a linearly independent set of k-linear functionals

such that L := Spank(L) is a P-module so that I := P(L) is a zero-dimensional ideal,

N(I) := t1, . . . , tr ,q := q1, . . . , qr ⊂ P be the set triangular to L,

which satisfy the conditions of Theorem 28.2.1. Let also

12 They are developping an appropriate strategy for this crucial choice.13 Which is also supported by old (Late Eighties) and more naıve experimentations by Sasaki who

proposed to compute the lexicographical Grobner basis of I ⊂ P induced by X1 < . . . , Xn , byperforming a sequence of Grobner bases of I wrt term ordering <1, <2, · · · <n on T , where, foreach i , the restricion of <i to T [1, i] coincide with the lex ordering induced by X1 < · · · Xi .

446 Lazard(q(h)

i j

)∈ kr2

, 1 ≤ k ≤ r , be the matrices defined by Xhqi = ∑j q(h)

i j q j ,

Λ := λ1, . . . , λr be the set biorthogonal 14 to q

By duality we have

Proposition 29.6.1. With the notation above, we have

Xhλ j =r∑

i=1

q(h)i j λi , for each i, j, h.

Proof. Since, by definition, Xhλ j (qi ) = λ j (Xhqi ), if we denote by b(h)jl the

values such that Xhλ j = ∑rl=1 b(h)

jl λl for each l, j, h, we have

b(h)j i =

r∑l=1

b(h)jl λl(qi )

= Xhλ j (qi )

= λ j (Xhqi )

= λ j

(r∑

l=1

q(h)il ql

)

=r∑

l=1

q(h)il λ j (ql)

= q(h)i j .

Algorithm 29.6.2. With this information, Remark 29.2.6 can be directly ap-plied to compute the structure of the P-module L := Spank(Λ) by a reformu-lation of Macaulay’s Algorithms 30.4.13 and 30.6.3, which had already beenapplied by Grobner to compute a reduced representation of a primary ideal(Section 32.3).

Writing, for each integer ν ∈ N

e1, . . . , eν for the canonical basis of Pν,

T (ν) := tei , t ∈ T , 1 ≤ i ≤ ν,≺ for any term ordering on T (ν) satisfying the condition

i < j ⇒ t1ei ≺ t2e j for each t1, t2 ∈ T ,

the algorithm returns

14 Which can be trivially deduced by Gaussian reduction.


an integer ν,a module U ⊂ Pν , by producing, w.r.t. ≺, its Grobner basis G(U ) and the set

N≺(U ) ⊃ e1, . . . , eν,a vectorspace isomorphism σ : Spank(N(U )) → Spank(Λ),

such that denoting Σ : Pν → Spank(Λ) the projection

Σ

(ν∑

i=1

fi ei

)=

ν∑i=1

fiσ(ei ),

we have

U = ker(Σ),σ is a splitting homomorphism for Σ, and so Pν/U ∼= Spank(N(U )) ∼=

Spank(Λ).

The algorithm (see Figure 29.6) initially sets ν := 0, N(U ) := ∅ and, byiteration on µ

chooses any element λ ∈ Λ, λ /∈ Σ(Pν)

sets ν := ν + 1, eν := σ(λ)

and, by a direct application of the scheme of the Moller algorithm, computesIm(Σ) and ker(Σ)

until Im(Σ) = L .

Example 29.6.3. Let us consider the ideal

I := (X22 + X2 − 2X2

1, X1 X2 − X1, X31 − X1) ⊂ k[X1, X2]

which is given by means of the reduced Grobner basis w.r.t. the lexicographicalordering induced by X1 < X2 so that deg(I) = 4 and Z(I) = ai , 1 ≤ i ≤ 4where

a1 := (0, 0), a2 := (1, 1), a3 := (−1, 1), a4 := (0, −1).

If we denote by i the evaluation at ai , the set biorthogonal to 1, X1, X2, X21

is λ1, λ2, λ3, λ4 where

λ1 := 1, λ2 := 12 (2 − 3),

λ3 := 1 − 4, λ4 := 12 (−41 + 2 + 3 + 24)

and the multiplication tables are⎛⎜⎜⎝

0 1 0 00 0 0 10 1 0 00 1 0 0

⎞⎟⎟⎠ and

⎛⎜⎜⎝

0 0 1 00 1 0 00 0 −1 20 0 0 1

⎞⎟⎟⎠ .

448 Lazard

Fig. 29.6. Canonical Module

(ν, N′, σ, G′) := CanonicalModule(Λ, <,Q)where

P := k[X1, . . . , Xn],T := Xa1

1 . . . Xann : (a1, . . . , an) ∈ N

n,Λ := λ1, . . . , λr ⊂ P∗ is a linearly independent set of k-linear functionals

such that L := Spank(Λ) is a P-module;< is a term ordering on PQ =

(a(h)

l j

)∈ kr2

, 1 ≤ h ≤ n

is the set of the square matrices defined by

Xhλi = ∑rj=1 a(h)

i j λ j , for each i, j, h.

ν ∈ N

e1, . . . , eν the canonical basis of Pν

T (ν) := tei , t ∈ T , 1 ≤ i ≤ ν ordered so thatt1ei ≺ t2e j ⇐⇒ i < j or i = j and t1 < t2

N′ ⊂ T (ν) an order module,σ : Spank(N′) → Spank(Λ) is a vector space isomorphismΣ : Pν → Spank(Λ) the projection

Σ(∑ν

i=1 fi ei ) = ∑νi=1 fi σ(ei ),

U := ker(Σ),N′ = N≺(U )G′ is the Grobner basis of U w.r.t. ≺

(r, G, N, Λ, q,B,N ,Q, B, B) := structure(L, <)t1, . . . , tr := N,ν := 0, N′ := G′ := ∅Until #N′ = r = dim(L) do

Let J := j : λ j /∈ σ(Spank(N′))Let i : ti := max<(t j : j ∈ J )

ν := ν + 1, N′ := N′ ∪ eν, σ(eν) := λi ,B := (Xheν, h, eν) : 1 ≤ h ≤ n,Until B = ∅ do

Let (υ, h, τ ) ∈ B : υ ≺ υ′ for each (υ′, h′, τ ′) ∈ BB := B \ (Xhτ, h, τ )If υ = Xhτ /∈ N′ ∪ T(G′) do

If Xhσ(τ) ∈ Spank(σ (N′)) thenLet Xhσ(τ) = ∑

ω∈N′ aωσ(ω) be a linear relationG′ := G′ ∪ υ − ∑

ω∈N′ aωωelse

N′ := N′ ∪ υ, B := B′ ∪ (Xhυ, h, υ) : 1 ≤ h ≤ n

Then we set

σ(e1) := λ4, N := e1,G = ∅, B := X1e1, X2e1,σ (X1e1) := λ2, N := e1, X1e1,G = ∅, B := X2

1e1, X2e1, X1 X2e1,σ(X2

1e1) := λ1 + λ3 + λ4, N := e1, X1e1, X21e1,

G = ∅, B := X31e1, X2e1, X1 X2e1, X2

1 X2e1,σ(X3

1e1) := λ2, N := e1, X1e1, X21e1,


G = (X31 − X1)e1, B := X2e1, X1 X2e1, X2

1 X2e1,σ(X2e1) := 2λ3 + λ4, N := e1, X1e1, X2

1e1, X2e1,G = (X3

1 − X1)e1, B := X1 X2e1, X21 X2e1, X2

2e1,σ(X1 X2e1) := λ2, N := e1, X1e1, X2

1e1, X2e1,G = (X3

1 − X1)e1, (X1 X2 − X1)e1, B := X22e1,

σ(X22e1) := 2λ1 + λ4, N := e1, X1e1, X2

1e1, X2e1,G = (X3

1 − X1)e1, (X1 X2 − X1)e1, (X22 + X2 − 2X2

1)e1, B := ∅.

The fact that L(I) I is expected, since I is Gorenstein.

Example 29.6.4. To show a less trivial example let us produce a reducibleprimary ideal at the origin, for which, following Grobner’s proposal (Sec-tion 32.3), the algorithm of Figure 29.6 returns a decomposition (see Exam-ple 32.3.7).

Let us consider the ideal

I := (X32 − X3

1, X1 X22, X3

1 X2, X41) ⊂ k[X1, X2]

which is given by means of the reduced Grobner basis w.r.t. the degree lexico-graphical ordering induced by X1 < X2.

Then let us denote by λi , 1 ≤ i ≤ 8 the set biorthogonal to theorder ideal N<(I)1, X1, X2, X2

1, X1 X2, X22, X3

1, X21 X2; the corresponding

multiplication tables are

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

and

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Then we compute 15

σ(e1) := λ7, N := e1,G = ∅, B := X1e1, X2e1,σ(X1e1) := λ4, N := e1, X1e1,G = ∅, B := X2e1, X2

1e1, X1 X2e1,

15 The algorithm, in fact, requires us to begin by setting σ(e1) := λ8. Relaxing, as I am doing inthis example, the instruction

Let i : ti := max<

(t j : j ∈ J )

gives the risk of getting a higher value of ν. This is made evident by setting σ(e1) := λ6!

450 Lazard

σ(X2e1) := λ6, N := e1, X1e1, X2e1,G = ∅, B := X2

1e1, X1 X2e1, X22e1,

σ(X21e1) := λ2, N := e1, X1e1, X2

1e1,G = ∅, B := X1 X2e1, X2

2e1, X31e1, X2

1 X2e1,σ(X1 X2e1) := 0, N := e1, X1e1, X2

1e1,G = X1 X2e1, B := X2

2e1, X31e1, X2

1 X2e1,σ(X2

2e1) := λ3, N := e1, X1e1, X21e1, X2

2e1,G = X1 X2e1, B := X3

1e1, X21 X2e1, X1 X2

2e1, X32e1,

σ(X31e1) := λ1, N := e1, X1e1, X2

1e1, X22e1, X3

1e1,G = X1 X2e1, B := X2

1 X2e1, X1 X22e1, X3

2e1, X41e1, X3

1 X2e1,X2

1 X2e1 ∈ T(G), X1 X22e1 ∈ T(G),

σ(X32e1) := λ1, N := e1, X1e1, X2

1e1, X22e1, X3

1e1,G = X1 X2e1, (X3

2 − X31)e1, B := X4

1e1, X31 X2e1,

σ(X41e1) := 0, N := e1, X1e1, X2

1e1, X22e1, X3

1e1,G = X1 X2e1, (X3

2 − X31)e1, X4

1e1, B := X31 X2e1,

X21 X2e1 ∈ T(G), B := ∅,

σ(e2) := λ8, N := e1, X1e1, X21e1, X2

2e1, X31e1, e2,

G = X1 X2e1, (X32 − X3

1)e1, X41e1, B := X1e2, X2e2,

σ(X1e2) := λ5, N := e1, X1e1, X21e1, X2

2e1, X31e1, e2, X1e2,

G = X1 X2e1, (X32 − X3

1)e1, X41e1, B := X2e2, X2

1e2, X1 X2e2,σ(X2e2) := λ4, N := e1, X1e1, X2

1e1, X22e1, X3

1e1, e2, X1e2 ,G = X1 X2e1, (X3

2 − X31)e1, X4

1e1, X2e2 − X1e1, B := X21e2, X1 X2e2,

σ(X21e2) := λ3, N := e1, X1e1, X2

1e1, X22e1, X3

1e1, e2, X1e2,G = X1 X2e1, (X3

2 − X31)e1, X4

1e1, X2e2 − X1e1, X21e2 − X2

2e1,B := X1 X2e2,

X1 X2e2 ∈ T(G), B := ∅.

30

Macaulay II

Many of the notions introduced in Section 29.3 in order to describe and applythe linear-algebra structure of the vector-space k[N<(I)] = Spank(N<(I)) ∼=P/I, where I ⊂ P , stemmed on the one hand from a deeper analysis of theMoller algorithm, on the other hand from a reconsideration of Grobner’s de-scription of Macaulay’s results within ideal duality.

The aim of this chapter is to survey that result by Macaulay: after present-ing Macaulay’s computational assumptions and terminology (Section 30.1),we discuss his notation and the basic properties of his inverse systems(Section 30.2).

Section 30.3 is devoted to his linear-algebra algorithms which compute theinverse system of homogeneous and affine ideals.

Macaulay then concentrated his consideration to m-primary1 ideals and m-closed ideals I, seen as the ‘limit’ of m-primaries – I = ⋂

d I + md . For them(Section 30.4) he

introduced the notion of Noetherian equations,gave algorithms to compute their Noetherian equations, and their P-module

structure,already hinted at the notion of canonical forms, linear representation, and

Grobner representation which he is able to read directly from theNoetherian equations.

His next step generalized this result from zero-dimensional primaries to thehigher-dimensional case by means of extension/contraction; in order to avoidthe risk of failing to explain his results, I quote in Section 30.5 that chapterof his book, limiting myself to supporting the reader by following Macaulay’sargument on a non-trivial example.

1 Where m is the maximal at the origin.

451

452 Macaulay II

The introduction of Noetherian equations allowed Macaulay to introduce(Section 30.6) the notion of multiplicity for a primary ideal q as the length of arefined chain linking q with

√q and to perform a deeper study on the structure

of primary ideals at the origin (Section 30.7).

30.1 The Linear Structure of an Ideal

Hilbert’s notion of the characteristic (Hilbert) function of an ideal I as

the number of independent conditions which must be satisfied by the coefficients of ahomogeneous polynomial of degree R, so that it be congruent to zero with respect to I

posed the natural question of describing (and explicitly producing) such ‘in-dependent conditions’, whose set, using the notion and notation introduced inthis part, is clearly L(I). The first person to attack this problem, viewing it as acomponent of a solving tool, was Macaulay.2

Historical Remark 30.1.1. It is interesting to consider the computational set-ting used and assumptions made by Macaulay; as he stated in a footnote 3

It is to be understood throughout that a given module means a module whose basis isgiven, and that to determine a module means to determine its basis.

Then for some given modules (i.e. ideals) the following computational abilityis needed:4

[...] for the carrying out of the resolution in general the following comprehensive as-sumptions are made:

(I) that the basis of the L.C.M. of any given set of modules is known,(II) that the basis of the residual of any given module with respect to another is known,

and(III) that a complete set of linearly independent members of any assigned degree (spec-

ified numerically) of a given H-module [i.e. a homogeneous ideal] can bewritten down and computed.

In other words, he assumes that

(I) given, through some bases, the ideals qi it is possible to compute a basis ofthe intersection ideal I = ∩iqi ;

(II) given, through some bases, the ideals b, a, it is possible to compute a basisof the quotient ideal I = a : b;

2 In his paper F. S. Macaulay, On the Resolution of a Given Modular System into Primary Sys-tems Including Some Properties of Hilbert Numbers, Math. Ann. 74 (1913) and in his book F.S. Macaulay, The Algebraic Theory of Modular Systems, Cambridge University Press (1916),where he expanded his previous result.

3 On the Resolution, op. cit. Section 1, p. 68.4 Ibid. Section 1 pp. 67–8.

30.1 The Linear Structure of an Ideal 453

(III) given, through some basis, the homogeneous ideal I ⊂ k[X0, . . . , Xn] itis possible, for each δ ∈ N, to explicitly list a k-basis of the k-vectorspace

Iδ := f ∈ I, homogeneous, deg( f ) = δ.His remarks on his assumptions are illuminating:5

It is possible to argue that all these assumptions are legitimate. (III) depends on a finitenumber of operations which can be actually performed, and from which it can also bedetermined whether a given polynomial is a member of a given H-module or not. (I) issolved for H-modules by Hilbert;[6] and although the basis of the L.C.M. found by thismethod includes many more members than necessary it can be reduced to a [minimalbasis] by (III). Also (II) is solved for H-modules by Hilbert when the basis of the firstgiven module consists of a single polynomial; and then can be solved generally, since 7

[a : (b1, . . . , bs) = ⋂i (a : bi )].

The impression is that the computational frame, as with the Kronecker Model,is a direct application of linear algebra tools; it is in this linear algebra settingthat the advanced ideal theoretical tools are stated and applied.

Algorithm 30.1.2 (Macaulay). It is important to stress that the solvability ofassumption (III) was not just a hope: assuming that an H -basis F of the (affine)ideal I ⊂ k[X1, . . . , Xn] can be computed, Macaulay had an easy algorithm 8

to perform: the algorithm iterates on increasing values d ∈ N producing thek-bases Bd of the vectorspace

I(d) := f ∈ I, deg( f ) ≤ dand it simply performs linear algebra on the set

Xi f : 1 ≤ i ≤ n, f ∈ Bd−1 ∪ f ∈ F : deg( f ) = d mod I(d − 1).

It is worth quoting this elementary approach since it is currently used inconnection with computational algorithms related to the subject of this partof this book, in order to improve both their practical performance and theirtheoretical complexity. Also, some advanced investigation of improvements tothis scheme is the basis of some interesting alternative approaches to Grobnerbasis computation.

Example 30.1.3. Let us consider the ideal I ⊂ k[X, Y, Z ] generated by theH-basis f1 := X2 − Y, f2 := XY − Z , f3 := X Z − Y 2; then we have

5 Ibid. Section 1, pp. 67–86 The reference is to Lemma 26.3.4.7 The scheme is the same as the one sketched in Lemma 26.3.4 and Proposition 26.3.5.8 Compare the quotation in Historical Remark 23.2.3.

454 Macaulay II

• B0 = B1 = ∅, B2 := f1, f2, f3;• B3 := gi , 1 ≤ i ≤ 7 where

X f1 = X3 − XY =: g1,

Y f1 = X2Y − Y 2 =: g2,

X f2 = X2Y − X Z = g2 − f3,

Z f1 = X2 Z − Y Z =: g3,

−X f3 + g3 = XY 2 − Y Z =: g4,

Y f2 = XY 2 − Y Z = g4,

Z f2 = XY Z − Z2 =: g5,

−Y f3 + g5 = Y 3 − Z2 =: g7,

Z f3 = X Z2 − Y 2 Z =: g6;• B4 := hi , 1 ≤ i ≤ 12 where

Xg1 + g2 = X4 − Y 2 =: h1,

Y g1 + g4 = X3Y − Y Z =: h2,

Xg2 = X3Y − XY 2 = h2 − g4,

Zg1 + g5 = X3 Z − Z2 =: h3,

Xg3 = X3 Z − XY Z = h3 − g5,

Xg4 + g5 = X2Y 2 − Z2 =: h4,

Y g2 = X2Y 2 − Y 3 = h4 − g7,

Xg5 + g6 = X2Y Z − Y 2 Z =: h5,

Zg2 = X2Y Z − Y 2 Z = h5 − g6,

Y g3 = X2Y Z − Y 2 Z = h5 − g6,

Xg6 = X2 Z2 − XY 2 Z =: h6,

Zg3 − h6 = XY 2 Z − Y Z2 =: h8,

Xg7 + g6 = XY 3 − Y 2 Z =: h7,

Y g4 = XY 3 − Y 2 Z = h7,

Zg4 = XY 2 Z − Y Z2 = h8,

Y g5 = XY 2 Z − Y Z2 = h8,

Zg5 = XY Z2 − Z3 =: h9,

−Y g6 + h9 = Y 3 Z − Z3 =: h12,

Zg6 = X Z3 − Y 2 Z2 =: h10,

Y g7 = Y 4 − Y Z2 =: h11,

Zg7 = Y 3 Z − Z3 = h12;• und so weiter.

Historical Remark 30.1.4. It is also worth reconsidering Macaulay’s remarkson H-bases (Historical Remark 23.2.3) in view of this construction: while the

30.1 The Linear Structure of an Ideal 455

notion is general 9 if one wants a ‘principal’, that is ‘minimal’, basis, the naturalway of constructing it proceeds by increasing degree d and extends the alreadyproduced basis Fγ−1 := f1, . . . , fh to an enlarged basis Fγ := Fγ−1 ∪ fh+1, . . . , fH , such that

Iγ = Spank(t fi , t ∈ T , fi ∈ Fγ−1, deg(t f ) ≤ γ ) Spank( fh+1, . . . , fH ).

Let, as usual, P := k[X1, . . . , Xn],

T := Xa11 . . . Xan

n : (a1, . . . , an) ∈ Nn,

m := (X1, . . . , Xn), the maximal ideal at the origin, and let us consider anideal I ⊂ P; in principle such an ideal is not necessarily homogeneous; wewill explicitly introduce such an assumption if and when we need it.

The obvious way to produce the required set of ‘independent conditions’that are to be satisfied by the members of the ideal I is to obtain them as asolution of the dual equations and this is Macaulay’s approach: let us thereforeconsider the infinite set of unknowns ξτ : τ ∈ T and let us introduce

Definition 30.1.5 (Macaulay). A dialytic equation of I is any linear combina-tion

∑τ∈T aτ ξτ ∈ k[ξτ ] satisfying

∑τ∈T aτ τ ∈ I.

For any υ ∈ T the υ-derivate of the dialytic equation∑

τ∈T aτ ξτ is thedialytic equation

∑τ∈T aτ ξτυ corresponding to the ideal member

∑τ∈T

aτ τυ = υ

(∑τ∈T

aτ τ

)∈ I.

The modular equations or inverse functions of I are the equations whichare identically satisfied by the coefficients of each and every member of I, thatis the elements∑

τ∈Tcτ ξτ ∈ k[[ξτ ]] :

∑τ∈T

cτ aτ = 0, for each∑τ∈T

aτ τ ∈ I ⊂ P.

The under-degree or order of the expression∑

τ∈T cτ ξτ is

mindeg(τ ) : τ ∈ T , cτ = 0.

9 The affinization a G of any homogeneous basis G of the homogenization h I of the affine ideal Ican be used as H-basis.

456 Macaulay II

Proposition 30.1.6 (Macaulay). If I is an ideal (respectively a homogeneousideal), then the set consisting of all inverse functions up to (respectively of)degree d and the one consisting of all dialytic equations up to (respectively of)the same degree are conjugate systems of linear equations, that is the solutionsof either system give the coefficients of the other system.

Proof. If∑

τ∈T cτ ξτ is an inverse function, then, for any∑

τ∈T aτ τ ∈P which is a member of I, we have, by definition of inverse function,∑

τ∈T cτ aτ = 0; this means that ξτ = cτ , for each τ ∈ T , is a solutionof all dialytic equations.

Conversely, any solution ξτ = cτ , for each τ ∈ T , of all dialytic equations∑τ∈T aτ ξτ satisfies the relations∑

τ∈Tcτ aτ = 0, for each

∑τ∈T

aτ τ ∈ I

so that∑

τ∈T cτ ξτ is an inverse function.The same argument proves that any solution ξτ = aτ for each τ ∈ T of

all inverse functions coincides with a dialytic equation∑

τ∈T aτ ξτ and con-versely.

30.2 Inverse System

To each inverse function∑

τ∈T cτ ξτ ∈ k[[ξτ ]] we can associate the linearfunctional γ : P → k defined by γ (τ) = cτ and which we have alreadyencoded (Remark 28.1.1) by the series

∑τ∈T cτ τ ∈ k[[X1, . . . , Xn]].

Conversely each such series∑

τ∈T cτ τ is associated to the inverse function∑τ∈T cτ ξτ .

Macaulay proposed a more illuminating notation and expressed such modu-lar equations or inverse functions or linear functionals or series as the Laurentseries∑

τ∈Tcτ τ

−1 =∑

(a1,...,an)∈Nn

ca1,...,an X−a11 . . . X−an

n ∈ k[[X−11 , . . . , X−1

n ]].

Definition 30.2.1 (Macaulay). The inverse system of the ideal I is the set ofall negative power series

∑τ∈T cτ τ

−1 which are inverse functions of I.

In general, in contrast to dialytic equations, which involve only a finite num-ber of variables ξτ , the inverse functions

∑τ∈T cτ ξτ = ∑

τ∈T cτ τ−1 can have

an infinite number of variables ξτ with a non-zero coefficient cτ = 0; in otherwords inverse functions are Laurent series.

30.2 Inverse System 457

It is clear that Macaulay’s notions are giving a linear-algebra encoding of theideal I (respectively the P-module L(I)), since each polynomial in I (respec-tively each linear functional in L(I)) is encoded by the corresponding dialytic(respectively modular) equation.

The main results discussed in this part were already formulated and provedby Macaulay in this encoded language and will be, when helpful, re-proposedhere.

Definition 30.2.2. For any υ ∈ T the υ-derivate of the inverse function E :=∑ω∈T cωω−1 is the inverse function

υE :=∑τ∈T

γτ τ−1 :=

∑τ∈T

cυτ τ−1 =

∑ω∈Tυ|ω

cωω−1υ.

Proposition 30.2.3. If E := ∑ω∈T cωω−1 is an inverse function of I such also

is its υ-derivate υE, for each υ ∈ T .

Proof. For any polynomial∑

τ∈T ατ τ ∈ I, if we set

aω :=

ατ if ω = υτ,

0 if υ ω,

we have ∑ω∈T

aωω =∑τ∈T

ατυτ = υ∑τ∈T

ατ τ ∈ I,

and ∑τ∈T

γτατ =∑τ∈T

cυτ aυτ =∑ω∈T

cωaω = 0.

In other words, via the notion of υ-derivation (Definitions 30.1.5 and30.2.2), the set of all inverse functions – as, trivially, that of all dialytic equa-tions – is a P-module. Therefore, for each f ∈ P the notion of f -derivate of adialytic (or modular) equation of I is well-defined, having the obvious meaning;moreover, such P-module structures are preserved by the subsets consisting ofall dialytic equations (respectively, inverse functions) of I .

Definition 30.2.4 (Macaulay). A zero-dimensional ideal I is said to have afinite basis E1, . . . , Eh of its inverse system if each inverse function E of Ican be expressed as a combination of derivates of the basis elements, that is as

E =h∑

i=1

E ′i =

h∑i=1

Pi Ei

458 Macaulay II

where each E ′i = Pi Ei is the Pi -derivate, Pi ∈ P , of Ei ; such a property is

denoted by

I = [E1, . . . , Eh].

The zero-dimensional ideal I is called a principal system 10 if there is a mod-ular equation E such that I = [E], that is the inverse system of I consists of themodular equation E and its derivates.

Proposition 30.2.5. Let E1, . . . , Eh be a finite set of negative power se-ries and assume that for each i , there is Fi ∈ P such that the Fi -derivateof Ei vanishes identically. Then the P-module k-generated by the generatingset

f Ei , f ∈ P, 1 ≤ i ≤ his the inverse system of an ideal [E1, . . . , Eh] = I ⊂ P.

Proof. If we set

I := f ∈ P : f Ei = 0, 1 ≤ i ≤ hthen I is a non-empty ideal since for each i and each f ∈ I, p ∈ P , we have(p f )Ei = p( f Ei ) = 0, and F := ∏

i Fi ∈ I.If we now consider the inverse system E of I, clearly [E1, . . . , Eh] ⊂ E and

we can deduce equality simply by k-dimensional argument.

Thus, the P-module (F1, . . . , Fs) (resp. [E1, . . . , Eh]) k-generated by thebasis consisting of all dialytic (resp. modular) equations Fi (resp. Ei ) and ofall their derivates defines an ideal I.

Corollary 27.12.8 implies that for a zero-dimensional affine ideal I,

dimk(P/I) =∞∑

t=1

H(t; I) = k0(I)

is finite, so that its inverse system has a finite basis.11

10 While Macaulay gives this definition in general, it is applied (and applicable) only to the case ofa simple K-N-module, that is in the case of a primary ideal at the origin. In this case, Macaulay’sdefinition has been re-labelled as ‘Gorenstein ideal’. For a formulation of the general notiongiven by Macaulay see Definition 30.5.1.

11 The study of an unmixed ideal I of rank r is reduced by Macaulay to the zero-dimensional caseby studying Iec = Ik(Yr+1, . . . , Yn)[Y1, . . . , Yr ] ∩ k[Y1, . . . , Yn ] where Yn , Yn−1, . . . , Y1is a Noether position for I. However, while the inverse system of Ie has a finite basis, thestate of the inverse system of the unmixed ideal I requires a deeper discussion (see Definition30.5.1).

30.2 Inverse System 459

When the inverse system of the ideal I has a finite basis, we can consider Ias represented either by a polynomial basis F1, . . . , Fs ⊂ P or by means ofa finite basis E1, . . . , Eh of its inverse system.12

The ideal I can therefore be seen both as the sum of the principal ideals (Fi ),

I = (F1, . . . , Fs) = (F1) + · · · + (Fs),

and as the intersection of the principal systems [E j ],

I = [E1, . . . , Eh] = [E1] ∩ · · · ∩ [Eh].

Both constructions can be seen as joining the k-linear bases of the compo-nents: the intersection is defined by joining the k-bases of the inverse functionsand the sum by joining the k-bases of the dialytic equations.

Historical Remark 30.2.6. This notation can be considered as natural becauseit goes back to Steinitz who, for a finite sequence of ideals I1, . . . , Is , denotedtheir sum (or greatest common divisor) by

I1 + · · · + Is = (I1, . . . , Is)

and their intersection (or least common multiple) by⋂i

Ii = [I1, . . . , Is].

Curiously one of the notations became common while the other is essentiallyforgotten.

One must also note that Macaulay applies the notation [E] to denote boththe module generated by all υ-derivates, υ ∈ T , of the modular equation E andthe ideal I whose inverse system consists of such a set of modular equations;we will consistently use this abuse of notation.

This dual representation has the obvious relation

Lemma 30.2.7. Let I = (F1, . . . , Fs) = [E1, . . . , Eh] be an ideal. Then, forany polynomial F ∈ P and each inverse function E, we have

(1) E ∈ [E1, . . . , Eh] iff for each i, i ≤ s, the Fi -derivate of E vanishes,that is Fi E = 0;

(2) F ∈ (F1, . . . , Fs) iff F Ei = 0 for each i, i ≤ h.

12 As Macaulay put it in The Algebraic Theory op. cit. Section 57, p. 65A module is completely determined by its system of modular equations no less than by itssystem of members. The two systems are alternative representations of the module.

460 Macaulay II

Proof.

(1) Let E = ∑ω∈T cωω−1 and Fi = ∑

τ∈T aτ τ .Since, for each υ ∈ T ,

∑τ∈T aτ τυ = υFi ∈ I, we have

E ∈ [E1, . . . , Eh] ⇒∑τ∈T

aτ cτυ = (Fiυ)E = 0

and

Fi E =∑τ∈T

aτ

∑υ∈T

cτυυ−1 =∑υ∈T

(∑τ∈T

aτ cτυ

)υ−1 = 0.

Conversely if Fi E = 0 for each i , then for each F ∈ I, F = ∑i Pi Fi ,

we have F E = ∑i Pi Fi E = 0.

(2) Let F = ∑τ∈T aτ τ and Ei = ∑

ω∈T cωω−1. Since, for each υ ∈ T ,

∑τ∈T

cυτ τ−1 =

(∑τ∈T

cυτ τ−1υ−1

)υ =

(∑ω∈T

cωω−1

)υ = Eiυ

we have

F ∈ (F1, . . . , F3) ⇒∑τ∈T

aτ cυτ = F(Eiυ) = 0.

and

F Ei =∑τ∈T

aτ

∑υ∈T

cυτ υ−1 =

∑υ∈T

(∑τ∈T

aτ cυτ

)υ−1 = 0.

Conversely if F Ei = 0 for each i , then for each E ∈ [E1, . . . , Eh],E = ∑

i Pi Ei , we have F E = ∑i Pi F Ei = 0.

This can be applied as a tool for computing colons:

Corollary 30.2.8. Let I := [E1, . . . , Ek] and J := (F1, . . . , Fs). Then

I : J = [E1, . . . , Ek] : (F1, . . . , Fs) = [. . . , Fi E j , . . .] =⋂i, j

[Fi E j ].

Proof. Let F ∈ P; we have

F ∈ (I : J) ⇐⇒ F Fi ∈ I, for each i,

⇐⇒ F Fi E j = 0, for each i, j,

⇐⇒ F ∈ [Fi E j ], for each i, j.

30.3 Linear Structure of an Ideal 461

30.3 Representing and Computing the Linear Structure of an Ideal

We have given the definitions of modular and dialytic equations without mak-ing explicit reference to their degree (or under-degree), but degree is crucial inMacaulay’s approach to constructing them.

We begin our discussion with the easiest case of a homogeneous ideal.

Algorithm 30.3.1 (Macaulay). Let us therefore assume that I is homogeneous.In this case, for each d ∈ N, the finite k-vectorspace

Id := f ∈ I : f is homogeneous, deg( f ) = dis a k-subvectorspace of the k-vectorspace

Pd := f ∈ P : f is homogeneous, deg( f ) = dgenerated by the basis

Td := τ ∈ T : deg(τ ) = d;moreover

#(Pd) =(

n + d − 1

d

), #(Id) =

(n + d − 1

d

)− hH(d; I).

Macaulay’s assumption (III) implies that for each d ∈ N

• one can explicitly list #(Id) linearly independent homogeneous polynomials∑τ∈Td

aτ τ ∈ Id ;• such polynomials provide directly a linearly indepenendent set of #(Id) ho-

mogeneous dialytic equations∑

τ∈Tdaτ ξτ = 0 of degree d;

• the solution of these dialytic equations gives h H(d; I) linearly independentvectors

ci , 1 ≤ i ≤ hH(d; I), ci = (ciτ : τ ∈ Td) ,

each giving a modular equation Ei := ∑τ∈Td

ciτ τ−1;

• the set E(d) := Ei , 1 ≤ i ≤ h H(d; I) provides a k-basis of the k-vectorspace of all homogeneous modular equations

∑τ∈Td

cτ τ−1 of degree d.

‘At least in imagination’ as Macaulay said, we can consider each such ho-mogeneous polynomial

∑τ∈Td

aτ τ as a row-vector in an infinite matrix, thedialytic array whose columns are indexed by the terms τ ∈ T and ordered bydegree-increasing value.

In the same way, the infinite set ∪d∈NE(d) can also be considered as aninfinite matrix, the inverse array, of the same kind.

462 Macaulay II

If in such matrices the rows are ordered by increasing value of degree, theseinfinite matrices would consist of ‘separate rectangular compartments whichdo not overlap one another’.[13]

Example 30.3.2. Continuing Example 30.1.3 let us consider the homogeneousideal J := H(I) ⊂ k[X, Y, Z ] generated by

H( f1) = X2, H( f2) = XY, H( f3) = X Z − Y 2.

Then we have

hH(d, H(I)) = (3+d3

), #(Id) = 0, 0 ≤ d ≤ 1

hH(d, H(I)) = 3, #(Id) = (3+d3

) − 3, 2 ≥ d

and, for d ≤ 3 we have the following dialytic and inverse arrays – whosecolumns are indexed by the ordered monomial set

1 ∩ X, Y, Z ∩ X2, XY, X Z , Y 2, Y Z , Z2∩ X3, X2Y, X2 Z , XY 2, XY Z , X Z2, Y 3, Y 2 Z , Y Z2, Z3 :

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

where the rows of the compartment of the dialytic array corresponding to de-gree 3 represent

X H( f1), Y H( f1), Z H( f1), X H( f3), Z H( f2), Y H( f3), Z H( f3);

if we apply Gaussian reduction to this compartment, the resulting rows would

13 On the Resolution, op. cit., Section 15, p. 74.


represent the polynomials

X H( f1) = X3 =: H(g1),

Y H( f1) = X2Y =: H(g2),

Z H( f1) = X2 Z =: H(g3),

−X H( f3) + H(g3) = XY 2 =: H(g4),

Z H( f2) = XY Z =: H(g5),

Z H( f3) = X Z2 − Y 2 Z =: H(g6),

−Y H( f3) + H(g5) = Y 3 =: H(g7),

If we now consider the generic case of a K-module, that is an affine ideal,I, we have technical difficulty due to the fact that, unlike for H-modules, thedialytic equations are series, instead of polynomials; while it is possible toapply again the representation of inverse functions and dialytic equations asproperly ordered row-vectors of infinite matrices whose columns are indexedby the terms τ ∈ T , some points must be explicitly addressed:

• For a homogeneous ideal, the rows encode homogeneous polynomials 14 andit is sufficient to order them by increasing value of their degree. In the non-homogeneous case, the rows of the inverse array must be ordered by increas-ing value of the under-degree of the inverse function encoded by them;15 asregards the rows of the dialytic array, Macaulay offers both ordering solu-tions:16 either via increasing value of the degree (as in the homogeneouscase) or via increasing value of the under-degree (as for inverse functions).

• The structure of the array is therefore modified: the backbone is still thesequence of ‘separate rectangular compartments which do not overlap oneanother’ each labelled by a degree (respectively: under-degree) d and con-sisting of linearly independent sub-vectors, but the vectors extend to thecolumns indexed by the terms having lower (respectively: higher) degreethan d.

• As in the homogeneous case, where the construction of both the dialytic andthe inverse arrays is performed by iteration on increasing value on the degreed,17 the same happens in the general case; but in this case, for each series

14 Notwithstanding whether they represent polynomials f ∈ I ⊂ P or the polynomial representa-tion

∑τ∈T γ (τ)τ of a linear functional γ : P → k.

15 Which is the same as the order of the series encoding the inverse function.16 Respectively in The Algebraic Theory op. cit., Section 59, p. 67 and in On the Resolution, op.

cit., Section 21, pp. 77.17 It could be argued that this construction could also be conceived in parallel on all degree blocks;

but, in the affine case, the scheme, which allowed Macaulay to ‘write down, at least in imagina-tion’, the basis required by his assumption (III) (see Algorithm 30.1.2), works only by iterationon increasing value of the degree d.

464 Macaulay II

represented by a row of the inverse array, one obtains only its truncation atdegree d; when the next degree is taken into consideration each truncatedseries must be extended.

The two alternative structural representations of the dialytic array (accord-ing to whether the rows are ordered by increasing value of their degree or theirunder-degree) are connected with two different computational approaches pro-posed by Macaulay in order to produce them.

We first discuss the computation which returns the dialytic array with therows ordered according to their degree, devoting the next section to the otherapproach.

Macaulay reduces the problem to the homogeneous one essentially adaptingAlgorithm 30.3.1 to h I by considering the infinite k-vectorspace chain

I(1) ⊂ I(2) ⊂ · · · ⊂ I(d) ⊂ I(d + 1) ⊂ · · · ⊂ I

where

I(d) := f ∈ I : deg( f ) ≤ d = I ∩ Spank(T (d))

and producing the inverse and dialytic matrices for each vector space I(d).The crucial point is that if we truncate the infinite dialytic and inverse ar-

rays of I at degree d by extracting only the finite principal minor 18 restrictedto the columns indexed by the terms of degree at most d and to the dialytic(respectively inverse) rows of degree (respectively under-degree) bounded byd, then

• the rows of the truncated dialytic array encode a k-basis of the vectorspaceI(d), and

• the rows of the truncated inverse array give a k-basis of the vectorspace ofthe truncations at degree d of the inverse functions of I.

Algorithm 30.3.3 (Macaulay). The computation of the inverse and dialytic ar-rays is performed by iteratively producing the structure of I(d); one beginsthe iteration with the minimal value µ for which

(n+µµ

) = H(µ; I) so that#(I(d)) > 0; assuming that the structure for I(d) is already available one has to

• extend the given k-basis of I(d) to one of I(d + 1), as explained in Algo-rithm 30.1.2,

• add to the dialytic array the rows representing the dialytic equations relatedto the new basis elements,

18 If we consider the inverse and dialytic arrays as two different matrices, this extraction gives justtwo submatrices. But if we interpret, as Macaulay did, the inverse and dialytic arrays as twocompartments of a single square matrix, then this extraction gives exactly the principal minorof this square matrix.


• continue each row of the inverse array up to degree d + 1 in such a way thattheir extensions also satisfy the new dialytic equations,

• extend these linear independent solutions of the dialytic equations in orderto produce a basis of the inverse functions of I(d + 1) and

• add such solutions as further rows of the inverse array.

In connection with this algorithm Macaulay remarks 19 that in the case ofa zero-dimensional ideal ‘the compartments of the dialytic array eventuallybecome square and the total number of rows of the inverse array is finite’, thusalso giving a termination condition.

In the higher-dimensional case, the dialytic and inverse arrays are infinite,as well as the k-basis; only their truncated version at degree d is computable ina finite number of loops, while ‘in theory’ the infinite computation would givethe required infinite array presentation.

Example 30.3.4. Continuing Example 30.1.3 let us now consider the ideal I ⊂k[X, Y, Z ] generated by the H-basis

f1 := X2 − Y, f2 := XY − Z , f3 := X Z − Y 2.

Then, for d = 3, we have the following dialytic and inverse arrays – whosecolumns are indexed by the ordered monomial set

1 ∩ X, Y, Z ∩ X2, XY, X Z , Y 2, Y Z , Z2∩ X3, X2Y, X2 Z , XY 2, XY Z , X Z2, Y 3, Y 2 Z , Y Z2, Z3

0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 00 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Note that, from the obvious solution for d = 2, we have to

• add to the dialytic array, the 7 arrays representing B3,• adapt the 4 rows of the modular array in order to also satisfy the new equa-

tions,20

19 The Algebraic Theory op. cit., Section 59, p. 68.20 For instance the dialytic array E := Y −1 Z−1 must be extended to E := Y −1 Z−1+X−2 Z−1+

X−1Y −2 in order to satisfy Eg3 = Eg4 = 0.

466 Macaulay II

• add the new(3+2

2

) − 7 dialytic arrays.

30.4 Noetherian Equations

The direct application of under-degree for polynomials in Macaulay’s al-gorithms requires a suitable notation, obtained by dualling the ones relatedwith H-bases: for any polynomial (or even series) f ∈ k[[X1, . . . , Xn]]we denote by L( f ) its lowest-degree non-zero homogeneous component andord( f ) := deg(L( f )) its order or under-degree; for an ideal I, L(I) denotesthe ideal L( f ) : f ∈ I; the same notation is implicitly extended to dialyticequations and inverse functions.

Macaulay’s approach using the under-degree of the dialytic equations be-gins with the remark that for any inverse function E representing a Laurentpolynomial of degree d and any polynomial f ∈ P

ord( f ) > d ⇒ E f = 0, and, more generally,E f = Eg for g := Can( f, md+1) ∈ Spank(T (d)), so thatE is therefore a modular equation for md+1, andthe set of all modular equations of I having degree bounded by d coincides

with the set of all modular equations of I + md+1.

This suggests the following notion characterizing, within the set of inversefunctions, those Laurent series which are just Laurent polynomials:

Definition 30.4.1 (Macaulay). An inverse function∑

τ∈Tcτ τ

−1 for which there

exists γ ∈ N such that

deg(τ ) > γ ⇒ cτ = 0

is called a Noetherian equation.

Historical Remark 30.4.2. Noether, of course, is Max and not Emmy. The his-tory of the terminology is quite curious.21 Macaulay introduced the terminol-ogy of

• H-module, or Hilbert-module, to characterize homogeneous ideal – whencethe notion of H-basis of the affine ideal I as the result of dehomogenizingany (homogeneous) basis of h I,

• K-module, or Kronecker-module, to characterize affine ideals, and• N-module, or Noether-module, to characterize the primaries at the origin.

21 Ibid. Sections 6–9, pp. 69–71; mainly the third footnote of p. 71.

30.4 Noetherian Equations 467

But Max Noether observed that the notion of ‘Noether-module’ had alreadybeen introduced by Lasker to characterize the ideals contained in the maximalideal at the origin.

As a consequence Macaulay was forced to use Moore’s notion of simplemodule, that is ‘primary associated to a linear maximal ideal’ and then label‘simple Noether-module’ the primaries associated at the origin.

Any primary q associated to the origin contains some power ofm := (X1, . . . , Xn), q ⊃ mρ , where ρ denotes its characteristic number; there-fore, for each τ ∈ T ,

deg(τ ) ≥ ρ ⇒ τ ∈ mρ ⊂ q

so that for each inverse function∑

τ∈T cτ τ−1 of q we have

cτ = 0, for each τ ∈ T , deg(τ ) ≥ ρ,

that is the equation is Noetherian and its degree is bounded by ρ − 1.In particular, the inverse system 1−1 is satisfied exactly by those polynomi-

als that vanish at the origin; therefore it defines the maximal ideal m and iscontained in the inverse system of any m-primary ideal, and in general, in anideal I ⊂ m.

On the other hand, an inverse function associated to a point b :=(b1, . . . , bn) ∈ kn different from the origin is not Noetherian, actually it canbe easily described as ∑

(a1,...,an)∈Nn

ba11 . . . ban

n X−a11 . . . X−an

n .

On the basis of these remarks, in order to compute the Noetherian equationsof I,22 Macaulay suggests considering the infinite k-vectorspace chain

I + m ⊃ I + m2 ⊃ · · · ⊃ I + md ⊃ I + md+1 ⊃ · · · ⊃ I

and iteratively computing inverse and dialytic arrays of each I + md .This approach which returns only the Noetherian equations of I does not

give the complete structure of I but it produces that of its m-closure I0 :=⋂d I + md ⊃ I. The whole structure, that is also the inverse functions related

to I1, can be found by means of a change of origin.23

22 All over this section, we are implicitly restricting our consideration to an ideal I ⊂ m.23 For which Macaulay proposed an ingenious notation: he suggested expressing each coefficient

cτ of an inverse function E := ∑τ∈T cτ τ−1 as

cτ := ca11 . . . can

n where τ = Xa11 . . . Xan

n .

Then if we move the origin to a point b := (b1, . . . , bn) ∈ kn for which we obtain the modular

468 Macaulay II

If we consider the infinite dialytic and inverse arrays of the m-closure of Iand we truncate them at degree d , extracting only the finite principal minorrestricted to the columns indexed by the terms of degree at most d and tothe dialytic (respectively inverse) rows of under-degree (respectively degree)bounded by d, then

• the rows of the truncated dialytic array encode, equivalently,

• a k-basis of the vectorspace(I + md+1

) ∩ Spank(T (d)),• a k-basis of the vectorspace of the truncations at degree d of the polyno-

mials f ∈ I,• a k-basis of the vectorspace f ∈ I + md+1, deg( f ) ≤ d;

• the rows of the truncated inverse array give

• a k-basis of the vectorspace of the truncations at degree d of the inversefunctions of I, and

• a k-basis of the vectorspace of the inverse functions of I + md+1.

It is sufficient to adapt Algorithm 30.3.3 operating on the chain of the idealsI(d +1), in order to deduce an algorithm operating along the chain of the idealsI + md ; for doing that, one has just to interchange the role of degree and order(under-degree).

Algorithm 30.4.3 (Macaulay). One begins the iteration with the minimal valuel1 := minord( f ), f ∈ I, f = 0 of the under-degree of the non-zero elementsof I, and, assuming that the structure for I+md has already been obtained, onehas to

and inverse equations

F := ∑τ∈T aτ τ = ∑

(a1,...,an )∈Nn a(a1,...,an ) Xa11 . . . Xan

n ,

E := ∑τ∈T cτ τ−1 = ∑

(a1,...,an )∈Nn ca11 . . . can

n X−a11 . . . X−an

n

and we move the origin back to its original position, that is to the point (−b1, . . . , −bn), F andE are to be transformed as

F ′ :=∑

(a1,...,an )∈Nna(a1,...,an )(X1 − b1)a1 . . . (Xn − bn)an ,

E ′ :=∑

(a1,...,an )∈Nn(c1 + b1)a1 . . . (cn + bn)an X

−a11 . . . X−an

n

where in the expansion each occurrence of ca11 . . . can

n is replaced by cτ where τ =X

a11 . . . Xan

n .

Note that if E = 1 then

E ′ =∑

(a1,...,an )∈Nnb

a11 . . . ban

n X−a11 . . . X−an

n

as we have previously remarked.


• extend (as will be explained in Algorithm 30.4.5) the given k-basis of I+md

to one of I + md+1, thus obtaining a finite set F of polynomials having theunder-degree d and such that L( f ) : f ∈ F is a k-linear basis of thevectorspace L( f ) : f ∈ I, ord( f ) = d,

• add to the dialytic array the rows representing the dialytic equations relatedto the new basis elements,

• find a k-basis of the set of all the inverse functions of degree exactly d whichsatisfy all the dialytic equations already found (this includes also those ofdegree less than d) and

• add such solutions in the form of further rows of the inverse array, thusproducing a basis of the inverse functions of I + md+1.

As Macaulay remarked,24 the situation is analogous to that of the previousalgorithm: a finite computation up to degree d returns only the Noetherianequations having a degree bounded by d , while the infinite computation wouldreturn the infinite complete set of Noetherian equations; the case of a simpleNoetherian module 25 is characterized by a termination condition and, at termi-nation, the algorithm returns the finite basis of the Noetherian equations, thatis the inverse system of the m-primary ideal

⋂I + md :

we can proceed similarly to find in theory the whole of the Noetherian array.. . .the whole system of modular equations of a Noetherian module can be expressed as asystem of Noetherian equations.. . .If . . . the rows of the compartment l1 + i of the dialytic array should be equal in numberto the power products of degree l1 + i there will be no Noetherian equation of absolutedegree ≥ l1 + i . In this case the Noetherian equations are then the modular equations ofthe simple Noetherian module contained in the given module. The simple module itselfis [I + ml1+i ] and l1 + i is its characteristic number.

Thus the simple modules at isolated points of a given module M can all be found bymoving the origin to each point in succession and finding its Noetherian equations andcharacteristic number.

Example 30.4.4. Continuing Example 30.3.4 let us now consider the ideal I ⊂k[X, Y, Z ] generated by the basis

f1 := X2 − Y, f2 := XY − Z , f3 := X Z − Y 2;

24 The Algebraic Theory op. cit., Section 65–66, p. 75.25 That is an ideal whose m-closure

⋂I + md is m-primary.

470 Macaulay II

clearly, for I + m3 we have the following dialytic and inverse arrays:0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Then:

• we obtain a k-basis of I + m4 by adding

X2Y − X4, X2 Z − XY 2, XY 2 − X3Y, XY Z − Y 3, X Z2 − Y 2 Z ,

Y 3 − X3Y, Y 2 Z − X2Y Z , Y Z2 − XY 2 Z , Z3 − XY Z2,

• so that we have to add a further inverse function, namely

X−3 + X−1Y −1 + Z−1,

and we obtain the following truncated arrays0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0

The crucial difference between Algorithm 30.4.3 and Algorithm 30.3.3 isthat the role of degree in the latter is played in the former by the order (under-degree). In both cases we need to produce a k-basis of I whose elements areordered by increasing value of their, respectively, degree and under-degree, orin other words, we need to deal with, respectively, H(I) and L(I).

However, the computation of an H-basis is needed by Macaulay in order toestablish not that H(I) = H(F) but just that no element f = ∑

i pi fi existsfor which deg( f ) < deg(pi fi ), so that Algorithm 30.1.2 produces a k-basisordered by increasing degree; but a k-basis ordered by increasing under-degreecan be 26

26 The Algebraic Theory op. cit., Section 65, p. 74.


obtained from any basis of M , which need not be an H-basis

since the non-existence of elements f = ∑i pi fi for which ord( f ) <

ord(pi fi ) is obvious.

Algorithm 30.4.5 (Macaulay). Thus there is no need of precomputation in theadaptation of Algorithm 30.1.2 which returns what Macaulay called 27 a ‘com-plete standard set’ of an m-primary ideal q whose characteristic number is ρ.

The algorithm, which is performed by increasing value of the order (orunder-degree) of the elements, requires knowledge of a basis F of q and, foreach d < ρ, returns a properly ordered k-basis Bd of(

q + md+1)

∩ Spank(T (d))

so that Bρ−1 ∪ τ ∈ T , deg(τ ) ≥ ρ is the k-basis of q.It consists of

initializing µ := minord( f ) : f ∈ F, Bi := ∅, i < µ;iterating on d = µ, . . . , ρ − 1 by

setting Cd := τ f : τ ∈ T , f ∈ F, ord(τ f ) = d,performing linear algebra on the set Dd := L( f ), f ∈ Cd, thusreturning a set B ⊂ Cd such that L( f ), f ∈ B is a k-basis of the

k-vectorspace Spank(Dd) andsetting Bd := Bd−1 ∪ B;

testing whether #B = (n+d−1d

) = #Td gives a termination condition, whichallows us to deduce that d = ρ and that the required basis of q isBd−1.

Obviously, an infinite computation would return a ‘complete standard set’of the m-closure of any ideal q.

Example 30.4.6. Continuing Example 30.3.4, let us produce a complete stan-dard set of the m-primary ideal I + m4 where I ⊂ k[X, Y, Z ] is generated bythe basis (see Example 30.4.4)

f1 := X2 − Y, f2 := XY − Z , f3 := X Z − Y 2 :

µ := 1, C1 := f1, f2, D1 := −Y, −Z, B1 := f1, f2,C2 := f3, X f1, Y f1, Z f1, X f2, Y f2, Z f2,D2 := X Z − Y 2, −XY, −Y 2, −Y Z , −X Z , −Y Z , −Z2,B := X Z − Y 2, −XY + X3, −Y 2 + X2Y, −Y Z + X2 Z , −Z2 + XY Z,

27 On the Resolution op. cit., Section 22, p. 78.

472 Macaulay II

a k-basis extracted by D3 is

X2Y, X2 Z−XY 2, XY 2, XY Z−Y 3, X Z2−Y 2 Z , Y 3, Y 2 Z , Y Z2, Z3,thus giving

B := X2Y − X4, X2 Z − XY 2, XY 2 − X3Y ∪ XY Z − Y 3, X Z2 − Y 2 Z , Y 3 − X3Y ∪ Y 2 Z − X2Y Z , Y Z2 − XY 2 Z , Z3 − XY Z2.

Let f1, . . . , fs be a basis of I and let f ∈ P be any polynomial whichsatisfies all Noetherian equations of I. For any d ∈ N, this implies that fsatisfies all Noetherian equations of I whose degree is bounded by d, that is allNoetherian equations of I+md+1, and that there are polynomials p1, . . . , ps ∈P such that f − ∑

i pi fi ∈ md+1. Since this holds for each d ∈ N, it impliesthe existence of series p1, . . . , ps ∈ k[[X1, . . . , Xn]] such that f = ∑

i pi fi .Therefore, by the Lasker Theorem (Corollary 27.7.6), the m-closure of I is

generated by f1, . . . , fs in k[[X1, . . . , Xn]], that is⋃d

I + md = ( f1, . . . , fh)k[[X1, . . . , Xn]] ∩ k[X1, . . . , Xn].

Macaulay formalized this property, giving the following

Definition 30.4.7 (Macaulay). For an (affine) ideal I ⊂ m an N-set is a basis f1, . . . , fh such that

for each f ∈ I, there exists p1, . . . , ph ∈ k[[X1, . . . , Xn]] : f =h∑

i=1

pi fi .

The N-set f1, . . . , fh is called principal if it is minimal in the sense thatno subset f1, . . . , fi−1, fi+1, . . . , fh has the same property.

Proposition 30.4.8. Any principal N-set of an affine ideal I ⊂ m is fixed innumber.

Proof. For each principal N-set f1, . . . , fh ⊂ I and for each

pi ∈ k[[X1, . . . , Xn]], 1 ≤ i ≤ h,

it clearly necessarily holds that∑i

pi fi = 0 ⇒ p1(0) = · · · = ph(0) = 0.

Let us consider any other principal N-set f ′1, . . . , f ′

h′ for I; our aim is toprove that h = h′, by producing a contradiction with the assumption h > h′.


Since for each i, 1 ≤ i ≤ h′, and each j, 1 ≤ j ≤ h, there are elementspi j , p′

j i ∈ k[[X1, . . . , Xn]] such that

f ′i =

∑j

pi j f j , f j =∑

i

p′j i f ′

i , for each i, j, 1 ≤ i ≤ h′, 1 ≤ j ≤ h,

if we set

ai j :=

pi j (0) if i ≤ h′,0 if h′ < i ≤ h

and a′j i :=

p′

j i (0) if i ≤ h′,0 if h′ < i ≤ h,

from the above relation we deduce

f j =∑

l

∑i

p′j i pil fl , for each i, j, 1 ≤ i ≤ h′, 1 ≤ j ≤ h,

and the inconsistent relations

h∑i=1

a′j i ail =

h′∑i=1

a′j i ail = δ jl

which claim the mutual invertibility of the matrices(

a′j i

)and

(ai j

)whose last

h − h′ columns (respectively, rows) are null-vectors.

Algorithm 30.4.9 (Macaulay). Knowledge of a complete standard set – that isa k-basis properly ordered via under-degree – for the m-primary ideal q allowsus to compute an N-set of q by dualling the computation discussed in HistoricalRemark 30.1.4 in order to produce a ‘principal’, that is minimal, basis of it:28

This will comprise all the members of lowest under-degree, and also any of a highestunder-degree i which (in respect to the terms of degree i) are independent of the otherprincipal members of under-degree i previously chosen combined with derivates ofprincipal members previously chosen of under-degree < i . . . . Any principal standardset of members of a given simple K-N-module comprises a fixed number of membersof each assigned under-degree (§ 48).[29] In the above we may of course substitute“dialytic equation” to “member”.

In other words, we begin with Fµ := Bµ and iterate, for d = µ + 1..ρ

enlarging the basis by setting Fd := Fd−1 ∪ fh+1, . . . , fk, so that

Spank(Dd) = Spank(τ L( f ), τ ∈ T , f ∈ Fd−1, ord(τ f ) = d Spank(L( fh+1), . . . , L( fk)).

where Dρ = Spank(T (ρ)) and Dd , d < ρ, is the output of Algorithm 30.4.5.

28 On the Resolution op. cit., Section 22, p. 79.29 The reference is to a formulation of Proposition 30.4.8 slightly stronger than that I presented

here.

474 Macaulay II

Algorithm 30.4.10 (Macaulay). Macaulay then provided 30 a procedurewhich, from an N-set

F1, . . . , Fk′ , ord(F1) ≤ ord(F2) ≤ · · · ord(Fk′),

of the m-primary ideal I + md+1 where d is sufficiently large,31 returns a prin-cipal N-set for I:

Modify F2, F3, . . . , Fk′ by means of F1 to F(1)2 , F(1)

3 , . . . , F(1)k′ so as to have under-

degree as high as possible. Let F(1)2 be of as low an under-degree as any one of

F(1)2 , F(1)

3 , . . . , F(1)k′ . Modify F(1)

3 , . . . , F(1)k′ by means of F1, F(1)

2 to F(2)3 , . . . , F(2)

k′so to have under-degree as high as possible.

Proceeding in this way we arrive at a set F(k−1)k , F(k−1)

k+1 , . . . , F(k−1)k′ such that when

F(k−1)k+1 , . . . , F(k−1)

k′ are modified by

F1, F(1)2 , . . . , F(k−1)

k

they all appear to admit of indefinitely high under-degree.

In order to terminate one needs, of course, to prove the correctness of what‘appears’, that is one needs to prove that

• each F (k−1)i , k + 1 ≤ i ≤ k′, has a representation

F (k−1)i =

k∑j=1

Pji F ( j−1)j , Pji ∈ k[[X1, . . . , Xn]].

This is equivalent to the statement that

• each Fi , k + 1 ≤ i ≤ k′, has a representation

Fi =k∑

j=1

Q ji Fj , Q ji ∈ k[[X1, . . . , Xn]].

Such a statement can be computationally tested since, by the Lasker Theorem(Corollary 27.7.6), this is equivalent to the statements that

• each ideal (F1, . . . , Fk) : Fi , k + 1 ≤ i ≤ k′, is not contained in m.

If this test succeeds then 32

30 On the Resolution op. cit., Section 23, p. 79.31 While Macaulay does not explicitly make this remark, it is clear that the N-set F1, . . . , Fk′

can be properly enlarged, adding the elements in the principal standard N-set of I+md+1 havingunder-degree d + 1 any time the current basis has been modified so as to have under-degree ashigh as d + 1.

Note that the iterative loops of both Algorithms 30.4.5 and 30.4.9 can be indefinitely extendedso as to enlarge both the k-basis and the N-set of I + md to ones of I + md+1.



F1, . . . , Fk is a principal N-set of [I] and F1, F(1)2 , . . . , F(k−1)

k a correspondingprincipal standard N-set.

In fact, for any polynomial F ∈ I and any polynomial representation F =∑k′i=1 Pi Fi simple iterated substitution allows us to obtain the series represen-

tation F = ∑ki=1 Si F (i−1)

i

Example 30.4.11. Continuing Example 30.4.6 it is clear that f1, f2, f3 is astandard N-set; the simple reduction

f3 − X f2 = X2Y − Y 2, f3 − X f2 + Y f1 = 0

gives that f1, f2 is a principal N-set.

Historical Remark 30.4.12. Although it is stimulating to imagine that this isthe first appearance of the notion of Hironaka’s standard bases, because wefind in the same paragraph both the concept and the name, this is not the case:in the definition, it is useless 33 to require the property ord( f ) ≤ ord(pi fi ) andMacaulay remarks 34 that there are instances in which ord( f ) > ord(pi fi ):

If F1, . . . , Fk is a principal standard N-set of members of M it is not necessarilytrue that every member F of M is of the form P1 F1 + P2 F2 + · · · + Pk Fk whereP1 F1, P2 F2, . . . , Pk Fk are all of under-degree as high as F .Ex. If

F1 = x1x3 + φ3, F2 = x2x3 + ψ3,

where φ3, ψ3 are of under-degree 3, then x1 F2 − x2 F1 is of under-degree 4 but not ofthe form P1 F1 + P2 F2 where P1 F1, P2 F2 are both of under-degree as high as 4.

Algorithm 30.4.13 (Macaulay). Macaulay 35 also dualed Algorithm 30.4.5 inorder to extract a ‘principal’, that is minimal, finite basis of the inverse systemof the m-primary ideal q from the finite set of its Noetherian equations:36

Similarly a complete standard set of N-equations of a simple K-N-module is any com-plete linearly independent set such that the number of equations of each and everydegree, starting with the highest degree γ − 1,[37] is made as small as possible. Forany complete standard set we can pick out a principal standard set. These will compriseall the equations of highest degree and also any of lowest degree i which (in respectto terms of degree i) are independent of the other principal equations of degree i pre-viously chosen combined with derivates of principal equations previously chosen of

33 As we remarked, for his applications, Macaulay needs to discard the possibility ord( f ) <

ord(pi fi ) but has no reason to require that L(I) = L(F).34 On the Resolution op. cit., Section 23, p. 80.35 On the Resolution op. cit., Section 22, p. 79.36 This is the original algorithm from which stemmed the algorithms discussed in Sections 32.3

and 29.6.37 For which γ denotes the characteristic number of q.

476 Macaulay II

degree > i . Any principal standard set of N-equations of a given simple K-N-modulecomprise a fixed number of equations of each assigned degree (§ 49).

In other words, denoting by

γ the characteristic number of q,E the complete set of N-equations of q produced by Algorithm 30.4.3,

and writingfor each N-equation E = ∑

τ∈T cτ τ−1 ∈ E,

δ(E) := max(deg(τ ) : cτ = 0), H(E) :=∑

τ∈T (δ(E))

cτ τ−1,

for each d < γ , Dd := e ∈ E : ord(e) = dwe begin by setting Fγ−1 := Dγ−1 and iterate, for d = γ − 2..0 enlarging thebasis by setting Fd := Fd+1 ∪ Eh+1, . . . , Ek, so that

Spank(Dd) = Spank(υH(E), υ ∈ T , E ∈ Fd+1, deg(υE) = d Spank(H(Eh+1), . . . , H(Ek)).

Example 30.4.14. Continuing Example 30.4.4 and writing

E0 := 1−1, E1 := X−1, E2 := X−2+Y −1, E3 := X−3+X−1Y −1+Z−1

we set

F3 := D3 = E3,F2 := F3, since D2 = E2 = X E3,F1 := F3, since D1 = E1 = X2 E3,F0 := F3, since D0 = E0 = X3 E3.

Algorithm 30.4.15 (Macaulay). An alternative algorithm for the computationof the structure of an m-primary ideal q whose characteristic number ρ isknown 38 can be performed by decreasing induction on the degree; it requiresthe knowledge of a ‘complete standard set’ B of q, which can be obtained viaAlgorithm 30.4.5.

It simply consists of producing these Noetherian equations as the solution,for d = ρ−1, . . . , 0, of the dialytic equations in Bd := f ∈ B : ord( f ) ≥ d.

For each d < ρ − 1 the solutions are of two different kinds:

• some are simply the continuations of solutions found in the previous (d + 1)step;

• the others are new Noetherian equations having degree d.

38 And equivalently to deduce all the Noetherian equations of degree bounded by ρ −1 of an idealI ⊂ m by computing those of I + mρ .


Solving the dialytic equations Bd is simplified by the remark that the υ-derivates of the Noetherian equations found in the previous step are solutions.

Example 30.4.16. This algorithm being quite trivial in the example we have sofar discussed,39 I will now consider the ideal

q := ( f, X2 Z , X Z2) + m4, f = Y − X2 − Z2

whose dialytic and inverse arrays are0 0 1 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 00 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 -1 00 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 -10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

where

solving the dialytic equations in B3 returns E1 := Z−3 and E2 := X−3;solving the dialytic equations in B2 means

including E3 := Z E1 := Z−2 and E5 := X E2 := X−2,extending E1 := Z−3 + Y −1 Z−1 and E2 := X−3 + X−1Y −1,adding the new solution E4 := X−1 Z−1;

solving the dialytic equations in B1 means

including E6 := Z2 E1 := Z−1 and E7 := X2 E2 := X−1, andextending E3 := Z−2 + Y −1 and E5 := X−2 + Y −1;

finally, solving B0 we include E7 := 1−1.

Historical Remark 30.4.17. It is interesting to note that the notions of canoni-cal form, Grobner and linear representation were present in Macaulay:40

If M is a module of rank n the number of its modular equations is finite and equal to thesum

∑µ of the multiplicities of its simple modules. In order that we may have F = 0

mod M the coefficients of F must satisfy∑

µ equations (which will not be independent

39 At the preliminary step we begin with a Noetherian equation X−ρ+1; the next step simplyintroduces, by Xρ−1−d -derivation, the new Noetherian equation X−d , and computes the termsof degree d of the Noetherian equations already found.


478 Macaulay II

unless F is of sufficiently high degree). Any set of∑

µ linearly independent polynomi-als such that no linear combination of them is a member of M is called a complete set ofremainders for M ; and has the property that any polynomial F which is not a memberof M is congruent mod M to a unique linear combination of the set of the remainders.The simplest way of choosing a complete set of remainders is to take the polynomial 1of degree 0, then as many power products of degree 1 as possible, then as many powerproducts of degree 2 as possible, and so on, till a set of

∑µ power products has been

obtained of which no linear combination is a member of M . We shall call any such seta simple complete set of remainders for M ,

and, more explicitly, 41

To a simple module M corresponds a set of µ [= #(N<(M))] polynomials of which nolinear combination is a member of M , and such that any other polynomial is congruentas regards M to a linear combination of the µ polynomials. For certain points of viewit may be considered preferable to make this property serve for the definition [42] of themultiplicity µ of M .

Algorithm 30.4.18. Macaulay 43 even gave an algorithm to produce, for an m-primary ideal, its Grobner representation from its inverse system:

If M = [E1, E2, . . . , Ek ] is a simple Noetherian module no member E of the system[E1, E2, . . . , Ek ] can have the same coefficients (assumed real) as a member F of M ;for if E and F had the same coefficients the sum of their squares would be zero. Henceif the members of the system [E1, E2, . . . , Ek ] have the power products changed fromnegative to positive 44 they will form a complete set of remainders for M .

Example 30.4.19. In Example 30.4.16 we have the Grobner representation

q := 1, X, Z , X2 + Y, X Z , Z2 + Y, X3 + XY, Z3 + Y Z.

30.5 Dialytic Arrays of M(r) and Perfect Ideals

Up to this point we have discussed the k-linear algebra structure of an ideal,mainly in relation with m-primary and m-closed ideals, where m is the maxi-mal ideal at the origin; of course up to the obvious translation which takes theorigin to another point this discussion covers all the zero-dimensional cases.

In principle, the same approach allows us to manage higher-dimensionalideals M ⊂ k[x1, . . . , xn], whose rank is r , by considering their exten-sions/contractions

M (r) := Mk(xr+1, . . . , xn)[x1, . . . , xr ] ∩ k[x1, . . . , xn]

where xr+1, . . . , xn is a maximal set of independent variables for M .

41 On the Resolution op. cit., Section 27, p. 82.42 Definition 27.12.9.43 The Algebraic Theory op. cit., Section 68, p. 79.44 That is the inverse system

∑τ∈T cτ τ−1 is changed to the polynomial

∑τ∈T cτ τ .

30.5 Dialytic Arrays and Perfect Ideals 479

In his discussion Macaulay assumes that he has performed a ‘generic’change of coordinates beforehand; this implies that, if M = ⋂s

i=1 qi is theprimary decomposition and ri = n − dim(qi) is the rank of each component,then

xn, xn−1, . . . , x1 is a Noether position for Mxri +1, . . . , xn is a maximal set of independent variables for qi

qi ∩ k[xr+1, . . . , xn] = (0) ⇐⇒ ri ≥ r.

Since, by assumption, r = n − dim(M) ≤ n − dim(qi), for each i , we have

qik(xr+1, . . . , xn)[x1, . . . , xr ] ∩ k[x1, . . . , xn] =

qi iff r = ri

(1) iff r < ri

and

M (r) =⋂

i

(qik(xr+1, . . . , xn)[x1, . . . , xr ] ∩ k[x1, . . . , xn]

) =⋂

i :ri =r

qi

is the top-dimensional component of M ; in particular if M is unmixed, thenM (r) = M .

The assumption of having performed a ‘generic’ change of coordinates hasanother consequence, namely that

• each member in the basis of M used is of the same degree in x1, x2, . . . , xr

as in x1, x2, . . . , xn .Therefore, if we denote by

∆ : k[xr+1, . . . , xn][x1, . . . , xr ] → N

the degree induced by the weight

∆(xi ) :=

1 i ≤ r,0 i > r,

the last statement can be formalized as

• ∆(Fi ) = deg(Fi ) for each member Fi of the given basis of M .

I report here Macaulay’s words, 45 limiting myself to following his argumenton an easy but not trivial example.

77. We have hitherto specially considered modules of rank n, that is, modules whichresolve into simple modules. The H-module of rank n is of special type, since it is itselfa simple module, and its equations are homogeneous. The general case of a moduleof rank n is therefore that of a module which is not an H-module. When however weconsider a module of rank < n it is of some advantage to replace it by its equivalentH-module, which is of the same rank but of greater dimensions by 1. We shall not avoid

45 The Algebraic Theory op. cit., Section, 77–82, pp. 85–91.

480 Macaulay II

by this means the consideration of modules which are not H-modules, but the resultsobtained will be expressed more conveniently. We shall therefore assume that the givenmodule M whose modular equations and properties are to be discussed is an H-modulein n variables x1, x2, . . . , xn .

By treating any H-module M of rank r (whether mixed or unmixed) as a module M(r)

in r variabales x1, x2, . . . , xr it will resolve into simple modules and have only a finitenumber of modular equations, viz. a number µ equal to the sum of the multiplicitiesof its simple modules. The unknowns in the modular equations will be represented bynegative powers products of x1, x2, . . . , xr while the coefficients will be whole func-tions of the parameters xr+1, . . . , xn .[46] The module determined by these modularequations will be unmixed, viz. the L.C.M. of all the primary modules of M of rank r(§ 43); and will be the module M itself if M is unmixed. We proceeed to discuss theseequations and shall call them the r-dimensional modular equations of M (or the modu-lar equations of M(r)) since they are obtained by regarding the module M as a moduleM(r) in space of r dimensions. M(r) is not an H-module.The dialytic array of M(r). We choose any basis [47] (F1, F2, . . . , Fk) of M as thebasis of M(r). This is not in general an H-basis of M(r).

46 In other words, the ideal M ⊂ k[x1, . . . , xn ] is considered as the integral ideal

M(r) := Mk(xr+1, . . . , xn)[x1, . . . , xr ] ∩ k[x1, . . . , xn ]

and the corresponding inverse functions as members of

k[xr+1, . . . , xn ][[x−11 , . . . , x−1

r ]].

47 Throughout these comments we will consider the ideal

M := (y1 y2, y22 , y3 y2, y3

1 , y3 y21 , y2

3 y1)

on which we will first perform the linear change of coordinates

y1 = x1 + x3, y2 = x2 + x3, y3 = −x1 − x2 + x3.

Therefore we have

M := (y1 y2, y22 , y3 y2, y3

1 , y3 y21 , y2

3 y1)

= (y1, y2) ∩ (y31 , y3 y2

1 , y1 y2, y22 , y3 y2, y2

3 )

= (x1 + x3, x2 + x3)

∩ ((x1 − 2x3)2, x3x2 + x23 , x2

2 − x23 , x1x2 + x3x1, x2

3 x1, x33 )

= (F1, F2, F3, F4, F5, F6)

where

F1 = x1x2 + x3x1 + x3x2 + x23 ,

F2 = x22 + 2x3x2 + x2

3 ,

F3 = −x1x2 − x22 − x3x1 + x2

3 ,

F4 = x31 + 3x3x2

1 + 3x23 x1 + x3

3 ,

F5 = −x31 − x2

1 x2 − x3x21 − 2x3x1x2 + x2

3 x1 − x23 x2 + x3

3 ,

F6 = x31 + 2x2

1 x2 + x1x22 − x3x2

1 + x3x22 − x2

3 x1 − 2x23 x2 + x3

3 ;


The module Mxr+1=...=xn=0[48] determined by the highest terms of the members of

the basis of M(r) is of rank r (assuming that x1, x2, . . . , xr have been subjected to a lin-ear homogeneous substitution beforehand) and is therefore a simple H-module whosecharacteristic number will be denoted by γ .

Construct [49] a dialytic array for M(r) whose elements are whole functions ofxr+1, . . . , xn in which each row represents an elementary member ωi Fj of M(r),

as a consequence, in the two frames we have

M(r) = M(2) := Mk(y3)[y1, y2] ∩ k[y1, y2, y3] = (y1, y2),

M(r) = M(2) := Mk(x3)[x1, x2] ∩ k[x1, x2, x3] = (x1 + x3, x2 + x3),

and µ = 1.

Note that in k[y1, y2, y3], for each primary component qi, ri = n − dim(qi), yri +1, . . . , yn

is a maximal set of independent variables for qi, but y3 y2, y3 y21 , and y2

3 y1 have a differentdegree in y1, y2 from that in y1, y2, y3.

On the other hand, for each Fi , deg(Fi ) = ∆(Fi ).48 Writing

W := xa11 . . . xar

r : (a1, . . . , ar ) ∈ Nr ,

each F ∈ k[xr+1, . . . , xn ][x1, . . . , xr ] can be uniquely expressed as

F =∑

τ∈Wc(F, τ )τ, c(F, τ ) ∈ k[xr+1, . . . , xn ].

Then, denoting by

π : k[xr+1, . . . , xn ][x1, . . . , xr ] → k[x1, . . . , xr ]

the projection defined by

π(F) = F(x1, . . . , xr , 0, . . . , 0], for each F(x1, . . . , xr , xr+1, . . . , xn),

we have Mxr+1=...=xn=0 := π(M).

Since we assume that we have performed a generic change of coordinates so that

xn , xn−1, . . . , x1 is a Noether position for M ,xr+1, . . . , xn is a maximal set of independent variables for M ,xi is integral over k[xr+1, . . . , xn ] for each i ≤ r ,

we know that, for each i ≤ r , there is a monic polynomial f ∈ k[xr+1, . . . , xn ][xi ] such that

f ∈ M ∩ k[xr+1, . . . , xn ][xi ], f /∈ k[xr+1, . . . , xn ],

so that M(r) is zero-dimensional.Also, since π(F) = ∑

τ∈W c(F, τ )(0)τ and, by assumption, the basis is homogeneous inx1, . . . , xr , xr+1, . . . , xn we have

c(π(F), τ ) = c(F, τ )(0) ∈ k \ 0 ⇐⇒ deg(τ ) = deg(F).

Therefore, the assumption that ∆(Fi ) = deg(Fi ) for each basis element Fi implies that π(Fi ) =0.Note that in the two frames we have

My3=0 := (y1 y2, y22 , y3

1 ),

Mx3=0 := (x1x2, x22 , x3

1 ),

and γ = 3.49 In the example, I follow the notation used by Macaulay in the examples presented in On the

482 Macaulay II

where ωi is a power product of x1, x2, . . . , xr (see § 59). The first set of rows willrepresent the members of the basis which are of lowest degree l, the next set a completeset of elementary members of degree l + 1 which are linearly independent of one an-other and of the complete rows in the first set, the next set a complete set of elementarymembers of degree l + 2 linearly independent of one another and of the complete rowsin the first two sets, and so on.[50]

Resolution, op. cit., Sections 44–46, pp. 93–96.Note that, if this notation is preserved while applying Grobner theory in an interpretation ofMacaulay’s notion of perfectness, it forces the aporetic ordering x3 < x1 < x2 of variables.We use this variable ordering in our comments, but we will choose a more consistent variableordering when discussing perfectness in Chapter 36.While we truncate this infinite computation at degree γ = 3, the complete solution will bediscussed in Example 32.7.5.

50 This set in degree 2 is:

1 x1 x2 x21 x1x2 x2

2r1 F1 x2

3 x3 x3 0 1 0r2 F2 x2

3 0 2x3 0 0 1r3 F3 x2

3 −x3 0 0 −1 −1

which can be transformed by Gaussian reduction to

1 x1 x2 x21 x1x2 x2

2t1 3x2

3 0 3x3 0 0 0r1 F1 x2

3 x3 x3 0 1 0r2 F2 x2

3 0 2x3 0 0 1

where t1 = r3 + r2 + r1.In degree 3 we have:

1 x1 x2 x21 x1x2 x2

2 x31 x2

1 x2 x1x22 x3

2t1 3x2

3 0 3x3 0 0 0 0 0 0 0

r1 F1 x23 x3 x3 0 1 0 0 0 0 0

r2 F2 x23 0 2x3 0 0 1 0 0 0 0

r4 F4 x33 3x2

3 0 3x3 0 0 1 0 0 0

r5 x1 F1 0 x23 0 x3 x3 0 0 1 0 0

r6 x2 F1 0 0 x23 0 x3 x3 0 0 1 0

r7 x2 F2 0 0 x23 0 0 2x3 0 0 0 1

r8 x1 F3 0 x23 0 −x3 0 0 0 −1 −1 0

r9 x2 F3 0 0 x23 0 −x3 0 0 0 −1 −1

r10 F5 x33 x2

3 −x23 −x3 −2x3 0 −1 −1 0 0

r11 F6 x33 −x2

3 −2x23 −x3 0 x3 1 2 1 0

r12 x1 F2 0 x23 0 0 2x3 0 0 0 1 0

which can be transformed by Gaussian reduction to

1 x1 x2 x21 x1x2 x2

2 x31 x2

1 x2 x1x22 x3

2t3 9x3

3 9x23 0 0 0 0 0 0 0 0

t1 3x23 0 3x3 0 0 0 0 0 0 0

t2 3x33 6x2

3 0 3x3 0 0 0 0 0 0

r1 F1 x23 x3 x3 0 1 0 0 0 0 0

r2 F2 x23 0 2x3 0 0 1 0 0 0 0

r4 F4 x33 3x2

3 0 3x3 0 0 1 0 0 0

r5 x1 F1 0 x23 0 x3 x3 0 0 1 0 0

r6 x2 F1 0 0 x23 0 x3 x3 0 0 1 0

r7 x2 F2 0 0 x23 0 0 2x3 0 0 0 1


In comparing this with the scheme of § 59 [51] there is the obvious difference that theelements of the array are whole functions of xr+1, . . . , xn instead of pure constants; andthere is the more important difference that the compartments l, l + 1, . . . do not neces-sarily consist of independent rows, because the array is not constructed from an H-basisof M(r). It is only the complete rows of the array that are independent. The elements inthe compartments are all pure constants independent of xr+1, . . . , xn . The diagram of§ 59 serves perfectly well to illustrate the dialytic array although its properties are nowdifferent.[52]

In each compartment we choose a set of independent rows such that all the remainingrows of the compartment are dependent on them,[53] and we name them regular rowsand extra rows respectively, and apply the same terms to the complete rows of whichthey form part. In the compartment γ the regular rows will form a square array, and thesame will be true of the compartments γ +1, γ +2, . . .. Eventually a compartment δ ≥γ will be reached such that the number of rows in the whole array for degree δ is exactlyµ less than the whole number of columns, where µ is the number of modular equationsof M(r) as mentioned above. After this all succeeding compartments δ + 1, δ + 2, . . .

will consist of square arrays only without any extra rows.[54]

We can now modify any extra row [55] of the array by regular rows so as to make allits elements which project beyond the columns of degree γ − 1 vanish, and this leavesits elements in the columns up to degree γ − 1 whole functions of xr+1, . . . , xn of thesame degree as they were before.[56] If this is done with all the extra rows projectingbeyond the columns of degree γ − 1 the array may be said to be brought to its regularform in which the whole number of rows of the array for degree γ − 1 is µ less than thewhole number of columns, and all the compartments γ, γ +1, . . . are made square. The

where

t1 = r3 + r2 + r1,

t2 = r10 + r4 + r5 + x3r1,

t3 = r11 − r4 − 2r5 − r6 + 3x3r1 + 2t2,

and

r8 = −r6 − r5 − x3r2 − 2x3r1 + x3t1,

r9 = −r7 − r6 + 3x3r2 − x3t1,

r12 = r6 − x3r2 + x3r1

are linearly dependent on the others.51 That is the structure obtained by Algorithm 30.3.3.52 It is sufficient to glance at the arrays computed in order to verify these claims.

Note that the statement that ‘the elements in the compartments are all pure constants’ holdsonly because ∆(Fi ) = deg(Fi ) for each Fi .

53 We chose the sets r1, r2 and r4, r5, r6, r7. Note also that the rows r8, r9, r12 are not part ofthe array; they were only added in order to allow us to check their dependence. The remainingthree rows r3, r10, r11 are the three extra rows.

54 In our example we have γ = 3 and µ = 1; the value δ is exactly 3: in fact we have 10 columns,indexed by the 10 terms of degree at most 3, and 9 independent rows, namely ri , 1 ≤ i ≤7, 10 ≤ i ≤ 11.

55 The extra rows are r3, r10 and r11 and we have already reduced them via regular rows whileperforming Gaussian reduction; this computation produces the ‘regular forms’ t1, t2, t3.

56 Because the elements are homogeneous and therefore in each ‘compartment’ the degree of theentries is determined.

484 Macaulay II

extra rows, modified so as to end at the columns of degree γ − 1, represent members ofM(r) of degree γ − 1 which are not elementary members ωi Fj .

We may further modify the regular forms of the complete array for degree γ − 1 soas to reduce the number of rows in each compartment γ − 1, γ − 2, . . . successivelyto independent rows. The elements of some of the rows of the array for degree γ − 1may thus become fractional in xr+1, . . . , xn and the whole number of compartmentswill in general be increased, so that the last (or first) compartment will be numberedl ′ < l.[57] Supposing this to be done we can choose a simple complete set of remain-ders for M(r) consisting of all power products of x1, . . . , xr of degree < l ′ and as manypower products of each degree l ′′ ≥ l ′ as the number of columns of the compartment l ′′exceeds the number of rows of the same. We denote these power products in ascend-ing degree by ω1, ω2, . . . , ωµ (so that ω1 = 1) and all remaining power products toinfinity in ascending degree by ωµ+1, ωµ+2, . . . The two series ω1, ω2, . . . , ωµ and

ωµ+1, ωµ+2, . . . overlap in respect to the degrees of their terms.[58]

The basis of M used for constructing the dialytic array of M(r) must be one in whicheach member is of the same degree in x1, x2, . . . , xr as in x1, x2, . . . , xn .[59] We shallsay that M is a perfect module if the array of M(r) as originally constructed has noextra rows, i.e. if the basis (F1, F2, . . . , Fk) is an H-basis of M(r).[60]

57 As in this example where 1 = l ′ < l = 2.58 In our case, we simply have ω1 = 1.59 If this property is not satisfied the crucial statement that ‘the elements in the compartments are

all pure constants’ does not hold.60 Here we reach the crucial point: Macaulay, for his further computations, needed the absence

of extra rows; such rows vanish in the compartment related to their degree; they representpolynomials

F =∑

τ∈Wc(F, τ )τ, c(F, τ ) ∈ k[xr+1, . . . , xn ],

for which

c(F, τ ) = 0 ⇒ deg(τ ) < deg(F) ⇒ c(F, τ ) ∈ (x1, . . . , xr ).

If we now consider any ordering <1 on T ∩ k[xr+1, . . . , xn ] and any degree-compatible order-ing <2 on W and < denotes the corresponding block ordering, and we assume that the columnsof the dialytic array are ordered by increasing value of their term-index, then, clearly, for anyrow r in the dialytic array, writing

F = ∑τ∈W c(F, τ )τ = ∑

ω∈T c(F, ω)ω, the polynomial represented by r ,ω := T<(F),τ := T<2 (F), that is the column-index corresponding to the rightmost non-vanishing entry

in r ,υ := T<1 ( fτ ) where fτ := c(F, τ ) ∈ k[xr+1, . . . , xn ],

then we have ω = υτ , that is T<(F) = T<1 ( fτ )T<2 (F).Now if F is represented by the regular form of an extra row, then F is obtained by lin-ear algebra reduction of a polynomial ωi Fj , ∆(ωi Fj ) > ∆(F), by means of polynomialsω′

i F ′j , ∆(ω′

i F ′j ) > ∆(F), so that we have a representation

F =∑

i

Gi Fi , ∆(F) < ∆(Gi Fi )

so that H(F) /∈ H(Fi ) : 1 ≤ i ≤ k.Therefore the following statements are equivalent:


78. Solution of the dialytic equations of M(r). We return to what has been calledabove the regular form of the dialytic array of M(r). Each row represents a member ofM(r) and supplies a congruence equation mod M(r). Solving these equations, regard-ing ωµ+1, ωµ+2, . . . as the unknowns, we have

Dωp + Dp1ω1 + Dp2ω2 + · · · + Dpµωµ = 0 mod M(r) (p = µ + 1, µ + 2, . . .).

There are two slightly different cases according as the degree of ωp < γ or ≥ γ . If ωpis of degree < γ we use the regular form of the array for degree γ − 1 for solving forωp . D is then the determinant of this array formed from the columns corresponding toωµ+1, ωµ+2, . . . , and Dpi the determinant formed from the columns corresponding to

ωµ+1, . . . , ωp−1, ωi , ωp+1, . . .. [61] If ωp is of degree ≥ γ we must use the array upto the degree of ωp in order to solve for ωp . D is the same as in the former case exceptfor a factor independent of xr+1, . . . , xn (since the compartments γ, γ + 1, . . . aresquare and all their elements are pure constants) by which the equation can be divided.

• there exists an extra row;• there exist υ ∈ T ∩ k[xr+1, . . . , xn ], τ ∈ W satisfying

υτ ∈ T<(M), τ ∈ T<(M);• (F1, . . . , Fk ) is not an H-basis of M(r).

Macaulay’s notion of perfectness for an ideal M ⊂ k[x1, . . . , xn ] of rank r given by a basis(F1, . . . , Fk ), where x1, . . . , xn , is a ‘generic’ frame satisfying

• xri +1, . . . , xn is a maximal set of independent variables for each primary component qi, ri =n − dim(qi),

• for each Fi , deg(Fi ) = ∆(Fi ),

is the following:

The module M is perfect if, for each υ ∈ T ∩ k[xr+1, . . . , xn ], τ ∈ W we have

υτ ∈ T<(M) ⇒ τ ∈ T<(M).

We will show in the next part that this definition is completely equivalent to depth(M) =dim(M).

61 Up to degree γ − 1 = 2 we can solve for

ω2 := x1, ω3 := x2, ω4 := x21 , ω5 := x1x2, ω6 := x2

2

in terms of ω1 = 1. We have

D =

∣∣∣∣∣∣∣∣∣∣∣

9x23 0 0 0 0

0 3x3 0 0 0

6x23 0 3x3 0 0

x3 x3 0 1 00 2x3 0 0 1

∣∣∣∣∣∣∣∣∣∣∣, D2 =

∣∣∣∣∣∣∣∣∣∣∣∣

9x33 0 0 0 0

3x23 3x3 0 0 0

3x33 0 3x3 0 0

x23 x3 0 1 0

x23 2x3 0 0 1

∣∣∣∣∣∣∣∣∣∣∣∣,

D3 =

∣∣∣∣∣∣∣∣∣∣∣∣

9x23 9x3

3 0 0 0

0 3x23 0 0 0

6x23 3x3

3 3x3 0 0

x3 x23 0 1 0

0 x23 0 0 1

∣∣∣∣∣∣∣∣∣∣∣∣, D4 =

∣∣∣∣∣∣∣∣∣∣∣∣

9x23 0 9x3

3 0 0

0 3x3 3x23 0 0

6x23 0 3x3

3 0 0

x3 x3 x23 1 0

0 2x3 x23 0 1

∣∣∣∣∣∣∣∣∣∣∣∣,

486 Macaulay II

Also Dpi is a sum of products of determinants of the regular form of the array for

degree γ − 1 with determinants from the remaining rows of the larger array,[62] so thatthe H.C.F. of the determinants of the array for degree γ − 1 can be divided out,[63] andwe obtain in both cases

(A) Rωp + Rp1ω1 + · · · + Rpµωµ = 0 mod M(r) (p = µ + 1, µ + 2, . . . , ).

This equation is homogeneous in x1, . . . , xn and each Rpi is homogeneous inxr+1, . . . , xn . Also, owing to the fact that the remainders ω1, ω2, . . . , ωµ are a simpleset [64] each ωp is congruent modM(r) to a linear combination of the power productsω1, ω2, . . . , ωµ which are of equal or less degree than ωp . Hence Rpi vanishes if the

degree of ωi exceeds the degree of ωp. Also R = 1 if M is perfect [65] (cf. § 81).

D5 =

∣∣∣∣∣∣∣∣∣∣∣∣

9x23 0 0 9x3

3 0

0 3x3 0 3x23 0

6x23 0 3x3 3x3

3 0

x3 x3 0 x23 0

0 2x3 0 x23 1

∣∣∣∣∣∣∣∣∣∣∣∣, D6 =

∣∣∣∣∣∣∣∣∣∣∣∣

9x23 0 0 0 9x3

30 3x3 0 0 3x2

36x2

3 0 3x3 0 3x33

x3 x3 0 1 x23

0 2x3 0 0 x23

∣∣∣∣∣∣∣∣∣∣∣∣,

so that

D = 81x43 , D2 = D3 = 81x5

3 , D4 = D5 = D6 = −81x63 .

62 For degree γ = 3 and

ω7 = x31 , ω8 = x2

1 x2, ω9 = x1x22 , ω10 = x3

2

we have

D =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

9x23 0 0 0 0 0 0 0 0

0 3x3 0 0 0 0 0 0 06x2

3 0 3x3 0 0 0 0 0 0x3 x3 0 1 0 0 0 0 00 2x3 0 0 1 0 0 0 0

3x23 0 3x3 0 0 1 0 0 0

x23 0 x3 x3 0 0 1 0 00 x2

3 0 x3 x3 0 0 1 00 x2

3 0 0 2x3 0 0 0 1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

,

so that

D = 81x43 and D7 = D8 = D9 = D10 = 81x7

3 .

Since the value of each entry of the new rows along the diagonal is 1, in this case D is exactlythe same as before.

63 Dividing out D = x43 we obtain the solution

ω2 = ω3 = −x3ω1,

ω4 = ω5 = ω6 = x23ω1,

ω7 = ω8 = ω9 = ω10 = −x33ω1.

64 That is give a linear representation (see Historical Remark 30.4.17).65 The leading term of the polynomial (A) is ωpT<(R) which is equal to ωp because M is perfect.


79. The modular equations of M(r). If the coefficient of ωp = x p11 x p2

2 . . . x prr

in the general member of M(r) of any degree is represented by ω−p =(x p1

1 x p22 . . . x pr

r )−1 we have

ω−1ω1 + ω−2ω2 + · · · + ω−pωp + · · · = 0 mod M(r),

and, by (A),

R(ω−1ω1 + . . . + ω−µωµ) =∞∑

p=µ+1

ω−p(Rp1ω1 + Rp2ω2 + · · · + Rpµωµ

)mod M (r).

Here coefficients of ω1, ω2, . . . , ωµ on both sides are equal, i.e. [66]

(B) Rω−i =∞∑

p=µ+1

Rpi ω−p (i = 1, 2, . . . , µ).

This is the complete system of modular equations of M(r), or r-dimensional modularequations of M, and the system includes all its own derivates. R and all the Rpi aredefinite whole functions of xr+1, . . . , xn. If any other complete system were given andsolved for ω−1, ω−2, . . . , ω−µ in terms of ω−µ−1, ω−µ−2, . . . the result would be theunique system (B).

Since in (A) Rωp and Rpi ωi are of the same degree in x1, x2, . . . xn , so in (B),Rω−i and Rpi ω−p are of the same degree, i.e. all terms in one equation (B) are of

the same degree in x1, x2, . . . , xn .[67] Also since (§ 78) Rpi vanishes if the degreeof ωi exceeds the degree of ωp there is no ω−p on the right-hand side of (B) of lessabsolute degree than ω−i ; but every ω−p of the same degree as ω−i and not amongω−1, ω−2, . . . , ω−µ will appear on the right-hand side of (B).

(B) is the complete system of r -dimensional equations of the L.C.M. of all the pri-mary modules of M of rank r ; and will decompose into separate distinct systems corre-sponding to the separate primary modules of rank r if M has more than one irreduciblespread of rank r .

The n-dimensional equations. We can obtain the whole system of n-dimensionalequations of M corresponding to the system (B) as follows: ω−p or (x p1

1 x p22 . . . x pr

r )−1

represents the whole coefficient of x p11 x p2

2 . . . x prr in the general member of M(r), i.e.

it stands for ∑(x p1

1 x p22 . . . x pn

n )−1xpr+1r+1 . . . x pn

n ,

the summation extending to all values of pr+1, . . . pn only. If this be substituted foreach (x p1

1 x p22 . . . x pr

r )−1 in each of the equations (B) the whole coefficients of the

66 In our, quite trivial, example we obtain

1−1 = x3x−11 + x3x−1

2 − x23 x−2

1 − x23 x−1

1 x−12 − x2

3 x−22

+ x33 x−3

1 + x33 x−2

1 x−12 + x3

3 x−11 x−2

2 + x33 x−3

2 + · · · .

67 These remarks on homogeneity can be verified within the example.

488 Macaulay II

power products of xr+1, . . . , xn will represent the n-dimensional equations.[68] Thiswill be the whole system of n-dimensional equations of M if M is umixed, as we shallassume hereafter is the case.

The whole system of modular equations of a mixed module may be regarded asconsisting of the separate systems corresponding to the primary modules into which itresolves.

80. The system of homogeneous equations

(C) Rω−i =∑

Rpi ω−p (i = 1, 2, . . . , µ)

obtained from the system (B) by retaining only those terms on the right hand in whichRpi and ω−p are of the same degree as R and ω−i respectively is the complete systemof equations of the simple H-module determined by the highest terms in x1, . . . , xr ofthe members of an H-basis of M(r).[69]

This can be seen by considering the diagram of § 59 [70] assuming that it had beenconstructed from an H-basis of M(r). The compartments l, l + 1, l + 2, . . . in the twoarrays in § 59 are the dialytic and inverse arrays of the simple H-module determinedby the highest terms of the members of the H-basis; and the modular equations ofthis simple H-module are represented by the compartments 0, 1, . . . , l, l + 1, . . . of theinverse array. The system (C) is that which is represented by the compartments of theinverse array.

81. If R = 1 the module M (assumed unmixed) is perfect. Since M is unmixed everywhole member of M(r) is a member of M (§ 43). Also, since R = 1, there is an inversearray of M(r) each of whose compartments consists of independent rows in which all

68 Continuing our example we obtain∑i

xi3(xi

3)−1 =∑

i

xi+13 (x1xi

3)−1 +∑

i

xi+13 (x2xi

3)−1 −∑

i

xi+23 (x2

1 xi3)−1

−∑

i

xi+23 (x1x2xi

3)−1 −∑

i

xi+23 (x2

2 xi3)−1 +

∑i

xi+33 (x3

1 xi3)−1

+∑

i

xi+33 (x2

1 x2xi3)−1 +

∑i

xi+33 (x1x2

2 xi3)−1 +

∑i

xi+33 (x3

2 xi3)−1 + · · · ,

whence, for each i :

0 = −x−i3 + x−i+1

3

(x−1

1 + x−12

)− x−i+2

3

(x−2

1 + x−11 x−1

2 + x−22

)+ x−i+3

3

(x−3

1 + x−21 x−1

2 + x−11 x−2

2 + x−32

)+ · · · .

In Example 32.7.5 I will be able to prove that the complete extension of these infinite modularequations is

0 =i∑

d=0

(−1)d x−i+d3

⎛⎝ ∑

τ∈Wd

τ−1

⎞⎠ =: Ei .

69 This remark allows the computation of the inverse functions of H(M(r)) as

M(r) = [E1, . . . , Er ] ⇒ H(M(r)) = [L(E1), . . . , L(Er )].

70 The structure obtained by Algorithm 30.3.3.


the elements are pure constants. Hence there is a corresponding dialytic array havingthe same property. From this it follows that M is perfect (§ 77).

82. The r -dimensional and n-dimensional equations of M . If the system (B) isa principal system, i.e. if all its equations are derivates of a single one of them, eachsimple module of M(r) is a principal system; for if F is a polynomial containing all thesimple modules of M(r) except one, then [M(r) : (F)] is the last one, and is a principalsystem (§ 62).[71] The converse is also true (see § 72). Also the unmixed module M inn variables is a principal system, as we proceed to prove.72

Let the r -dimensional equation of which all the equations of the system (B) arederivates be

∞∑Rp1,p2,...,pr (x p1

1 x p22 . . . x pr

r )−1 = 0,

where Rp1,p2,...,pr is a homogeneous polynomial in xr+1, . . . , xn of degree p1 + p2 +· · · + pr + δ. The integer δ may be negative, but the more unfavourable case for theproof is that in which it is positive. Let cp1,p2,...,pn be the coefficient of x

pr+1r+1 . . . x pn

nin Rp1,p2,...,pr , so that pr+1 + · · · + pn = p1 + p2 + · · · + pr + δ. To convert theequation into an n-dimensional equation we put

(x p11 x p2

2 . . . x prr )−1 =

∞∑q

xqr+1r+1 . . . xqn

n (x p11 x p2

2 . . . x prr x

qr+1r+1 . . . xqn

n )−1

as in § 79, and we have∑p

cp1,...,pn xpr+1r+1 . . . x pn

n∑

qx

qr+1r+1 . . . xqn

n (x p11 . . . x pr

r xqr+1r+1 . . . xqn

n )−1 = 0, (1)

or, equating the whole coefficient of xlr+1r+1 . . . xln

n to zero,[73]

∑p

cp1,p2,...,pn

(x p1

1 . . . x prr x

lr+1−pr+1r+1 . . . xln−pn

n

)−1 = 0, (2)

71 This is a trival consequence of Corollary 30.2.8: the assumptions are

M(r) =⋂

iqi = [E], F ∈

⋂i = j

qi , F /∈ q j

which imply

q j = M(r) : F = [F E]

proving that q j is a principal system.72 The result below states that if M(r) is a principal system, so also is M , but this requires the

notion of ‘principal system’ for a non-zero-dimensional ideal to be specified. Such a definitionwill be provided in Definition 30.5.1 below.

73 From the principal equation, we already deduced

0 = 1−1.

0 = −x−13 + x−1

1 + x−12 ,

0 = −x−23 + x−1

1 x−13 + x−1

2 x−13 − x−2

1 − x−11 x−1

2 − x−22 ,

0 = −x−33 + x−1

1 x−23 + x−1

2 x−23 − x−2

1 x−13 − x−1

1 x−12 x−1

3 − x−22 x−1

3

+ x−31 + x−2

1 x−12 + x−1

1 x−22 + x−3

2· · ·

490 Macaulay II

which is homogeneous and of absolute degree lr+1 +· · ·+ ln −δ. Similarly the generaln-dimensional equation obtained from the coefficient of x

mr+1r+1 . . . xmn

n in the

xt11 . . . xtr

r -derivate of (1) is

∑p

cp1,p2,...,pn

(x p1−t1

1 . . . x pr −trr x

mr+1−pr+1r+1 . . . xmn−pn

n

)−1 = 0, (3)

where t1, . . . , tr , mr+1, . . . , mn are any n fixed positive integers (including ze-ros) such that t1 + · · · + tr ≤ a fixed limit τ (since there are only a finitenumber of linearly independent derivates of the original r -dimensional equation)

and(

x p1−t11 . . . x pr −tr

r xmr+1−pr+1r+1 . . . xmn−pn

n

)−1is zero if any one of the indices

p1 − t1, . . . , pr − tr , mr+1 − pr+1, . . . , mn − pn is negative.Consider all the n-dimensional modular equations of degree l, that is, all the equa-

tions of the system (3) of absolute degree l. The absolute degree of (3) is

mr+1 + · · · + mn − δ − t1 − · · · − tr = l.

Hence each of mr+1, . . . , mn is equal to or less than l + δ + τ ; and every equation(3) of absolute degree l is a derivate of the single equation (2) if lr+1, . . . , ln are allchosen as high as l + δ + τ. Hence there is a single equation of which all the modularequations of M of degree l are derivates, and any equation (2) in which lr+1, . . . , ln arenot numerically specified will serve for the single equation.[74]

The result hinted at in the last section can be formalized as follows:

Definition 30.5.1 (Macaulay). An ideal I, dim(I) > 0, is called a principalsystem if there is a chain of zero-dimensional principal systems Ii := [Ei ]such that I = ⋂

i Ii and I1 ⊃ I2 ⊃ · · · ⊃ Ii ⊃ Ii+1 ⊃ · · · ⊃ I.

As a consequence of this definition, if we avoid the ambiguity in the no-tation, interpreting each of the modules [Ei ] as the module generated by allυ-derivates, υ ∈ T , of the modular equation Ei and we denote by E the in-verse system of I, we have

Corollary 30.5.2. For an ideal I, dim(I) > 0, which is a principal system,using the notation above, the following hold:

0 =i∑

d=0

(−1)d x−i+d3

⎛⎝ ∑

τ∈Wd

τ−1

⎞⎠ .

74 If we set, for each i ,

Ei :=i∑

d=0

(−1)d x−i+d3

⎛⎝ ∑

τ∈Wd

τ−1

⎞⎠

it is easy to verify that, for each i and each υ ∈ W , we have υEi = Ei−deg(υ). Thus, anyequation Ei of ‘unspecified’ value i returns the partial section E j , 0 ≤ j ≤ i of the setE j , j ∈ N which however does not return a basis of the complete inverse system which in

fact is ∑∞i=1 λi Ei :

∑∞i=1 λi Xi ∈ k[[X ]].


for each i, j, i < j, Ei ∈ [E j ],for each i, j, i < j, there is P ∈ P such that Ei = P E j ,[E1] ⊂ [E2] ⊂ · · · ⊂ [Ei ] ⊂ [Ei+1] ⊂ · · · ⊂ E ,there is an infinite sequence e1, . . . , er , . . . of modular equations such that

for each i , [Ei ] = Spank(e1, . . . , eri ), where ri := deg(Ii ), andE = ∑∞

i=1 λi ei :∑∞

i=1 λi T i ∈ k[[T ]].

The example we have computed throughout this section illustrates the struc-ture of Macaulay’s definition. Another illuminating example will be discussedin Example 32.7.4.

We conclude this section by illustrating the result hinted at by Macaulay inthe first paragraph of Section 72:75

Proposition 30.5.3 (Macaulay). Let I be a zero-dimensional ideal and let I =∩r

i=1qi be its irredundant primary decomposition.Then I is a principal system if and only if each qi is such.

Proof. Let us assume that each component qi is a principal system and, foreach i , let us denote by

Ei the modular equation such that qi = [Ei ],γi the characteristic number of qi ,ai the first coordinate of its root.

Our aim is to prove that, for E := E1 + E2 + · · · + Er , [E] = [E1, . . . , Er ]holds; since one inclusion is trivial, it is sufficient to prove that E1 ∈ [E].

Up to a change of coordinates, we can wlog assume a1 = ai for each i = 1and, for simplicity, let us assume a1 = 0. Let

∑∞j=1 c j X j

1 ∈ k[[X1]] be theseries such that

r∏i=2

(X1 − ai )γi

∞∑j=1

c j X j1 = 1

and write

P(X1) := ∏ri=2(X1 − ai )

γi∑γ1−1

j=1 c j X j1 ∈ k[X1],

S(X1) := ∑∞j=γ1

d j X j1 = ∏r

i=2(X1 − ai )γi

∑∞j=γ1

c j X j1 ∈ k[[X1]],

which satisfy P(X1) = 1 − S(X1).Since, for each i , i ≥ 1, and each γ ≥ γi , we have (X1 − ai )

γ ∈ qi sothat (X1 − ai )

γ Ei = 0, we can deduce both P(X1)Ei = 0 for i ≥ 2 and


492 Macaulay II

S(X1)E1 = 0 so that

E1 = (1 − S(X1)) E1 = P(X1)E1 = P(X1)E .

Conversely, assume I = [E] is a principal system and let F ∈ P be suchthat F ∈ ⋂

i = j qi , F /∈ q j ; then, as a consequence of Corollary 30.2.8 we haveq j = I : F = [F E].

30.6 Multiplicity of Primary Ideals

Following Definition 27.12.9 we recall that for a zero-dimensional ideal

I ⊂ k[X1, . . . , Xn] = P

its degree, or multiplicity, deg(I) can be equivalently characterized (Corol-lary 27.12.8) as

• the k-dimension of P/I,• the constant value k0(I) = HI(T ),• #(N<(I)), w.r.t. any term ordering <.

Finally, for a higher-dimensional unmixed ideal I, its degree is (Defini-tion 27.13.7) that of Ie = Ik(Xr+1, . . . , Xn)[X1, . . . , Xr ] where

dim(I) = n − r, I ∩ k[Xr+1, . . . , Xn] = (0).

Consideration of the inverse system associated to a p-primary ideal q al-lowed Macaulay to introduce the notion of multiplicity of q, in terms of thelength deg(q) := µ of an ascending refined chain 76 of p-primary ideals

p = q1 ⊃ · · · ⊃ qi ⊃ qi+1 · · · ⊃ qµ = q :

A primary module can be shown to be built up of a certain number of what maybe called layers (illustrated roughly by the multiple layers of wrappings in which asolid object may be enveloped), and its resolution consists in removing the layers,one at a time. The number of these layers is called the multiplicity of the primarymodule.[77]

We will discuss here its illustration, restricting ourselves to the case of aprimary at the origin; the restriction is wlog since

• if I belongs to a maximal ideal, other than the origin, the result is obtainedby just performing translation;

76 It is ‘refined’ in the sense that it cannot be further refined by inserting another primary qi ⊃q′ ⊃ qi+1.


30.6 Multiplicity of Primary Ideals 493

• if I has rank r the refined chain

p(r) = q1 ⊃ q2 ⊃ · · · ⊃ qµ−1 ⊃ qµ = Ie

returns the refined chain

p = qc1 ⊃ qc

2 ⊃ · · · ⊃ qcµ−1 ⊂ qc

µ = I

where qci := qi ∩ k[X1, . . . , Xn] for each i .

Lemma 30.6.1 (Macaulay). Let q be a primary at the origin, deg(q) = µ.Then there is an ordered set of inverse functions e1, . . . , eµ such that

• q = [e1, . . . , eµ],• for each i ≤ µ,

• Spank(e1, . . . , ei ) is closed under derivation,• dimk(Spank(e1, . . . , ei )) = i.

Proof. Let us consider any finite basis E1, . . . , Eh of the inverse system ofq and let γ be the characteristic number of q.

Therefore the inverse system is generated by the set E of all the υ-derivatesof each Ei , υ ∈ T (γ ), and has dimension µ:

E := υEi , υ ∈ T (γ ), 1 ≤ i ≤ h, dimk(Spank(E)) = µ.

One can order E so that

τ Ei ωE j ⇐⇒ i < j or i = j and deg(τ ) > deg(ω),

so that, for each E ∈ E and each τ ∈ T , τ E = 0 implies τ E E .

One can then trim E , removing each element E such that

E ∈ Spank(E ′ ∈ E, E ′ E).

The result is a sequence of inverse functions e1, . . . , eµ which satisfies therequired properties.

Corollary 30.6.2. Let q be a primary at the origin, deg(q) = µ.Let e1, . . . , eµ be any ordered set of inverse functions satisfying the prop-

erties above and, for each i , define qi := [e1, . . . , ei ]. Then

• qi is a primary ideal at the origin, for each i;• deg(qi ) = i, for each i;• p = q1 ⊃ q2 ⊃ · · · ⊃ qµ−1 ⊃ qµ = q.

494 Macaulay II

The intermediate elements qi in the chain, as Macaulay 78 put it:

can be chosen in any order and with a considerable amount of latitude, being subjectonly to the conditions that each one chosen must contain [79] the one of the nearestlower multiplicity previously chosen (i.e. its modular equations must include those ofthe other) and must be contained in the one of nearest higher multiplicity previouslychosen.

Algorithm 30.6.3 (Macaulay). Conversely, given any finite set e1, . . . , eµ ofµ linearly independent modular equations of the primary ideal at the origin q,deg(q) = µ, Macaulay shows 80 how to extract a subset

E1, . . . , Et ⊂ e1, . . . , eµsuch that q = [E1, . . . , Et ].

It is sufficient to

• enumerate the set in such a way that

ord(e1) ≥ · · · ≥ ord(ei ) ≥ ord(ei+1) ≥ · · · ≥ ord(eµ);• initialize t := 1, Et := e1, ν := 1;• then:

• set ν := ν + 1,• check whether eν ∈ [E1, . . . , Et ],• if this is not the case, set t := t + 1, Et := eν

until ν > µ.

30.7 The Structure of Primary Ideals at the Origin

In connection with the result of Algorithm 30.4.18, let us introduce the follow-ing notation:

• for any Noetherian inverse system E := ∑τ∈T cτ τ

−1, F(E) denotes thepolynomial F(E) := ∑

τ∈T cτ τ ∈ P;• dually, for any polynomial F := ∑

τ∈T cτ τ ∈ P , E(F) denotes the Noethe-rian inverse system E(F) := ∑

τ∈T cτ τ−1,

so that E(F) : F ∈ P is the set of all Noetherian inverse systems at theorigin.81

78 On the Resolution op. cit., Section 29, p. 83.79 In Macaulay’s (geometric) terminology: ‘the ideal I contains the ideal J’ means that I ⊂ J.80 This result is essentially a reformulation of Algorithm 30.4.13.81 But not the whole set of inverse systems which also contains inverse systems represented not

just by polynomials but also by series.

30.7 Primary Ideals at the Origin 495

Proposition 30.7.1 (Macaulay). Let q = [E1] be a Noetherian principal sys-tem (i.e. an m-primary ideal) and F1, . . . , Fµ be a complete set of remain-ders (a Grobner representation) of q; write q1 := q : m.

Then there is a unique polynomial

F ∈ q1 ∩ Spank(F1, . . . , Fµ)

such that q = [E(F)].

Proof. We can wlog assume that

[E1] consists of a k-basis E1, E2, . . . , Eµ, whereE2, . . . , Eµ = [E1] \ E1 consists of derivates of E1, so thatord(Ei ) < ord(E1), for each i , and thatFi = F(Ei ), for each i .

With these assumptions, since each Ei is a derivate, we have

Spank(E2, . . . , Eµ) = [X1 E1, . . . , Xn E1] = [E1] : (X1, . . . , Xn) = q : m.

As a consequence F2, . . . , Fµ is a Grobner representation of q1. This impliesthat F1 has a Grobner description F1 = ∑µ

i=2 λi Fi mod q1 and

F := F1 −µ∑

i=2

λi Fi ∈ q1 ∩ Spank(F1, . . . , Fµ).

Also E(F) = E1−∑µi=2 λi Ei and the assumption ord(Ei ) < ord(E1) allow

us to conclude that [E(F)] = [E1].

Corollary 30.7.2. Let e1, . . . , eµ be any set of µ linearly independent mod-ular equations of the primary ideal at the origin q, deg(q) = µ, and let

E1, . . . , Et ⊂ e1, . . . , eµ

be the subset extracted from it by Algorithm 30.6.3. Then, writing

E ′ := υEi , υ ∈ T \ 1, 1 ≤ i ≤ t,

496 Macaulay II

we have

• q = [E1, . . . , Et ]• q : m = SpanK (E ′).

Proof. Lemma 30.6.1 and Algorithm 30.6.3 allow us to deduce that

[E1, . . . , Et ] = q.

Also we have

q : m =⋂

i

[Ei ] : m =⋂

i

[Ei ] \ Ei

where the last equality follows by Proposition 30.7.1.

Algorithm 30.7.3 (Macaulay). This principal standard set E1, . . . , Et ofN-equations produced by Algorithm 30.6.3 has a role in Macaulay’s approachto determining embedded primaries. The scenario is the following: we have anideal M = ⋂s

i=1 qi ⊂ P and we have deduced all the isolated prime compo-nents whose intersection we denote M (0) = ⋂s

i=r+1√

qi ; the colon operation

returns M : M (0) = ⋂ri=1 q′

i where, for each j ≤ r ,√

q j =√

q′j , but, in gen-

eral, deg(q′j ) =: µ′ < µ = deg(q j ); once we deduce

√q j , j ≤ r , the problem

is to deduce an embedded component M ′µ such that M = M ′

µ ∩(⋂s

i=1i = j

qi

).

Macaulay’s solution is the following,82 where wlog one assumes√

q j = mand γ ′ is the characteristic number of M ′

µ:

If M ′ [:= q j ] is a simple H-N-module, γ ′ − 1 is the highest degree of a member of thebasis of [M : m] which is not a member of M .The basis of [M : m] comprises a certain definite number t of members of which nolinear combination is a member of M ,[83] and a certain number (or the whole) of themembers of the basis of M . It is clear also that any member of [M : m] is a linearcombination of the t members and a member of M . To the t members of [M : m]which are not members of M will also correspond t N-equations [84] of M which arenot N-equations of [M : m] . . . .The t N-equations are the only modular equations in respect to which M and [M :m] differ . . . . The t N-equations are the principal equations of the required imbeddedsimple N-H-module M ′

µ and since they can be found they determine M ′µ. The t N-

equations are not unique since they may be modified in any manner by the N-equationsof [M : m]; if it can be seen how to choose them so that µ may be a minimum [85] itwould be the best choice to make.

82 On the Resolution op. cit., Section 41, p. 91.83 With the notation here they are F(E1), . . . ,F(Et ).84 That is E1, . . . , Et .85 Macaulay is here posing the question of how to determine a reduced

√q j -primary component.


Proposition 30.7.4. Let I = [E] be a homogeneous principal system and letl := deg(E). Then

• the numbers of linearly independent derivates of E of degree and l − arethe same;

• h H(; I) = hH(l − ; I), for each , 0 ≤ ≤ l.

Proof. Let q = q1, . . . , qµ be a Grobner representation of I consisting ofhomogeneous polynomials; then 86

P = I ⊕ Spank(qi ∈ q, deg(qi ) = ).Let then f1, . . . , f J be a k-basis of I, e1, . . . , eI be a k-basis of

[E] := e ∈ [E] homogeneous, deg(e) = ,gi := F(ei ), for each i so that f1, . . . , f J , g1, . . . , gI is a k-linear basis ofP; therefore f1 E, . . . , f J E, g1 E, . . . , gI E generates

[E]l− := e ∈ [E] homogeneous, deg(e) = l − .Since, for each j , f j E = 0, and, for each i ,

∑i

λi gi E = 0 ⇒∑

i

λi gi ∈ I ⇒(∑

i

λi gi

)ei = 0 ⇒ λi = 0,

we deduce that g1 E, . . . , gs E is a k-basis of [E]l−, whence the first claim.The second follows by the equality hH(; I) = dimK ([E]l−).

Proposition 30.7.5. Let q = [E] be a principal primary ideal at the origin, sothat the characteristic number γ of q is γ = ord(E) + 1.

Let q′ ⊃ q be another primary ideal at the origin,

deg(q) = µ > µ′ = deg(q′)

and let q′′ := q : q′. Then:

• q′ = q : q′′,• deg(q′′) = µ′′ = µ − µ′,• if moreover q, q′ and, therefore, also q′′ are homogeneous, we have, for each

≤ deg(E) = γ − 1,

h H(; q′) + h H(γ − 1 − ; q′′) = h H(; q) = h H(γ − 1 − ; q).

86 In the language of Buchberger theory we would consider as Grobner representation the setN<(I) w.r.t. some term ordering < and we would apply the result of Lemma 22.2.12, obtainingP

∼= I ⊕ k[N<(I)].

498 Macaulay II

Proof. Let e1, . . . , eµ′ be a linearly independent ordered set of inversefunctions such that [e1, . . . , eµ′ ] = q′ and which satisfies the properties ofLemma 30.6.1. We can complete it to a set eµ′+1, . . . , eµ in such a way thatE = eµ and e1, . . . , eµ satisfy the properties of Lemma 30.6.1. We also writefi := F(ei ), for each i , so that f1, . . . , fµ and f1, . . . , fµ′ are Grobnerrepresentations of (respectively) q and q′.

Up to reducing each fi , µ′ < i ≤ µ, via a linear combination of

f1, . . . , fµ′ , it is possible to assume wlog that fi e j = 0 for each i, j, j ≤µ′ < i ≤ µ so that fi ∈ q′, for each i, µ′ < i ≤ µ.

If we then write

E ′′j := f j+µ′ E for each j, 1 ≤ j ≤ µ − µ′ = µ′′

then

• each E ′′j is a modular equation of q′′ since, for each polynomial f ∈ P

f ∈ q′′ ⇐⇒ f f j+µ′ ∈ q, for each j

⇐⇒ 0 = f f j+µ′ E = f E ′′j , for each j;

• E ′′j : 1 ≤ j ≤ µ′′ is a linearly independent set since

∑j

λ j E ′′j = 0 ⇐⇒

(∑j

λ j f j+µ′

)E = 0

⇐⇒∑

j

λ j f j+µ′ ∈ q

⇐⇒(∑

j

λ j f j+µ′

)ei = 0 for each i, 1 ≤ i ≤ µ

⇐⇒ λ j = λ j f j+µ′e j+µ′ = 0 for each j, 1 ≤ j ≤ µ′′;• for each F ∈ q′ such that F E = 0, there exist λ1, . . . λµ′′ satisfying F =∑

λ j f j+µ′ since, for suitable λ1, . . . λµ

F −µ∑

j=1

λ j f j ∈ q ⊂ q′ ⇒µ′∑

j=1

λ j f j ∈ q′

⇒⎛⎝ µ′∑

j=1

λ j f j

⎞⎠ ei = 0, for each i, 1 ≤ i ≤ µ′

⇒ λ j = λ j f j e j = 0, for each j, 1 ≤ j ≤ µ′

⇒ F =µ∑

j=µ′+1

λ j f j ,


so that;• q′′ = [E ′′

1 , . . . , E ′′µ′′ ] and deg(q′′) = µ′′;

• finally from q ⊂ q′q′′ we have q′ ⊃ q : q′′ whence the equality followsfrom the degree formula.

If we now consider the linearly independent k-basis f1, . . . , fν′ , (respec-tively, e1, . . . , eν′′ ) of the dialytic equations of q′ (respectively, modularequations of q′′) having degree γ − 1 − (respectively ), the same argumentas in Proposition 30.7.5 proves that

e1, . . . , eν′′ = f1 E, . . . , fν′ Eso that

hH(; q′′) = ν′′ = ν′ = hH(γ − 1 − ; q) − hH(γ − 1 − ; q′).

Algorithm 30.7.6. Note that the procedure contained in the proof of the propo-sition above is an effective algorithm for computing the colon of two primaryideals at the origin.

31

Grobner II

Grobner, always interested by the possible interplay between polynomial ide-als and differential equations, gave in some papers but mainly in his trea-tises an illuminating reformulation of Macaulay’s Noetherian equations interms of differential equations: the result of the application to a polynomialf (X1, . . . , Xr ) ∈ Q := K [X1, . . . , Xr ] of a Noetherian equation of an m-primary or m-closed ideal, where m is the maximal ideal at the point b ∈ K r ,is read by Grobner as the evaluation at b of a proper derivate of f .

His characterization, which of course is applicable only to fields of charac-teristic 0, was connected in the 1990s with the Moller algorithm and deeplystudied, under the probably inappropriate label of Grobner duality; such studyled to an algorithm with good complexity – O(s3r3) – for computing for anideal I ⊂ m, given through a finite set of generators, the Noetherian equationsof its m-primary component q, deg(q) = s, if m is an isolated maximal of I.1

In this and in the next chapter, I reformulate these results in terms ofMacaulay’s duality, dropping Grobner’s formulation, thus removing the use-less restriction on the characteristic of the field.

I begin by introducing (Section 31.1) a proper notation for describing theQ-module of the Noetherian equations and the subsets which are stable un-der each Xi -derivation (using Macaulay’s terminology), that is which are Q-submodules (Section 31.2).

I then discuss (Section 31.3) the corresponding duality 2 between m-closedideals and stable vectorspaces of Noetherian equations, thus dropping the as-sumptions on finite dimensionality. In Section 31.4 I translate in the context

1 If we consider that we are allowed to apply the same limiting consideration performed ‘at leastin imagination’ by Macaulay himself, such algorithms in principle allow us to deduce the infiniteset of the Noetherian equations of the m-closure

⋂ρ I + mρ of I.

2 We have kept the inappropriate label of Grobner Duality.

500


of Noetherian equations, the Leibnitz Formula which is a natural tool inGrobner’s formulation.

Sections 31.5 and 31.6 are devoted to Grobner’s interpretation of Noetherianequations in terms of differential conditions.

31.1 Noetherian Equations

Let Q := K [X1, . . . , Xr ], W : Xa11 . . . Xar

r : (a1, . . . , ar ) ∈ Nr and let

m := (X1, . . . , Xr ) be the maximal at the origin.For each τ := Xa1

1 . . . Xarr ∈ W denote M(τ ) : Q → K the morphism

defined by M(τ ) = c( f, τ ) for each f = ∑t∈W c( f, t)t ∈ Q.

Writing M := M(τ ) : τ ∈ W we have

Corollary 31.1.1. For any

f :=∑t∈W

at t ∈ Q and :=∑τ∈W

cτ M(τ ) ∈ SpanK (M)

we have

( f ) =∑t∈W

at ct .

Therefore SpanK (M) ⊂ Q∗ := HomK (Q, K ) is the set of all the Noethe-rian equations. In particular for each m-primary ideal q, we have L(q) ⊂SpanK (M).

Let us denote, for each K -subvectorspace Λ ⊂ SpanK (M),

I(Λ) := P(Λ) = f ∈ Q : ( f ) = 0, for each ∈ Λand, for each K -subvectorspace P ⊂ Q,

M(P) := L(P)∩SpanK (M) = ∈ SpanK (M) : ( f ) = 0, for each f ∈ P.Any semigroup ordering 3 < on Q induces also the corresponding ordering

on M defined by

M(τ ) ≤ M(ω) ⇐⇒ τ ≤ ω.

Remark 31.1.2. The discussion of Macaulay’s results shows that whenever thedialytic equations, that is the polynomials, are ordered according to their de-gree, the corresponding inverse functions are ordered according to their order(or under-degree) and conversely. This suggests that, if we want to extend thenotation of Buchberger’s theory to inverse functions, it is advisable to relax

3 Not necessarily only a term ordering.

502 Grobner II

the assumptions and consider any semigroup ordering and not just the well-ordering case. Actually, we will systematically reverse the ordering.

Definition 31.1.3. For any element

:=∑

i

ci M(τi ) ∈ Spank(M) : ci ∈ k\0, τi ∈ W, τ1 < τ2 < · · · < τi < · · ·

• the leading term of is T<() := τ1,• the order (or under-degree) of is ord() :=mini (deg(τi ));• the degree of is deg() :=maxi (deg(τi )).

For a set Λ ⊂ Spank(M), T<Λ := T<(), ∈ Λ.Note that, if < is a degree-compatible term ordering, we have

ord() = deg(T<()), for each ∈ SpanK (M).

31.2 Stability

Definition 31.2.1. For each j = 1, . . . , r ,

σ j := σX j : SpanK (M) → SpanK (M) is the linear map such that

σX j (M(τ )) =

M(ω) if τ = X jω

0 if X j τfor each τ ∈ W;

ρ j := ρX j : SpanK (M) → SpanK (M) is the linear map such that

ρX j (M(τ )) = M(X jτ) for each τ ∈ W;λ j := ρ jσ j : SpanK (M) → SpanK (M) is the linear map such that

λ j (M(τ )) =

M(τ ) if X j | τ

0 if X j τfor each τ ∈ W.

Note that

σ jρ j = Id, for each j,ρ jσ j = λ j , for each j,σkρ j = ρ jσk, for each j, k, j = k.

Since, for each i, j , σX j σXi = σXi σX j , a linear map σt : SpanK (M) →SpanK (M) is inductively defined for each t ∈ W by σX j t := σX j σt so that foreach τ, ω ∈ W we have

στ (M(ω)) =

M(υ) if ω = τυ,0 if τ ω.

31.2 Stability 503

Therefore, for each f = ∑i ci ti ∈ Q, a map σ f : SpanK (M) → SpanK (M)

is uniquely defined as σ f () = ∑i ciσti ().

Under this definition, the vectorspace SpanK (M) is naturally endowed withthe Q-module structure defined by

f = σ f (), for each f ∈ Q, ∈ SpanK (M).

Note also that for each ∈ SpanK (M) and each f ∈ Q, σ f () is exactly thef -derivative of .

This leads directly to the following:

Definition 31.2.2. A subvectorspace Λ ⊂ SpanK (M) is called

• X j -stable if for each ∈ Λ, σX j () ∈ Λ;• stable if for each ∈ Λ and each f ∈ Q, σ f () ∈ Λ.

Lemma 31.2.3. For any subvectorspaces Λ, Λ1, Λ2 ⊂ SpanK (M) the fol-lowing holds:

(1) For any change of coordinates Y1, . . . , Yn, the following conditionsare equivalent:

• Λ is stable,• Λ is X j -stable, for each j ,• Λ is Yi -stable, for each i.

(2) If Λ = 0 is stable then M(1) ∈ Λ.

(3) If Λ1 and Λ2 are stable so also are Λ1 ∩ Λ2 and Λ1 + Λ2.

Lemma 31.2.4. Let Λ ⊂ SpanK (M) be a subvectorspace; for each ∈ Λ,each f ∈ I(Λ) and each i , we have

(Xi f ) = σi ()( f ).

Theorem 31.2.5. Let Λ ⊂ SpanK (M) ⊂ Q∗ be any finite-dimensional K -subvectorspace.

Then, the following conditions are equivalent:

(1) Λ is stable.(2) The vectorspace I(Λ) is an ideal and I(Λ) ⊂ m.

504 Grobner II

Proof.

(1) ⇒ (2) For any ∈ Λ, any f ∈ I(Λ) and any i , we have σXi () ∈ Λ

so that (Xi f ) = σi ()( f ) = 0 thus proving that

Xi f ∈ I(Λ) for each f ∈ I(Λ) and each i,

that is that I(Λ) is an ideal.Moreover, since Λ is stable, by Lemma 31.2.3 we have M(1) ∈ Λ sothat

f (0) = M(1)( f ) = 0, for each f ∈ I(Λ),

that is I(Λ) ⊂ m.(2) ⇒ (1) Since Λ ⊂ Q∗ is finite dimensional we have Λ = LP(Λ).

For each f ∈ I(Λ), ∈ Λ, i ≤ r , since I(Λ) is an ideal we haveXi f ∈ I(Λ) so that σi ()( f ) = (Xi f ) = 0 and

σi () ∈ LI(Λ) = LP(Λ) = Λ.

31.3 Grobner Duality

Proposition 31.3.1. For each K -vector-subspace I ⊂ Q and each K -vector-subspace Λ ⊂ SpanK (M), we have

(1) Λ ⊂ MI(Λ),(2) if Λ is finite-dimensional, then Λ = MI(Λ),(3) I ⊂ IM(I).

Proof.

(1) By Proposition 28.1.6 we have Λ ⊂ LP(Λ) so that

Λ = Λ ∩ SpanK (M)

⊂ LP(Λ) ∩ SpanK (M)

= L(I(Λ)) ∩ SpanK (M)

= M(I(Λ)).

(2) In fact, by Corollary 28.1.12, Λ = LP(Λ) and in the proof above wecan substitute equality to inclusion.

(3) Since M(I) ⊂ L(I) we have P(M(I) ⊃ P(L(I); so that using Proposi-tion 28.1.6 again we have

I ⊂ PL(I) ⊂ PM(I) = IM(I).

31.3 Grobner Duality 505

For each ρ ∈ N, writing ∇ρ := SpanK (M(τ )( · ) : τ ∈ W(ρ)), we have

Lemma 31.3.2. For each ρ ∈ N we have

• I(∇ρ) = P(∇ρ) = mρ,

• M(mρ) = L(mρ) = ∇ρ.

Proof. Trivially we have

P(∇ρ) = I(∇ρ) ⊃ mρ and L(mρ) ⊃ M(mρ) ⊃ ∇ρ,

and the equalities follow by dim(∇ρ) = (rρ

) = deg(mρ).

Corollary 31.3.3. For each m-primary q we have

• M(q) = L(q),• q = IM(q).

Proof. Since q is m-primary we have q ⊃ mρ for some ρ and

L(q) ⊂ L(mρ) = ∇ρ ⊂ SpanK (M),

so that M(q) = L(q). Hence

q = PL(q) = PM(q) = IM(q).

Proposition 31.3.4. For each finite-dimensional stable subvectorspace Λ ⊂SpanK (M) we have

• I(Λ) ⊂ m is an m-primary ideal,• dim(Λ) = deg(I(Λ)).

Proof. Theorem 31.2.5 gives that I(Λ) ⊂ m is an ideal. Since Λ is finite thereis ρ ∈ N such that Λ ⊂ ∇ρ so that I(Λ) ⊃ mρ is primary.

Also dim(Λ) = deg(P(Λ)) = deg(I(Λ)).

Proposition 31.3.5. For each m-primary q we have

• M(q) is stable;• dim(M(q)) = deg(q).

Proof. Since IM(q) ⊂ m is an ideal, Theorem 31.2.5 gives the stability ofM(q).

Also, since M(q) = L(q) we have

dim(M(q)) = dim(L(q)) = deg(PL(q)) = deg(q).

506 Grobner II

Lemma 31.3.6. For each m-primary ideals q1 and q2 and each finite dimen-sional stable K -vector subspaces Λ1, Λ2 ⊂ SpanK (M) we have

(1) q1 ⊂ q2 ⇒ M(q1) ⊃ M(q2);(2) Λ1 ⊂ Λ2 ⇒ I(Λ1) ⊃ I(Λ2);(3) M(q1 + q2) = M(q1) ∩ M(q2);(4) I(Λ1 + Λ2) = I(Λ1) ∩ I(Λ2);(5) M(q1 ∩ q2) = M(q1) + M(q2);(6) I(Λ1 ∩ Λ2) = I(Λ1) + I(Λ2).

Proof. This is a reformulation of Lemma 28.1.5 and Corollary 28.1.16.

Corollary 31.3.7. The mutually inverse maps I( · ) and M( · ) are the re-strictions of, respectively, P( · ) to m-primary ideals, and L( · ) to finite-K -dimensional stable K -subvectorspace.

They give a biunivocal, inclusion reversing, correspondence between the setof the m-primary ideals q ⊂ Q and the set of the finite-K-dimensional stableK -subvectorspaces Λ ⊂ SpanK (M).

Moreover, for any q ⊂ Q we have deg(q) = dimK (M(q)) and, for anyfinite-K -dimensional stable K -subvectorspace Λ ⊂ SpanK (M) we havedimK (Λ) = deg(I(Λ)).

Lemma 31.3.8. Let Pρ , ρ ∈ N, be zero-dimensional ideals and Lρ ⊂ Q∗,ρ ∈ N, be finite-K -dimensional Q-modules. Then

(1) L(∑

ρ Pρ) = ⋂ρ L(Pρ);

(2) P(∑

ρ Lρ) = ⋂ρ P(Lρ);

(3) L(⋂

ρ Pρ) = ∑ρ L(Pρ);

(4) P(⋂

ρ Lρ) = ∑ρ P(Lρ).

Proof.

(1) From∑

ρ Pρ ⊃ Pρ, for each ρ, we have L(∑

ρ Pρ) ⊂ L(Pρ) and

L

(∑ρ

Pρ

)⊂

⋂ρ

L(Pρ);

conversely, for any ∈ ∩ρL(Pρ) and any f ∈ ∑ρ Pρ , f = ∑ν

i=1 fi ,with fi ∈ Pi , we have ( f ) = ∑ν

i=1 ( fi ) = 0 so that

L

(∑ρ

Pρ

)=

⋂ρ

L(Pρ).

31.3 Grobner Duality 507

(2) From∑

ρ Lρ ⊃ Lρ, for each ρ, we have P(∑

ρ Lρ) ⊂ P(Lρ) and

P

(∑ρ

Lρ

)⊂

⋂ρ

P(Lρ);

conversely, for any f ∈ ⋂ρ P(Lρ) and any ∈ ∑

ρ Lρ , = ∑νi=1 i ,

with i ∈ Li , we have ( f ) = ∑νi=1 i ( f ) = 0 so that⋂

ρ

P(Lρ) ⊂ P

(∑ρ

Lρ

).

(3) L(⋂

ρ Pρ) = L(⋂

ρ PL(Pρ) = LP(∑

ρ L(Pρ)) = ∑ρ L(Pρ).

(4) P(⋂

ρ Lρ) = P(⋂

ρ LP(Lρ) = PL(∑

ρ P(Lρ)) = ∑ρ P(Lρ).

Corollary 31.3.9. Let qρ , ρ ∈ N, be m-primary ideals and Λρ ⊂ SpanK (M),ρ ∈ N, be finite-dimensional stable K -vectorsubspaces. Then

(1) M(∑

ρ qρ) = ⋂ρ M(qρ);

(2) I(∑

ρ Λρ) = ⋂ρ I(Λρ);

(3) M(⋂

ρ qρ) = ∑ρ M(qρ);

(4) I(⋂

ρ Λρ) = ∑ρ I(Λρ).

Lemma 31.3.10. Let Λ ⊂ SpanK (M) be a (not necessarily finite-dimensional) stable subvectorspace and let, for each ρ ∈ N, Λρ := Λ ∩ ∇ρ.

Then we have:

(1) Λ1 ⊂ · · · ⊂ Λρ ⊂ Λρ+1 ⊂ · · · ⊂ Λ,(2) I(Λ1) ⊃ · · · ⊃ I(Λρ) ⊃ I(Λρ+1) ⊃ · · · ⊃ I(Λ),(3) Λ = ∑

ρ Λρ ,(4) I(Λ) = ⋂

ρ I(Λρ),(5) I(Λ) is an m-closed ideal,(6) Λ = MI(Λ).

Proof. (1), (2) and (3) are trivial and (4) follows by the lemma above.Ad (5): we have

I(Λ) =⋂ρ

I(Λρ)

=⋂ρ

I(Λ ∩ ∇ρ)

=⋂ρ

I(Λ) + I(∇ρ)

=⋂ρ

I(Λ) + mρ.

Ad (6): we have

Λ =∑ρ

Λρ =∑ρ

MI(Λρ) = M(⋂ρ

I(Λρ)) = MI(Λ).

508 Grobner II

Corollary 31.3.11. For each stable subvectorspace Λ ⊂ SpanK (M) we have:

• I(Λ) is an m-closed ideal,• Λ = MI(Λ).

Proposition 31.3.12. For each m-closed I we have

• I = IM(I);• M(I) is stable.

Proof. Setting Iρ := I + mρ, for each ρ, we have

I =⋂ρ

Iρ =⋂ρ

IM(Iρ) = I

(∑ρ

M(Iρ)

)= IM

(⋂ρ

Iρ

)= IM(I).

Let ∈ M(I) and let ρ = deg() so that ∈ ∇ρ = M(mρ); therefore

∈ M(I) ∩ M(mρ) = M(I + mρ);since M(I + mρ) is stable, for each f ∈ Q,

σ f () ∈ M(I) ∩ M(mρ) ⊂ M(I).

Theorem 31.3.13. The mutually inverse maps I( · ) and M( · ) give a biu-nivocal, inclusion-reversing, correspondence between the set of the m-closedideals I ⊂ Q and the set of the stable K -vectorsubspaces Λ ⊂ SpanK (M).

31.4 Leibniz Formula

Proposition 31.4.1. For any f, g ∈ Q and ω ∈ W we have

M(ω)( f g) =∑υ∈Wυτ=ω

M(υ)( f )M(τ )(g)

Proof. For

f = ∑υ∈W c( f, υ)υ = ∑

υ∈W M(υ)( f )υ,

g = ∑τ∈W c(g, τ )τ = ∑

τ∈W M(τ )(g)τ,

f g = ∑ω∈W c( f g, ω)ω = ∑

ω∈W M(ω)( f g)ω

and, for each ω ∈ W , we have

M(ω)( f g) = c( f g, ω) =∑υ∈Wυτ=ω

c( f, υ)c(g, τ )

=∑υ∈Wυτ=ω

M(υ)( f )M(τ )(g).

31.5 Differential Inverse Functions at the Origin 509

Corollary 31.4.2. For any f, g ∈ Q and any ∈ SpanK (M) we have

( f g) =∑υ∈W

M(υ)( f )συ()(g).

Corollary 31.4.3. For all f ∈ Q, ∈ SpanK (M), 1 ≤ i ≤ r , we have

(Xi f ) = Xi( f ) + σXi ()( f ).

Proposition 31.4.4 (Moller–Stetter). Let

1, . . . , s be any K -basis of a stable K -vectorspace Λ ⊂ SpanK (M),I ⊂ Q an ideal,g1, . . . , gt any finite basis of I.

Then

i (g j ) = 0, for each i, j ⇒ ( f ) = 0, for each ∈ Λ, f ∈ I.

Proof. Let f = ∑tj=1 f j g j ∈ I and let ∈ Λ. Then, for each υ ∈ W ,

συ() ∈ Λ because Λ is stable, and therefore συ()(g j ) = 0 for each j andeach υ ∈ W . By the Leibniz Formula, we have

( f ) =t∑

j=1

( f j g j ) =t∑

j=1

∑υ∈W

M(υ)( f j )συ()(g j ) = 0.

Corollary 31.4.5. With the same notation as above

i (g j ) = 0, for each i, j ⇒ Λ ⊂ M(I).

31.5 Differential Inverse Functions at the Origin

A nice interpretation of the set SpanK (M) of all the Noetherian equations at theorigin in terms of differential operators was proposed by Grobner, assumingchar(K ) = 0.

Let

Q := K [X1, . . . , Xr ], char(K ) = 0,

W := Xa11 . . . Xar

r : (a1, . . . , ar ) ∈ Nr ,

m := (X1, . . . , Xr ) be the maximal at the origin.

510 Grobner II

For each (i1, . . . , ir ) ∈ Nr , setting τ := Xi1

1 . . . Xirr , we denote by

D(τ ) := D(i1, . . . , ir ) : Q → Q

the differential operator

D(τ ) := D(i1, . . . , ir ) = 1

i1! . . . ir !

∂ i1+···+ir

∂ Xi11 . . . ∂ Xir

r

.

Also, for τ := Xd11 . . . Xdr

r ∈ W , and t := Xe11 . . . Xer

r ∈ W such that τ | tso that di ≤ ei , we will use the shorthand

( tτ

)to denote(

t

τ

):=

(e1

d1

). . .

(er

dr

).

Proposition 31.5.1. Let τ := Xd11 . . . Xdr

r ∈ W , and t := Xe11 . . . Xer

r ∈ W.

Then

D(τ )(t) := ( t

τ

)Xe1−d1

1 . . . Xer −drr if τ divides t,

0 if τ does not divide t.

Proof. In fact, if there exists i such that ei < di

D(τ )(t) = 1

d1! . . . dr !

∂d1+···+dr

∂ Xd11 . . . ∂ Xdr

r

Xe11 . . . Xer

r = 0

while, if ei ≥ di , for each i, we have

D(τ )(t) = 1

d1! . . . dr !

∂d1+···+dr

∂ Xd11 . . . ∂ Xdr

r

Xe11 . . . Xer

r

=∏d1

i=1 e1 − i + 1

d1!· · ·

∏dri=1 er − i + 1

dr !Xe1−d1

1 . . . Xer −drr

= e1!

d1!(e1 − d1)!· · · er !

dr !(er − dr )!Xe1−d1

1 . . . Xer −drr

=(

t

τ

)Xe1−d1

1 . . . Xer −drr .

Note that, for each τ ∈ W , D(τ )( · )(0, . . . , 0) = M(τ ), so that if we denoteD := D(τ ) : τ ∈ W and we set ev : SpanK (D) → SpanK (M) the morphismdefined by ev(D(τ )) = M(τ ) for each τ ∈ W we have

ev(δ)(·) = δ(·)(0, . . . , 0)=∑τ∈W

cτ M(τ )(·) for each δ :=∑τ∈W

cτ D(τ )(·) ∈ D

31.5 Differential Inverse Functions at the Origin 511

so that the set

δ( · )(0, . . . , 0) : δ ∈ SpanK (D) ⊂ Q∗ := HomK (Q, K )

coincides with the set of all the Noetherian equations at the origin and, inparticular, for each m-primary ideal q, we have

L(q) ⊂ δ( · )(0, . . . , 0) : δ ∈ SpanK (D).We impose on D the same semigroup ordering < as induced on M so that

D(τ ) ≤ D(ω) ⇐⇒ M(τ ) ≤ M(ω) ⇐⇒ τ ≤ ω

and we set

T<(δ) := T<(ev(δ)), ord(δ) := ord(ev(δ)), deg(δ) := deg(ev(δ)).

We can also impose on D the semigroup structure isomorphic to that ofW , setting D(τ1) · D(τ2) := D(τ1τ2), which coincides with the compositionof the two isomorphisms up to a normalizing coefficient:

Lemma 31.5.2. For υ := Xd11 . . . Xdr

r , and τ := Xe11 . . . Xer

r , we have

D(υ) (D(τ )( · )) =(

υτ

τ

)D(υτ)( · ).

Proof. In fact(υτ

τ

)D(υτ)( · ) = 1

d1! . . . dr !

1

e1! . . . er !

∂d1+e1+ ··· +dr +er

∂ Xd1+e11 · · · ∂ Xdr +er

r

( · )

= 1

d1! . . . dr !

∂d1+···+dr

∂ Xd11 . . . ∂ Xdr

r

(1

e1! . . . er !

∂e1+···+er

∂ Xe11 . . . ∂ Xer

r( · )

)

= D(υ) (D(τ )( · )) .

We can extend the notation σ f : SpanK (D) → SpanK (D), for each f ∈ Qsetting

στ (D(ω)) =

D(υ) if ω = τυ,

0 if τ ω,for each τ, ω ∈ W,

σ f (δ) = ∑i ciσti (δ) for each f = ∑

i ci ti ∈ Q.

Definition 31.5.3. A subvectorspace ∆ ⊂ SpanK (D) is called

• X j -stable if for each δ ∈ ∆, σX j (δ) ∈ ∆;• stable if for each δ ∈ ∆ and each f ∈ Q, σ f (δ) ∈ ∆.

512 Grobner II

31.6 Taylor Formula and Grobner Duality

Let

b := (b1, . . . , br ) ∈ K r ,m := (X1 − b1, . . . , Xr − br ) ⊂ Q,λb : Q → Q be the translation λb(Xi ) = Xi + bi , for each i .

Then λb(m) = m and, for each m-closed ideal i, I := λb(i) is an m-closedideal. Therefore

λb( · ) : ∈ SpanK (M) = δ( · )(b) : δ ∈ SpanK (D) ⊂ Q

is the set of all the Noetherian inverse equations w.r.t. m-closed ideals and, inparticular

L(q) ⊂ λb( · ) : ∈ SpanK (M),for each m-primary ideal q.

Remark 31.6.1. Let p ⊂ P = k[X1, . . . , Xn], dim(p) = n − r, be a primeideal and let us assume, up to a suitable change of coordinates, that p ∩k[Xr+1, . . . , Xn] = 0, so that p := pk(Xr+1, . . . , Xn)[X1, . . . , Xr ] is max-imal and has a prime decomposition p = ⋂s

i=1 ni in Ω(k)[X1, . . . , Xr ] := Qwhere Ω(k) is the universal field (Section 9.4) of k so that p = ni ∩k[X1, . . . , Xn], for each i.

If ai := (ai1, . . . , air ) ∈ Ω(k)r is the root for which

ni = (X1 − ai1, . . . , Xr − air ),

then, via the translation λai : Q → Q, we are in the situation discussed above.In particular:

• the set λai ( · ) : ∈ SpanK (M) ⊂ Q∗ consists of all the Noetherianinverse equations w.r.t. ni -closed ideals;

• if q⊂k[X1, . . . , Xn] is p-primary, then

q := qk(Xr+1, . . . , Xn)[X1, . . . , Xr ]

is p-primary and has a decomposition q = ⋂si=1 si into simple primary

components in Ω(k)[X1, . . . , Xr ], which satisfy

• √si = ni ,

• q = si ∩ k[X1, . . . , Xn] for each i ,• L(si ) ⊂ λai ( · ) : ∈ SpanK (M);

• if i is a p-closed ideal, then J := ik(Xr+1, . . . , Xn)[X1, . . . , Xr ] has a de-composition J = ⋂s

i=1 Ji where Ji is ni -closed and i = Ji ∩k[X1, . . . , Xn],for each i .

31.6 Taylor Formula and Grobner Duality 513

Lemma 31.6.2. For each b := (b1, . . . , br ) ∈ K r and f := ∑µi=1 c( f, ti )ti ∈

Q, we have

c(τ, λb( f )) = M(τ )λb( f ) = D(τ )λb( f )(0, . . . , 0) = D(τ )( f )(b).

Corollary 31.6.3 (Taylor formula). For each b := (b1, . . . , br ) ∈ K r andeach f := ∑µ

i=1 c( f, ti )ti ∈ Q, we have

λb( f ) = f (X1 + b1, . . . , Xr + br )

=∑τ∈W

D(τ )( f )(b)τ.

Corollary 31.6.4. Let ∆ ⊂ SpanK (D) be any K -vectorsubspace.Then, the following conditions are equivalent:

(1) ∆ is stable,(2) Λ := ev(∆) is stable,(3) the vectorspace I(∆) := f ∈ Q : δ( f )(b) = 0, for each δ ∈ ∆ is

an ideal and I(∆) ⊂ m.

Proof. Clearly (1) ⇐⇒ (2).The equivalence with (3) is a consequence of the obvious equality

δ( f )(b) = δλb( f )(0, . . . , 0) = ev(δ)λb( f ).

Let us write, for each K -vectorsubspace ∆ ⊂ SpanK (D),

Im(∆) := f ∈ Q : δ( f )(b) = 0, for each δ ∈ ∆and, for each K -vector subspace P ⊂ Q,

Dm(P) := δ ∈ SpanK (D) : δ( f )(b) = 0, for each f ∈ P.Lemma 31.6.5. For any stable K -vectorspace ∆ ⊂ SpanK (D), we haveIm(∆) = λ−1

b (I(ev(∆))).

Proof. Writing Λ := ev(∆), we have

Im(∆) = f ∈ Q : δ( f )(b) = 0, for each δ ∈ ∆= f ∈ Q : ev(δ)λb( f ) = 0, for each δ ∈ ∆= λ−1

b (g) : g ∈ Q, ev(δ)(g) = 0, for each δ ∈ ∆= λ−1

b (g : g ∈ Q, (g) = 0, for each ∈ Λ)

514 Grobner II

= λ−1b (P(Λ))

= λ−1b (I(Λ))

= λ−1b (I(ev(∆))).

Lemma 31.6.6. For P ⊂ Q, we have Dm(λ−1b (P)) = ev−1(M(P)).

Proof. We have

Dm(λ−1b (P)) = δ ∈ SpanK (D) : δ( f )(b) = 0, for each f ∈ λ−1

b (P)= δ ∈ SpanK (D) : δλ−1

b (g)(b) = 0, for each g ∈ P= δ ∈ SpanK (D) : ev(δ)λb(λ−1

b (g)) = 0, for each g ∈ P= δ ∈ SpanK (D) : ev(δ)( · ) ∈ L(P)= δ ∈ SpanK (D) : ev(δ)( · ) ∈ L(P) ∩ SpanK (M)= δ ∈ SpanK (D) : ev(δ)( · ) ∈ M(P)= ev−1(M(P)).

Corollary 31.6.7. Each m-closed ideal I ⊂ Q and each of the stable K -sub-vectorspaces ∆ ⊂ SpanK (D) satisfy

ImDm(I) = I and DmIm(∆) = ∆.

Proof. We have

ImDm(I) = λ−1b (I(ev(Dm(I))))

= λ−1b (I(ev ev−1(M(λb(I)))))

= λ−1b (IM(λb(I)))

= λ−1b λb(I)

= I

and

DmIm(∆)=Dm(λ−1b (I(ev(∆))))=ev−1(M(I(ev(∆))))=ev−1 ev(∆)=∆.

This allows us to conclude that:

Theorem 31.6.8 (Grobner). The mutually inverse maps Im( · ) and Dm( · )

give a biunivocal, inclusion-reversing, correspondence between the set of the

31.6 Taylor Formula and Grobner Duality 515

m-closed ideals I ⊂ Q and the set of the stable K -vectorsubspaces ∆ ⊂SpanK (D).

Moreover, to any m-primary ideal q⊂Q corresponds a finite K -dimensionalstable K -subvectorspace so that deg(q) = dimK (Dm(q)); and to any finiteK -dimensional stable K -subvectorspace ∆ ⊂ SpanK (D) corresponds anm-primary ideal so that dimK (∆) = deg(Im(∆)).

The application of ev allows us to interpret Proposition 31.4.1 as a formu-lation of the Leibniz Formula

Corollary 31.6.9 (Leibniz Formula). For any f, g ∈ Q and ω ∈ W wehave

D(ω)( f g) =∑υ∈Wυτ=ω

D(υ)( f )D(τ )(g)

and to reformulate its corollaries as

Corollary 31.6.10. For any f, g ∈ Q and any δ ∈ SpanK (D) we have

δ( f g) =∑υ∈W

D(υ)( f )συ(δ)(g).

Corollary 31.6.11. For all f ∈ Q, δ ∈ SpanK (D), 1 ≤ i ≤ r , we have

δ(Xi f ) = Xiδ( f ) + σXi (δ)( f ).

Corollary 31.6.12. Let

b := (b1, . . . , br ) ∈ K r and m := (X1 − b1, . . . , Xr − br ) ⊂ Q;

for any δ ∈ SpanK (D) we have

δ(Xi f )(b) = biδ( f )(b) + σXi (δ)( f )(b).

Corollary 31.6.13 (Moller–Stetter). Let

δ1, . . . , δs be any K -basis of a stable K -vectorspace ∆ ⊂ SpanK (D),b := (b1, . . . , br ) ∈ K r ,m := (X1 − b1, . . . , Xr − br ) ⊂ Q,I ⊂ Q be an ideal,g1, . . . , gt any finite basis of I.

516 Grobner II

Then

δi (g j )(b) = 0, for each i, j ⇒ δ( f )(b) = 0, for each δ ∈ ∆, f ∈ I.

Corollary 31.6.14. With the same notation as above

δi (g j )(b) = 0, for each i, j ⇒ ∆ ⊂ Dm(I).

32

Grobner III

The definition a la Hironaka of leading term for Noetherian equations, mainlydue to the necessity of reversing the ordering, has the effect that, for anm-closed ideal I ⊂ Q and the dual stable K -vectorspace Λ := M(I) ⊂SpanK (M), we have the relation

T<Λ = N<(I) and N<Λ = T<(I).

This naturally leads me to follow Macaulay’s suggestion and select, as K -basisfor Λ, what, in Buchberger terminology, would be called the set of the canon-ical forms of the terms belonging to T<Λ; such concepts have been labelledas Macaulay bases (Section 32.1) and have a natural relation (Section 32.2)with Grobner and natural representations.

The easiest example (see Example 32.1.5) is able to show that if one wantsto make effective use of Macaulay bases, since the obvious representation isexponentially space consuming, one needs an efficient and compact represen-tation; in Section 32.4 an O(s2r) representation (the Horner representation) issuggested.

The aim of this chapter is to present (Section 32.7) the algorithm, alreadypromised in Chapter 31, which, given any finite set F := f1, . . . , fn ⊂ m ⊂Q, and denoting by I the ideal generated by F , returns the Noetherian equationsof

the m-primary component q of I, in case m is an isolated maximal of I, withcomplexity O(s3r3), s = deg(q), and

I + mρ , for each ρ ∈ N, thusby an infinite limiting computation, one can iteratively list the ordered infinite

set of the Noetherian equations of the m-closure⋂

ρ I + mρ of I.

This requires the introduction of some preliminary tools, mainly

517

518 Grobner III

an efficient algorithm to evaluate a polynomial at a Macaulay basis which canbe performed with complexity O(s2r2) if both the polynomial andthe Macaulay basis are given by means of a Horner representation(Section 32.5);

the notion of continuation (Section 32.6).

In the chapter I also discuss (Section 32.3) a reformulation of Macaulay’sAlgorithms 30.4.13 and 30.6.3 (see also Section 29.6) due to Grobner whichallowed him to decompose primary ideals into reduced and irreducible compo-nents, thus allowing him to produce a reduced primary decomposition of anyideal.

32.1 Macaulay Bases

Let us consider

the maximal ideal at the origin,

m = (X1, . . . , Xr ) ⊂ Q := K [X1, . . . , Xr ] ⊂ K [[X1, . . . , Xr ]],

the set W := Xa11 . . . Xar

r : (a1, . . . , ar ) ∈ Nr ,

an m-closed ideal I.

Let us impose on both Q and K [[X1, . . . , Xr ]] the W-valuation which as-sociates to each series f = ∑

t∈W c( f, t)t the valuation

v( f ) := max<

t ∈ W : c( f, t) = 0

where < is an inf-limited and Noetherian ordering;1 then Theorem 24.6.16and the assumption I ⊂ m imply that 1 ∈ N<(I) and that each t ∈ T<(I) has acanonical form – or a Grobner description in terms of the linear representationof I w.r.t. <:

Can(t, I, <) =∑

τ∈N<(I)

γ (t, τ, <)τ

=∑

τ∈N<(I)

γ (t, τ, N<(I))τ ∈ K [[N<(I)]] ⊂ K [[X1, . . . , Xr ]]

1 The requirement that < is Noetherian can be dropped if we restrict our considerations either toan m-primary ideal, or to an ideal which is homogeneous w.r.t. the valuation vw, where w is theweight function

w := (w1, . . . , wr ) ∈ Rr , wi > 0.

Most of our examples will be of this kind.

32.1 Macaulay Bases 519

so that

t − ∑τ∈N<(I) γ (t, τ, N<(I))τ ∈ I,

t < τ ⇒ γ (t, τ, N<(I)) = 0.

Let us number the elements in N<(I) and define, for each τi ∈ N<(I),

i := (τi ) := M(τi ) +∑

t∈T<(I)

γ (t, τi , N<(I))M(t) ∈ SpanK (M).

Proposition 32.1.1. With the notation above, we have

M(I) = Dm(I) = SpanK (τi ), τi ∈ N<(I).Proof. Writing

ft := t −∑τ j <t

γ (t, τ j , N<(I))τ j , for each t ∈ T<(I),

a dialytic array, that is a K -linear basis of I, is the set ft : t ∈ T<(I); in orderto deduce the result it is therefore sufficient to prove that

(τ )( ft ) = 0, for each t ∈ T<(I), τ ∈ N<(I),

which is true since

(τ )( ft ) = M(τ )( ft ) +∑

v∈T<(I)

γ (v, τ, N<(I))M(v)( ft )

= −∑τ j <t

γ (t, τ j , N<(I))M(τ )(τ j ) +∑

v∈T<(I)

γ (v, τ, N<(I))M(v)(t)

= −γ (t, τ, N<(I)) + γ (t, τ, N<(I))

= 0.

Corollary 32.1.2. Let ρ ∈ N and, for each τi , deg(τi ) < ρ, write

′i := (τi ) := M(τi ) +

∑t∈T<(I)deg(t)<ρ

γ (t, τi , N<(I))M(t).

Then

• N<(I + mρ) = τi , deg(τi ) < ρ,• M(I + mρ) = SpanK ′

i , τi ∈ N<(I), deg(τi ) < ρ.

520 Grobner III

Definition 32.1.3. With reference to Definition 31.1.3 and setting

N<(Λ) := W \ T<Λ,a basis 1, 2, . . . , i , . . . of a stable K -subspace Λ ⊂ SpanK (M) is calledthe Macaulay basis of Λ w.r.t. < if

• T<Λ := T<(i ) ⊂ W is an order ideal;• i = M(T<(i )) + ∑

v∈N<(Λ) ξ(v, T<(i ))M(v), for suitable ξ(v, T<(i ))

∈ K and for each i .

Note that a Macaulay basis is nothing more than a reduced Gauss basis.

Corollary 32.1.4. With the notation above, if we set Λ := M(I) we have

• (τi ), τi ∈ N<(I) is a Macaulay basis of Λ,• T<Λ = N<(I).

Proof. For each i and each t ∈ T<(I),

γ (t, τi , N<(I)) = 0 ⇒ t > τi

so that T<((τi )) = τi .

Example 32.1.5. Let us consider the m-closed ideal

I := (X2 − X21, X3 − X3

1, . . . , Xr − Xr1),

the weight vector w := (1, 2, . . . , r) ∈ Rr and the corresponding valuation

vw : W → R satisfying vw(Xi ) = i, for each i, under which I is homogeneous;let us write, for each i ∈ N, i := ∑

τ∈Wvw(τ )=i

M(τ ).

Then it is easy to verify that, for each ρ ∈ N :

I + mρ = (Xρ1 , X2 − X2

1, X3 − X31, . . . , Xr − Xr

1),

deg(I + mρ) = ρ,

M(I + mρ) = SpanK i , 0 ≤ i < ρ,M(I) = SpanK i , i ∈ N.

Moreover, if < denotes the refinement of vw by the lexicographical orderinginduced by X1 ≺ · · · ≺ Xr ,

• for each ρ ∈ N, (Xρ1 , X2, X3, . . . , Xr ) = T<(I + mρ);

• for each ρ ∈ N, Xρ1 , X2 − X2

1, X3 − X31, . . . , Xr − Xr

1, is the Grobnerbasis of I + mρ w.r.t. <;

• for each i ∈ N, T<(i ) = Xi1;

• for each ρ ∈ N, T<M(I + mρ) = 1, X1, . . . , Xρ−11 ;

32.2 Grobner Representation 521

• the Grobner basis of I w.r.t. < is X2 − X21, X3 − X3

1, . . . , Xr − Xr1;

• N<(I) = Xi1, i ∈ N = T<M(I).

32.2 Macaulay Basis and Grobner Representation

Proposition 32.2.1. Let Λ ⊂ SpanK (M) be a stable K -subspace. Then TΛis stable.

Moreover, if i , 1 ≤ i ≤ s is a Macaulay basis of Λ, then T<(i ), 1 ≤i ≤ s is a Macaulay basis of T(Λ).

Proof. Since, by assumption, Λ is stable, then, for each ∈ Λ eitherσXi (T()) = 0 or σXi (T()) = T(σXi ()).

Proposition 32.2.2. Let Λ ⊂ SpanK (M) be a stable K -subspace.Let:

• 1, 2, . . . , i , . . . be its Macaulay basis w.r.t. <, where, for each i

i = M(T<(i )) +∑

v∈N<(Λ)

ξ(v, τi )M(v),

and τi = T<(i );• t1, . . . , ts a minimal basis of N<(Λ);• g j := t j − ∑

τi ∈T<Λ ξ(t j , τi )τi , for each j.

Then (g1, . . . , gs) is the Grobner basis of Im(Λ) w.r.t. <.

Proof. It is sufficient to show that

i (g j ) = M(τi )(g j ) +∑

v∈N<(Λ)

ξ(v, τi )M(v)(g j )

= −ξ(t j , τi )M(τi )(τi ) + ξ(t j , τi )M(t j )(t j )

= 0.

Let

< be any term ordering on W ,I ⊂ Q an m-primary ideal,N<(I) := τ1, . . . , τs, andi := (τi ) := M(τi )+

∑t∈T<(I) γ (t, τi , N<(I))M(t) ∈ SpanK (M) as above;

then:

Proposition 32.2.3. With the notation above, Λ := SpanK 1, . . . , s andN<(I) are biorthogonal.

522 Grobner III

Example 32.2.4. Note that the assumption that τi < τ j for each i < j doesnot imply that, for each i , Λi := SpanK 1, . . . , i , is a Q-module.

An easy example is the following

Q := K [X1, X2],< is any term ordering on W such that X2 > X2

1,

I := (X22 − X2

1, X1 X2, X31) so that

N<(I) := 1, X1, X21, X2, and

1 = (1) = M(1), 2 = (X1) = M(X1),3 = (X2

1) = M(X21) + M(X2

2), 4 = (X2) = M(X2)

and Λ3 := SpanK 1, 2, 3, is not a Q-module since P(Λ3) is not an ideal:

3(X22) = 1, X2

2 /∈ P(Λ3), while X2 ∈ P(Λ3).

32.3 Macaulay Basis and Decomposition of Primary Ideals

Let us now consider

the maximal ideal at the origin,

m = (X1, . . . , Xr ) ⊂ Q := K [X1, . . . , Xr ],

the set W := Xa11 . . . Xar

r : (a1, . . . , ar ) ∈ Nr ,

a Noetherian inf-limited ordering < on W ,an m-closed ideal I,the finite corner set C<(I) := ω1, . . . , ωs,the (not necessarily finite) set N<(I),the corresponding Macaulay basis (τ ) : τ ∈ N<(I) andthe K -vectorspace Λ ⊂ SpanK (M) generated by it.

For each j, 1 ≤ j ≤ s write

Λ j := SpanK υ(ω j ) : υ ∈ W and q j := I(Λ j ).

Moreover let J ⊂ 1, . . . , s be the set such that q j : j ∈ J is the set ofthe minimal elements of q j : 1 ≤ j ≤ s and note that

qi ⊂ q j ⇐⇒ Λi ⊃ Λ j .

We can reformulate Macaulay’s argument in Proposition 30.7.1 as

32.3 Decomposition of Primary Ideals 523

Lemma 32.3.1 (Macaulay). With the notation above, for each j , writing

Λ′j := SpanK υ(ω j ) : υ ∈ W ∩ m

we have

dimK (Λ′j ) = dimK (Λ j ) − 1,

(ω j ) /∈ Λ′j = M(q j : m),

q′ ⊃ q j ⇒ M(q′) ⊆ Λ′j for each m-primary ideal q

Proof. For each h, 1 ≤ h ≤ r , writing λh := Xh(ω j ), we have

Λ′j ⊂

∑h

SpanK υλh : υ ∈ W =⋂

M(q j : Xh)

= M(⋂

hq j : Xh

)= M(q j : m).

Since q j : m = q j we have

dimK (Λ j ) > dimK (M(q j : m)) ≥ dimK (Λ′j ) ≥ dimK (Λ j ) − 1,

whence the claim.

Corollary 32.3.2. With the notation above, if I is an m-primary ideal, then itis possible to enumerate the set N<(I) := τ1, . . . , τs so that

each subvectorspace Lσ := SpanK ((τ1), . . . , (τσ )) is a P-module so thateach Iσ = P(Lσ ) is a zero-dimensional ideal andthere is a chain I1 ⊃ I2 ⊃ · · · ⊃ Is = I.

Proof. The proof can be done by induction on s := #N<(I), being trivial if#N<(I) = 1, that is N<(I) = 1.

Let us choose any element ω j ∈ C<(I), j ∈ J , and let us set

τs := ω j , Ls−1 := SpanK ((ω), ω ∈ N<(I), ω = τs).Then

(ω j ) /∈ Ls−1,

dimK (Ls−1) = s − 1,#N<(Is−1) = s − 1, so that#N<(Is−1) = ω ∈ N<(I), ω = τs

and the claim follows by induction.

Corollary 32.3.3. Let I be a zero-dimensional ideal, deg(I) = s and assumethat Z(I) ⊂ K n. Then there is an ordered linearly independent set of K -linearfunctionals L = 1, . . . , s such that

524 Grobner III

L := SpanK (L) = L(I),each subvectorspace Lσ := SpanK (1, . . . , σ ) is a P-module so thateach Iσ = P(Lσ ) is a zero-dimensional ideal andthere is a chain I1 ⊃ I2 ⊃ · · · ⊃ Is = I.

Proof. Let us fix any term ordering < and let us consider the irredundant pri-mary decomposition I = ⋂r

i=1 qi ; then, for each i , let us write

mi := √qi = (X1 − ai1, . . . , Xr − air ),

ai := (ai1, . . . , air ) ∈ K r ,λi : Q → Q for the translation λi (X j ) = X j + ai j , for each j ,τi1, . . . , τiµi = N<(λi (qi )) enumerated in order to satisfy the properties of

Corollary 32.3.2,

Then if we set

L := (τi j )λi ( · ), 1 ≤ i ≤ t, 1 ≤ j ≤ µi = 1, . . . , s

we have deg(I) = ∑ri=1 µi = ∑r

i=1 deg(qi ) and L := SpanK (L) = L(I).The claim is obtained by Corollary 32.3.2 if we enumerate the set L so

that for each α, β, α = (τiα jα )λiα ( · ), β = (τiβ jβ )λiβ ( · ), we have iα =iβ, jα < jβ ⇒ α < β.

Theorem 32.3.4 (Grobner). If I is m-primary, then, with the notation above,we have

(1) each Λ j is a finite-dimensional stable vectorspace,(2) each q j is an m-primary ideal,(3) is reduced,(4) and irreducible,(5) I := ⋂

j∈J q j is a reduced representation of q.

Proof.

(1) This is trivial by construction.(2) This is a direct consequence of (1).(3) If q j is not reduced, then exists q′ ⊃ q j such that I = ⋂

i = j qi ∩ q′ andLemma 32.3.1 implies (ω j ) /∈ Λ′

j ⊇ M(q′), that is the contradiction

(ω j ) /∈∑

i = jSpanK (Λi ) + M(q′) = M(I) = Λ.

(4) If q j = q′ ∩ q′′ is reducible, Lemma 32.3.1 implies (ω j ) /∈ M(q′) +M(q′′), that is, again, the contradiction (ω j ) /∈ Λ.

32.3 Decomposition of Primary Ideals 525

(5) Since M(I) = Λ = ∑j Λ j = ∑

j∈J Λ j = ∑j∈J M(q j ) we have the

representation I := ⋂j∈J q j which is reduced since redundant compo-

nents have been removed by the restriction of the indices to J and thecomponents are reduced by (3).

Example 32.3.5. In Example 27.3.6 we have

I = m2 = (X2, XY, Y 2) ⊂ K [X, Y ],Λ = SpanK M(1), M(X), M(Y ),C<(I) = X, Y

and

ω1 := X, Λ1 = SpanK M(1), M(X), q1 = (X2, Y ),ω2 := Y, Λ2 = SpanK M(1), M(Y ), q1 = (X, Y 2),

whence (X2, XY, Y 2) = (X2, Y ) ∩ (X, Y 2).


I = (X22 − X2

1, X1 X2, X31),

Λ = SpanK M(1), M(X1), M(X21) + M(X2

2), M(X2),C<(I) = X2

1, X2and

ω1 := X2, Λ2 = SpanK M(1), M(X2), q1 = (X1, X22),

ω2 := X21, Λ2 = Λ, q2 = I because

X23 = M(X2), X13 = M(X1), X213 = X2

23 = M(1),

so that I is irreducible.In connection with Corollary 32.3.2 we have therefore to set

τ4 := X21, L3 := SpanK M(1), M(X1), M(X2),

obtaining I3 = (X21, X1 X2, X2

2) = (X1, X22) ∩ (X2

1, X2).

There are therefore two possible orderings of N<(I) satisfying Corol-lary 32.3.2:

N<(I) = 1, X1, X2, X21 which returns the chain

(X1, X2) ⊃ (X21, X2) ⊃ I3 ⊃ I,

and N<(I) = 1, X2, X1, X21 which returns the chain

(X1, X2) ⊃ (X1, X22) ⊃ I3 ⊃ I

526 Grobner III

Example 32.3.7. In Example 29.6.4 we obtain

C<(I) = X31, X2

1 X2, X22,

Λ1 := SpanK ωλ8, ω ∈ W = λ8, λ5, λ4, λ3, λ2, λ1,q1 := I(Λ1) = (X2

2, X31),

Λ2 := SpanK ωλ7, ω ∈ W = λ7, λ4, λ6, λ2, λ3, λ1,q2 := I(Λ2) = (X4

1, X1 X2, X32 − X3

1),

Λ3 := SpanK ωλ6, ω ∈ W = λ6, λ3, λ1 ⊂ Λ2,

q3 := I(Λ3) = (X1, X32) ⊃ q2,

I = q1 ∩ q2.

If I is not m-primary, let

ρ := maxdeg(ω j ) + 1 : ω j ∈ C<(I) so thatq′ := I + mρ is an m-primary component of I,Λ ∩ ∇ρ = M(q′),I = ⋂r

i=1 qi be an irredundant primary representation of I where√

q1 = m,J := ⋂r

i=2 qi , which can be deduced by means of Algorithm 30.7.3,J = ⋂u

i=1 ii , a reduced representation of J,q1 := ⋂s

j=1 q j a reduced representation of q1 which is wlog ordered so thatqi ⊃ J ⇐⇒ i > t,

q := ⋂tj=1 q j .

Then:


(1) q is a reduced m-primary component of I(2) q := ⋂t

j=1 q j is a reduced representation of q,

(3) I = ⋂ui=1 ii ∩ ⋂t

j=1 q j is a reduced representation of I.

Example 32.3.9. In Example 27.4.4(1) we have

I := (X2, XY ),

Λ = SpanK (M(1), M(X) ∪ M(Y i ), i ∈ N),C<(I) = X;

then

ρ = 2, I ∩ m2 = (X2, Y ) ∩ (X, Y 2),

I : m∞ = (X) ⊂ (X, Y 2);

whence (X2, XY ) = (X) ∩ (X2, Y ).

32.4 Horner Representation 527

Example 32.3.10. Example 27.4.4(2) shows that these results (and even thenotion of Macaulay basis) strongly depend on the choice of a frame of coordi-nates. In fact, it is sufficient in the same example to choose, for each a ∈ Q,

a = 0, Λ = SpanK (M(1), M(X) − aM(Y ) ∪ M(Y i ), i ∈ N) to obtain

ρ = 2, Λ ∩ ∇ρ = M(1), M(X) − aM(Y ), M(Y ),ω1 := X, Λ1 = M(1), M(X) − aM(Y ), q1 = (X2, Y + aX),ω2 := Y, Λ2 = M(1), M(Y ), q2 = (X, Y 2);

whence (X2, XY ) = (X) ∩ (X2, Y + aX).

32.4 Horner Representation of Macaulay Bases

Example 32.1.5 shows that the description of the Noether equations necessarilyrequires a compact and less-consuming form.

If we denote, for each j ,

M[ j, r ] := M(τ ) : τ = Xa11 . . . Xar

r ∈ W, a1 = · · · = a j−1 = 0 = a j ⊂ M,

then each element ∈ SpanK (M \ Id) can be uniquely expressed as

= (1) + · · · + ( j) + · · · + (r),

where ( j) ∈ SpanK (M[ j, r ]) for each j ; in this context we will also introducethe notation

(≥ j) :=r∑

i= j

(i).

Lemma 32.4.1. Let = (1) + · · · + (r) ∈ SpanK (M \ Id). The followinghold:

(1) λi () = λi ((1)) + · · · + λi (

(i−1)) + (i);

(2) (λi ())( j) =

⎧⎨⎩

λi (( j)) if j i ;

(3) (i) = (λi ())(≥i) = λi (

(≥i)).

Proof.

(1) The relations

λi () = λi ((1)) + · · · + λi (

(r))

and

λi (( j)) =

(i) if j = i,0 if j > i

hold trivially.

528 Grobner III

(2) This follows from the easy fact that, for each i, j

λi (( j)) ∈ SpanK (M[ j, r ]),

λi (( j)) = 0 ⇐⇒ i < j.

(3) Since λi (( j)) = (λi ())

( j) = 0 for j > i , we trivially have

(i) = λi ((i)) =

r∑j=i

λi (( j)) = λi

(r∑

j=i

( j)

)= λi (

(≥i))

and

(i) =r∑

j=i

(λi ())( j) = (λi ())

(≥i)).

This notation allows us to reformulate Macaulay’s Proposition 30.7.1 as

Corollary 32.4.2 (Macaulay). Let Λ ⊂ SpanK (M) be a finite-dimensionalstable K -subvectorspace and let B := 1, . . . , s, 1 = Id, be a basis of Λ.

Let ∈ SpanK (M) be such that the K -subvectorspace generated by B ∪ is stable.

Then there are ci j ∈ K , 1 ≤ j ≤ r, 1 ≤ i ≤ s, such that

( j) =s∑

i=1

ci jρ j ((≥ j)i ).

Proof. Since SpanK (B ∪ ) is stable, for each j , σ j () ∈ Λ and there existci j ∈ K such that

σ j () =s∑

i=1

ci ji .

Therefore,

( j) = (λ j ())(≥ j)

= (ρ j (σ j ())

)(≥ j)

=(

ρ j

(s∑

i=1

ci ji

))(≥ j)

=s∑

i=1

ci j(ρ j (i )

)(≥ j)

=s∑

i=1

ci jρ j

((≥ j)i

).

32.4 Horner Representation 529

Corollary 32.4.3. Let Λ ⊂ SpanK (M) be a finite-dimensional stable K -subvectorspace, dimK (Λ) = s, then there are (rs(s + 1))/2 elements ci jh ∈K , 1 ≤ j ≤ r, 1 ≤ i < h ≤ s, such that setting

1 := Id,

( j)h := ∑h−1

i=1 ci jhρ j ((≥ j)i ), 1 < h ≤ s, 1 ≤ j ≤ r,

h := ∑rj=1

( j)h , 1 < h ≤ s,

we have

Λ = SpanK (h, 1 ≤ h ≤ s).


0 := Id,

h := ∑h

j=1 ρ j (( j)h− j ), 1 ≤ h ≤ r,∑r

j=1 ρ j (( j)h− j ), r ≤ h.

Example 32.4.5. Let us now consider the Noetherian inf-limited ordering <

defined by

Xa11 Xa2

2 < Xb11 Xb2

2 ⇐⇒

a1 + a2 > b1 + b2 ora1 + a2 = b1 + b2 and a1 > b1

and the m-closed ideal I := (X22 − X2

1 − X31), for which, writing2

0 := Id,

1 := ρ2((≥2)0 )

= M(X2),

2 := ρ1((≥10 )

= M(X1),

3 := ρ1((≥1)1 )

= M(X1 X2),

2 The strange enumeration of the set 1, 2, . . . , i , . . . needs a justification; the algorithm(Figure 32.1) producing it is such that, given an ideal I, it returns a basis which, for each i ,writing Λi := 1, 2, . . . , i , satisfies

TΛi is an ordered ideal, so thatN(Λi ) is a monomial ideal,T(i+1) is the maximal generator of N(Λi ) which is not a member of T(I).

530 Grobner III

4 := ρ1((≥1)2 ) + ρ2(

(≥2)1 )

= M(X21) + M(X2

2),

5 := ρ1((≥1)3 ) + ρ2(

(≥2)4 )

= M(X21 X2) + M(X3

2),

6 := ρ1((≥1)4 ) + ρ2(

(≥2)1 )

= M(X31) + M(X1 X2

2) + M(X22),

7 := ρ1((≥1)5 ) + ρ2(

(≥2)6 )

= M(X31 X2) + M(X1 X3

2) + M(X32),

8 := ρ1((≥1)6 ) + ρ2(

(≥2)5 )

= M(X41) + M(X2

1 X2) + M(X1 X22) + M(X4

2),

· · ·2i := ρ1(

(≥1)2i−2) + ρ2(

(≥2)2i−3) + ρ2(

(≥2)2i−5)

= M(Xi1) + · · · , 4 < i,

2i+1 := ρ1((≥1)2i−1) + ρ2(

(≥2)2i )

= M(Xi1 X2) + · · · , 3 < i,

· · · ,

we have:

• for each ρ ∈ N, (X22, Xρ−1

1 X2, Xρ1 ), = T<(I + mρ),

• for each ρ ∈ N, X22 −X2

1 −X31, Xρ−1

1 X2, Xρ1 is the Grobner basis of I+mρ

w.r.t. <,• for each i ∈ N, N<(2i ) = Xi

1, N<(2i+1) = Xi1 X2,

• for each ρ ∈ N, M(I + mρ) = SpanK i , 0 ≤ i ≤ 2ρ,• for each ρ ∈ N,

N<(I + mρ) = 1 ∪ Xi1, Xi−1

1 X2, 1 ≤ i < ρ = T<M(I + mρ),

• the Grobner basis of I is X22 − X2

1 − X31,

• M(I) = SpanK (i , i ∈ N),• N<(I) = 1 ∪ Xi

1, Xi−11 X2, i ≥ 1 = T<M(I)

and each i satisfies the statement of Corollary 32.4.2.

32.5 Polynomial Evaluation 531

32.5 Polynomial Evaluation at Macaulay Bases

Each polynomial f ∈ K [X1, . . . , Xr ] := Q can be uniquely represented, viarecursive Horner representation (see Definition 29.3.5) as

f (X1, . . . , Xr ) = H0( f ) +r∑

j=1

X jH j ( f )(X1, . . . , X j ),

where H0( f ) = f (0) ∈ K , and, for each j , H j ( f ) ∈ K [X1, . . . , X j ].Let us now assume we are given, via recursive Horner representation, a

polynomial f ∈ Q and a Macaulay basis 1, . . . , s via the elementsci jh ∈ K , 1 ≤ j ≤ r, 1 ≤ i < h ≤ s, such that, for each h and j ,

( j)h =

h−1∑i=1

ci jhρ j ((≥ j)i ). (32.1)

Proposition 32.5.1. For each h, j, 1 ≤ j ≤ r, 1 ≤ h ≤ s there are polynomi-als fh j ∈ K [X1, . . . , X j ] such that

fh j = ∑h−1i=1

∑rν= j ci jhH j ( fiν);

( j)h ( f ) = fh j (0) = ∑h−1

i=1∑r

ν= j ci jh(H j ( fiν))(0) or, equivalently,

( j)h ( f ) = H0( fh j ) = ∑h−1

i=1∑r

ν= j ci jhH0(H j ( fiν)).

Proof. Let us express f and each ( j)h as

f =∑t∈W

c( f, t)t, ( j)h =

∑τ∈W

αhjτ M(τ )

and let us remark that, for each h, j, τ ,

( j)h ( f ) = ∑

τ∈W αhjτ c( f, τ );

αhjτ =

0 if τ is not a multiple of X j ,∑h−1i=1

∑rν= j ci jhαiνω, if τ = X j · ω

The first formula follows directly from the definition of M(τ ); the second justrequires us to expand the formula of Equation (32.1).

Let us then define, for each j and h,

fh j :=∑

υ∈W[1, j]

υ∑τ∈W

c( f, τυ)αhjτ

where we have set

W[1, j] = W ∩ K [X1, . . . , X j ].

532 Grobner III

Then we claim that, for each h, j :

(1) ( j)h ( f ) = fh j (0) = H0( fh j );

(2) H j ( fiν) =

⎧⎪⎪⎨⎪⎪⎩

0 if j > ν,∑υ∈W[1,ν]

υ∑

τ∈Wc( f, τ X jυ)αiντ if ν ≥ j > 0,∑

τ∈Wc( f, τ )αiντ if j = 0;

(3) fh j = ∑h−1i=1

∑rν= j ci jhH j ( fiν);

(4) H0( fh j ) = ∑h−1i=1

∑rν= j ci jhH0(H j ( fiν)).

In fact:

(1) ( j)h ( f ) = ∑

τ∈W αhjτ c( f, τ ) = ∑υ∈W[1, j]

υ(0)∑

τ∈Wc( f, τυ)αhjτ =

fh j (0);(2) obvious;(3) as a consequence of (2), one has

fh j =∑

υ∈W[1, j]

υ∑τ∈W

c( f, τυ)αhjτ

=∑

υ∈W[1, j]

υ∑ω∈W

c( f, ωX jυ)

h−1∑i=1

r∑ν= j

ci jhαiνω

=h−1∑i=1

r∑ν= j

ci jh

∑υ∈W[1, j]

υ∑ω∈W

c( f, ωX jυ)αiνω

=h−1∑i=1

r∑ν= j

ci jhH j ( fiν);

(4) obvious.

Corollary 32.5.2. With the notation and assumptions above, it is possible tocompute

( j)h ( f ) for each h, j, 1 ≤ j ≤ r, 1 ≤ h ≤ s, with complexity

O(r2s2).

Proof. We need to compute each H0( fh j ) but each such element fh j is aHorner component of the recursive Horner representation of f since each fh j

is a combination of Horner components of fiν, i < h, and

f1 j := H0( f ) +j∑

i=1

XiHi ( f )

for each j , because 1 = Id.

32.6 Continuations 533

32.6 Continuations

Let < be an inf-limited ordering, I ⊂ Q be an m-primary ideal, V := M(I),Λ := 1, . . . , s be a Macaulay basis of V .

As a direct consequence of Corollary 32.4.3, the K -basis

Γ := ρ j ((≥ j)i ), 1 ≤ j ≤ r, 1 ≤ i ≤ s

satisfies the following result.

Theorem 32.6.1. Let ∈ SpanK (M) \ V be such that

U := λ + a : λ ∈ V, a ∈ K is stable. Then ∈ SpanK (Γ ).

Our aim here is to discuss the structure both of V and of each stable exten-sion

U := λ + a : λ ∈ V, a ∈ K in view of Corollary 32.4.3 and Theorem 32.6.1; for doing that we will system-atically study the example introduced in Example 32.1.5 when r = 3 under therefinement of vw by the lexicographical ordering induced by X1 ≺ · · · ≺ Xr .

Example 32.6.2. Let us set

f1 := X2 − X21, f2 := X3 − X3

1,

I := ( f1, f2) and let us consider the refinement < of vw by the reversed lexi-cographical ordering induced by X1 X2 X3. Then we have

• the Grobner basis of I w.r.t. < is X21 − X2, X1 X2 − X3, X2

2 − X1 X3,• X2

1, X1 X2, X22 = T<(I),

• N<(I)1 ∪ X1 Xi−13 , X2 Xi−1

3 , Xi3, i ∈ N = T<M(I),

• for each i ∈ N,

T<(3i−2) = Xi−13 , T<(3i−1) = X1 Xi−1

3 , T<(3i ) = X2 Xi−13 ,

• for each ρ ∈ N,

(X21, X1 X2, X2

2, X1 Xρ−13 , X2 Xρ−1

3 , Xρ3 ) = T<(I + mρ),

• for each ρ ∈ N,

X21 − X2, X1 X2 − X3, X2

2 − X1 X3, X1 Xρ−13 , X2 Xρ−1

3 , Xρ3

is the Grobner basis of I + mρ w.r.t. <,• N<(I) = 1 ∪ X1 Xi−1

3 , X2 Xi−13 , Xi

3, i < ρ = T<M(I).

534 Grobner III

In particular,

1 := M(1),

2 := M(X1),

3 := M(X2) + M(X21),

4 := M(X3) + M(X1 X2) + M(X31),

5 := M(X1 X3) + M(X22) + M(X2

1 X2) + M(X41),

6 := M(X2 X3) + M(X21 X3) + M(X1 X2

2) + M(X31 X2) + M(X5

1),

7 := M(X23) + M(X1 X2 X3) + M(X3

1 X3) + M(X32)

+ M(X21 X2

2) + M(X41 X2) + M(X6

1);

as a consequence we have

ρ1(1) := M(X1),

ρ2(1) := M(X2),

ρ3(1) := M(X3),

ρ1(2) := M(X21),

ρ1(3) := M(X1 X2) + M(X31),

ρ2(3(2)) := M(X2

2),

ρ1(4) := M(X1 X3) + M(X21 X2) + M(X4

1),

ρ2(4(≥2)) := M(X2 X3),

ρ3(4(3)) := M(X2

3),

ρ1(5) := M(X21 X3) + M(X1 X2

2) + M(X31 X2) + M(X5

1),

ρ2(5(2)) := M(X3

2),

ρ1(6) := M(X1 X2 X3) + M(X31 X3) + M(X2

1 X22)

+ M(X41 X2) + M(X6

1),

ρ2(6(2)) := M(X2

2 X3),

ρ1(7) := M(X1 X23) + M(X2

1 X2 X3) + M(X41 X3) + M(X1 X3

2)

+ M(X31 X2

2) + M(X51 X2) + M(X7

1),

ρ2(7(≥2)) := M(X2 X2

3) + M(X42),

ρ3(7(3)) := M(X3

3).

This information (and other information which will be deduced during thefollowing discussion) can be summarized in the following tables:


Remark 32.6.3. The structure described in Corollary 32.4.3 and Theorem32.6.1 implies that it is easy to iteratively compute, for each j, h, i, 1 ≤ j, h ≤r, 1 ≤ i ≤ s, σh(i ) and σh(ρ j (

(≥ j)i )) = ρ jσh(

(≥ j)i ), since

i =r∑

j=1

i−1∑ι=1

cιj iρ j ((≥ j)ι ) ⇒ σh(i ) =

r∑j=1

i−1∑ι=1

cιj iσhρ j ((≥ j)ι ),

and

σh(i ) =i−1∑ι=1

cιι ⇒ σh(ρ j ((≥ j)i )) =

⎧⎨⎩

0 if h < j ,(≥ j)i if h = j ,∑i−1

ι=1 cιρ j ((≥ j)ι ) if h > j .

For instance, in the example we are discussing, we have

σ1(7) = σ1ρ1(6) + σ1ρ2((2)5 ) + σ1ρ3(

(3)4 ) = 6 + 0 + 0 = 6,

σ2(7) = σ2ρ1(6) + σ2ρ2((2)5 ) + σ2ρ3(

(3)4 ) =

(1)5 +

(2)5 + 0 = 5,

σ3(7) = σ3ρ1(6) + σ3ρ2((2)5 ) + σ3ρ3(

(3)4 ) =

(1)4 + 0 +

(3)4 = 4,

σ2ρ1(7) = ρ1(5) = (1)6 ,

σ3ρ1(7) = ρ1(4) = (1)5 ,

σ3ρ2((≥2)7 ) = ρ2(

(≥2)4 ) = ρ2(

(3)4 ) =

(2)6 .

The complete table is obtained by means of this recursive evaluation.

As a consequence of Corollary 32.4.3 and Theorem 32.6.1, we know notonly that there exist ci j ∈ K , 1 ≤ j ≤ r, 1 ≤ i ≤ s, such that

λ =r∑

j=1

s∑i=1

ci jρ j ((≥ j)i ),

but also that

σh(λ) ∈ V, for each h,

because, by assumption, U is stable; therefore, since Λ is Macaulay, if t ∈ Wis the term defined by T<(λ) = M(t), then necessarily

M(T<(λ)) ∈ T<U , σi (M(T<(λ))) ∈ T<U for each i.

On the basis of these remarks I introduce

536 Grobner III

λ λ(1) λ(2) λ(3) σ1(λ) σ2(λ) σ3(λ)

1 1 M(1) 0 0 0

X1 2 ρ1(1) 1 0 0

X21 ρ1(2) 2 0 0

X2 ρ2(1) 0 1 0

3 ρ1(2) + ρ2(1) 2 1 0

X1 X2 ρ1(3) 3 2 0

X22 ρ2(

(2)3 ) 0

(2)3 0

X3 ρ3(1) 0 0 1

4 ρ1(3) + ρ3(1) 3 2 1

X1 X3 ρ1(4) 4 (1)3 2

5 ρ1(4) + ρ2((2)3 ) 4 3 2

X2 X3 ρ2((3)4 ) 0

(3)4

(2)3

6 ρ1(5) + ρ2((3)4 ) 5 4 3

X23 ρ3(

(3)4 ) 0 0

(3)4

7 ρ1(6) + ρ2((2)5 ) + ρ3(

(3)4 ) 6 5 4

.

.

....

.

.

....

.

.

....

.

.

....

X1 Xi3 ρ1(3i+1) 3i+1

(1)3i

(1)3i−1

3i+2 ρ1(3i+1) + ρ2((≥2)3i ) 3i+1 3i 3i−1

X2 Xi3 ρ2(

(≥2)3i+1) 0

(≥2)3i+1

(≥2)3i

3i+3 ρ1(3i+2) + ρ2((≥2)3i+1) 3i+2 3i+1 3i

Xi+13 ρ3(

(3)3i+1) 0 0

(3)3i+1

3i+4 ρ1(3i+3) + ρ2((≥2)3i+2) + ρ3(

(3)3i+1) 3i+3 3i+2 3i+1

.

.

....

.

.

....

.

.

....

.

.

....

Definition 32.6.4. The corner set of V (see Definition 29.3.1) is the set

C<(V ) := τ ∈ W : M(τ ) ∈ N<(V ), σi (M(τ )) ∈ T<V for each i= τ ∈ T<(I(V )) : all its predecessors ω ∈ N<(I(V ))= G<(I(V )).

Any element

:= M(T<()) +∑ω∈W

cω M(ω) ∈ SpanK (M) \ V

such that


λ σ1(λ) σ2(λ) σ3(λ) λ( f1)(0) λ( f2)(0) T<(λ)

1 = M(1) 0 0 0 0 0 1

2 = ρ1(1) 1 0 0 0 0 X1

(2)3 = ρ2(1) 0 1 0 1 0 X2

(3)4 = ρ3(1) 0 0 1 0 1 X3

(1)3 = ρ1(2) 2 0 0 -1 0 X2

1

(1)4 = ρ1(3) 3 2 0 0 -1 X1 X2

(2)5 = ρ2(3

(≥2)) 0 3(≥2) 0 0 0 X2

2

(1)5 = ρ1(4) 4 3

(1) 2 0 0 X1 X3

(2)6 = ρ2(4

(≥2)) 0 4(≥3) 3

(≥2) 0 0 X2 X3

(3)7 = ρ3(4

(≥3)) 0 0 4(≥3) 0 0 X2

3

(1)6 = ρ1(5) 5 4

(1) (1)3 0 0 X2

1 X3

(2)7 = ρ2(5

(≥2)) 0 5(≥2) 0 0 0 X3

2

(1)7 = ρ1(6) 6 5

(1) (1)4 0 0 X3

1 X3

(2)8 = ρ2(6

(≥2)) 0 6(≥2) 5

(≥2) 0 0 X22 X3

(1)8 = ρ1(7) 7 6

(1) (1)5 0 0 X1 X2

3

(2)9 = ρ2(7

(≥2)) 0 7(≥2) 6

(≥2) 0 0 X2 X23

(3)10 = ρ3(7

(≥3)) 0 0 7(≥3) 0 0 X3

3

.

.

....

.

.

....

.

.

....

.

.

....

(1)3i+3 = ρ1(3i+2) 3i+2

(1)3i+1

(1)3i 0 0 X1 Xi

3

(2)3i+4 = ρ2(

(≥2)3i+2) 0

(≥2)3i+2

(≥2)3i+1 0 0 X2 Xi

3

(1)3i+4 = ρ1(3i+3) 3i+3

(1)3i+2

(1)3i+1 0 0 X1 X2 Xi

3

(2)3i+5 = ρ2(

(≥2)3i+3) 0

(≥2)3i+3

(≥2)3i+2 0 0 X2

2 Xi3

(1)3i+5 = ρ1(3i+4) 3i+4

(1)3i+3

(1)3i+2 0 0 X1 Xi+1

3

(2)3i+6 = ρ2(

(≥2)3i+4) 0

(≥2)3i+4

(≥2)3i+3 0 0 X2 Xi+1

3

(3)3i+7 = ρ3(

(≥3)3i+4) 0 0

(≥3)3i+4 0 0 Xi+2

3

(c1) T<() ∈ C<(V ),(c2) σ j () ∈ V for each j ,(c3) cω = 0 ⇒ ω /∈ T<V

is called a continuation of V at τ := T<().

An elementary continuation of V at τ ∈ C<(V ) is a continuation

:= M(T<()) +∑ω∈W

cω M(ω)

538 Grobner III

which, moreover, satisfies

(c4) if M(ω) ∈ C<(V ), cω = 0, then there is no continuation of V at ω.

Lemma 32.6.5. The following conditions are equivalent:

(1) U is stable and Λ ∪ is its Macaulay basis,(2) τ := T<() ∈ C<(V ) and is a continuation of V at τ.

Example 32.6.6. For instance, for ρ = 3, we have

V := M(I3) = SpanK (1, 2, 3),

and

C<(V ) := X21, X1 X2, X2

2, X3;the elementary continuations of V at

X21 are λ := aρ1(2), a ∈ K \ 0,

X1 X2 are λ := aρ1(3), a ∈ K \ 0,X3 are λ := aρ3(1), a ∈ K \ 0,

and the continuations of V at

X21 are λ := aρ1(2), a ∈ K \ 0,

X1 X2 are λ := aρ1(3) + bρ1(2), a, b ∈ K , a = 0,

X3 are λ := aρ3(1) + bρ1(3) + cρ1(2), a, b, c ∈ K , a = 0.

On the other hand there is no continuation at X22 since for each λ : T<(λ) = X2

2there necessarily holds

σ2(λ) = a3(2) + b2 + c1, a, b, c ∈ K , a = 0,

so that σ2(λ) /∈ V .

For ρ = 4, we have

V := M(I4) = SpanK (1, 2, 3, 4),

where 4 = ρ3(1) + ρ1(3) is a continuation of M(I3) at X3, and

C<(V ) := X21, X1 X2, X2

2, X1 X3, X2 X3, X23;

the elementary continuations of V at

X21 are λ := aρ1(2), a ∈ K \ 0,

X1 X2 are λ := aρ1(3), a ∈ K \ 0,X1 X3 are3 λ := aρ1(4) + aρ2(3

(2)), a ∈ K \ 0,

3 Since σ2ρ1(4) = (1)3 , in order to give that σ2(λ) ∈ V we must add aρ2(3

(2)) to aρ1(4) sothat σ2(λ) = a3.


and there is no continuation at

X2 X3 because for each4 λ : T<(λ) = X2 X3 both

σ2(λ) = a4(3) +b3

(1) +c3(2) +d2 +e1, a, b, c, d, e ∈ k, a = 0,

and

σ3(λ) = a3(2) + b2 + c1, a, b, c ∈ K , a = 0,

are not members of V ;X2

3 because for each λ : T<(λ) = X23

σ3(λ) = a4(3) + b3

(2) + c2 + d1, a, b, c, d ∈ K , a = 0,

is not a member of V .

The relation between elementary and other continuations is clarified by thefollowing.

Lemma 32.6.7. Let ′ and ′′ be two different continuations of V at τ . Then′ − ′′ is a continuation of V at some ω > τ , ω ∈ C<(V ).

Proof. Under the assumptions, ′ − ′′ clearly satisfies (c2) and (c3). Alsoω := T<(′ − ′′) > τ is such that ω /∈ T<V – because both and ′′ satisfy(c3) – and, for each i ,

σi (M(ω)) = T<

(σi (

′) − σi (′′)

) ∈ T<V so that ω ∈ C<(V ) and ′ − ′′ also satisfies (c1).

Corollary 32.6.8. If a continuation of V at t exists, then there is exactly oneelementary continuation of V at t which we will denote by CV,t .

Proof. Let ′ = M(t) + ∑ω∈T cω M(ω), cω = 0 ⇒ ω > t , be a continua-

tion of V at t . Let τ be the lowest term such that

• cτ = 0,• M(τ ) ∈ C<(V ), and• there is a continuation ′′ of V at τ .

4 With reference to the tables of Example 32.6.2 we are considering all linear combinations∑i ci σh(λi ) where the λi s run among those elements

( j)κ , 1 ≤ κ ≤ 4, 1 ≤ j ≤ r which

satisfy T<(( j)κ ) ≥ X2 X3.

540 Grobner III

Then ′ − ′′ is a continuation of V at t since it obviously satisfies (c1), (c2),(c3). Moreover, setting

′ − ′′ = M(t) +∑ω∈T

dω M(ω)

with dω = 0 ⇒ ω > t , if there is ω such that dω = 0 and M(ω) ∈ C<(V ),

then ω > τ . So, since C<(V ) is finite, an inductive argument allows us toconclude the proof.

Theorem 32.6.9. The following conditions are equivalent:

(1) U := λ + a : λ ∈ V, a ∈ K is stable and ∆ ∪ is its Macaulaybasis.

(2) There are t0 < · · · < tv , M(ti ) ∈ C<(V ) and ci ∈ K \ 0, 1 ≤ i ≤ v,such that

= CV,t0 +v∑

i=1

ci CV,ti .

Proof. (1) is satisfied if and only if is a continuation of V . The assertionfollows then from the easy remark that any continuation of V is a linear com-bination of elementary continuations of it.

Lemma 32.6.10. Let M(t) ∈ C<(V ) ∩ M[κ, r ] and let ι(κ) be such that

ρκ(T<((κ)ι )) = M(t).

For κ ≤ j ≤ r let J ( j) denote the set of indices i such that

(a) T<(ρ j (( j)i )) ∈ T<V ,

(b) T<(ρ j (( j)i )) > M(t),

(c) if T<(ρ j (( j)i )) ∈ C<(V ) then there is no elementary continuation of V at

T<(ρ j (( j)i )).

The following conditions are equivalent:

(1) the elementary continuation CV,t exists;(2) there are values a ji ∈ K such that, for each µ,

σµρκ((κ)ι ) +

r∑j=1

∑i∈J ( j)

a jiσµρ j (( j)i ) ∈ V .

Moreover, if the above conditions are satisfied,

CV,t = ρκ((κ)ι ) +

r∑j=1

∑i∈J ( j)

a jiρ j (( j)i ).


Proof. If

CV,t = ρκ((κ)ι ) +

r∑j=1

s∑i=1

a jiρ j (( j)i ),

then σµ(CV,t ) ∈ V and a ji = 0 unless i ∈ J ( j) since in the expansion of CV,t

there is no term in T<V nor are there terms in C<(V ) having elementarycontinuations, and moreover

T<(CV,t ) = M(t) = ρκ(T<((κ)ι ) > T<(ρ j (

( j)i )),

for each pair ( j, i) such that a ji = 0.Conversely let

C = ρκ((κ)ι ) +

r∑j=1

∑i∈J ( j)

a jiρ j (( j)i ),

be such that σµ(C) ∈ V ; therefore U := λ + aC : λ ∈ V, a ∈ K is stable.Since the sum is restricted on J ( j), C is the continuation of V at t .

If one knows the values of σh(ρ j ((≥ j)i )), for each j, h, i, 1 ≤ j, h ≤ r, 1 ≤

i ≤ s – which can be elementarily computed as explained in Remark 32.6.3 –the computation of all the continuations of V at each element t in the cornerset of V requires nothing more than efficient book-keeping.

Example 32.6.11. For instance in the cases we discussed in Example 32.6.6we have

X21: λ := ρ1(2) is a continuation since σh(λ) ∈ V, for each h;

X1 X2: λ := ρ1(3) is a continuation for the same reason;X2

2: λ := ρ2(3(2)) is not a continuation since, for each a, b ∈ K ,

σ2(λ) = σ2(λ + aρ1(2) + bρ1(3)) = 3(2) /∈ V ;

X3: λ := ρ3(1) is a continuation since σh(λ) ∈ V for each h;X1 X3: λ := ρ1(4) is not a continuation since σ2(λ) /∈ V ; however, for

λ := ρ1(4) + aρ2(3(2))

we have σ1(λ) = 4 ∈ V, σ3(λ) = 2 ∈ V and

σ2(λ) = 3(1) + a3

(2) ∈ V ⇐⇒ a = 1;

so ρ1(4) + ρ2(3(2)) is a continuation;

542 Grobner III

X2 X3: there is no continuation since, for

λ := ρ2(4(3)) + aρ1(4) + bρ2(3

(2)), a, b ∈ K ,

we have

σ1(λ) = a4 ∈ V,

σ2(λ) = 4(3) + a

(1)3 + b

(2)3 ∈ V,

σ3(λ) = 3(2) + a2 ∈ V ;

X23: there is no continuation since, for

λ := ρ3(4(3)) + aρ2(4

(3)) + bρ2(3(2)), a, b ∈ K ,

we have

σ1(λ) = 0 ∈ V, for each a, b ∈ K ,

σ2(λ) = a4(3) + b

(2)3 ∈ V ⇐⇒ a = b = 0,

σ3(λ) = 4(3) + a

(2)3 + b2 ∈ V, for each a, b ∈ K .

32.7 Computing a Macaulay Basis

We now show how to use the structure of the continuations of m-primary idealsin order to compute the Macaulay basis w.r.t. an inf-limited ordering < of anideal I ⊂ m which

is given by means of any set of generators F := f1, . . . , ft ⊂ m andwhose m-closure is an m-primary ideal.

In particular if we are given any finite set of polynomials F := f1, . . . , ft and we denote by I the ideal generated by F , it is just sufficient, for any ρ ∈ N,to enlarge F by adding all monomials of degree ρ and to apply the algorithmwe are now presenting in order to obtain the Macaulay basis w.r.t. < of the m-primary ideal I + mρ , thus, producing, ‘at least in imagination’, as Macaulayput it, the infinite Macaulay basis of the m-closed ideal

⋂ρ I + mρ .

The only tool we need is the following obvious remark: for each ∈SpanK (M), let us write

ev() := (( f1), . . . , ( ft )) ∈ K t .

If 1, 2, . . . , s denotes the ordered Macaulay basis w.r.t. < of I, whichwe aim to compute, and, for any i < s, we set

32.7 Computing a Macaulay Basis 543

Fig. 32.1. Macaulay basis from any basis

(Λ,M) := MacaulayBasis(F, <)where

F := f1, . . . , ft ⊂ Q,I := (F) an m-primary ideal,< an inf-limited ordering,Λ := 1, . . . , s is the Macaulay basis of M(I)

M = (

b(h)i j

)∈ K s2

, 1 ≤ h ≤ n is the set of the square matrices(

b(h)i j

)defined by σh(i ) = ∑s

j=1 b(h)i j j .

i := 1, 1 := Id, Λ := Id V := SpanK (Λ), C := C := ∅,B := G := X j , 1 ≤ j ≤ r,For j, h, 1 ≤ j, h ≤ r compute σh(M(X j )).Repeat

t := max<(G \ C), B := B \ tCompute (if it exists) CU,tIf CU,t exists then

If there exist cτ such that ev(CU,t ) = ∑τ∈C cτ ev(CU,τ ) then

i := i + 1, i := CU,t − ∑τ∈C cτ CU,τ

For h, j, 1 ≤ h ≤ r , 1 ≤ j < i doCompute b(h)

i j : σh(i ) = ∑sj=1 b(h)

i j j ;B := B ∪ T<(ρ j (

(≥ j)i )), 1 ≤ j ≤ r

G be the minimal basis of the monomial ideal generated by B ∪ C

For j, h, 1 ≤ j, h ≤ r compute σhρ j ((≥ j)i )

elseC := C ∪ CU,t C := C ∪ t

until G \ C := ∅

• Vi := 1, 2, . . . , i ,• Ci := τ ∈ C<(Vi ) : there is an elementary continuation of Vi at τ ,we know that, for each i, there exists cτ ∈ K such that i+1 = ∑

τ∈Cicτ CVi ,τ .

Since

i+1 ∈ M(I) ⇐⇒ ev(i+1) =∑τ∈Ci

cτ ev(CVi ,τ ) = 0,

i+1 can be obtained by solving this linear equation, since each ev(CVi ,τ ) canbe computed by the scheme described in Section 32.4.

All the other auxiliary tools having already been described in the previoussections, we can limit ourselves to describing the algorithm in Figure 32.1; be-cause it essentially consists of linear algebra reduction of sr vectors in K sr+t ,its complexity is O(s3r3).

Example 32.7.1. Let us now consider Example 32.4.5, where our knowledgeof 2i and 2i+1 and the results of the formulas of Remark 32.6.3 can be

544 Grobner III

summarized as

(1)2i = ρ1(

(≥1)2i−2),

(1)2i+1 = ρ1(

(≥1)2i−1),

(2)2i = ρ2(

(≥2)2i−3) + ρ2(

(≥2)2i−5),

(2)2i+1 = ρ2(

(≥2)2i ) + ρ2(

(≥2)2i−2),

σ1(2i ) = 2i−2,

σ1(2i+1) = 2i−1,

σ2(2i ) = 2i−3 + 2i−5,

σ2(2i+1) = 2i + 2i−2,

σ2ρ1((≥1)2i ) = ρ1(

(≥1)2i−3) + ρ1(

(≥1)2i−5),

σ2ρ1((≥1)2i+1) = ρ1(

(≥1)2i ) + ρ1(

(≥1)2i−2);

from which we can deduce

σ2ρ1(2i ) = ρ1((≥1)2i−3) + ρ1(

(≥1)2i−5)

= (1)2i−1 +

(1)2i−3,

(1)2i+2 = ρ1(

(≥1)2i ),

σ2((2)2i+2) = σ2(2i+2) − σ2ρ1(

(≥1)2i )

= σ2(2i+2) − (1)2i−1 −

(1)2i−3

= (2)2i−1 +

(2)2i−3,

(2)2i+2 = ρ2σ2(

(2)2i+2)

= ρ2((2)2i−1) + ρ2(

(2)2i−3),

2i+2 = ρ1((≥1)2i ) + ρ2(

(2)2i−1) + ρ2(

(2)2i−3),

and, similarly,

σ2ρ1(2i+1) = ρ1((≥1)2i ) + ρ1(

(≥1)2i−2)

= (1)2i+2 +

(1)2i ,

(1)2i+3 = ρ1(

(≥1)2i+1),

σ2((2)2i+3) = σ2(2i+3) − σ2ρ1(

(≥1)2i+1)

= σ2(2i+3) − (1)2i+2 −

(1)2i

= (2)2i+2 +

(2)2i ,

(2)2i+3 = ρ2σ2(

(2)2i+3)


= ρ2((2)2i+2) + ρ2(

(2)2i ),

2i+3 = ρ1((≥1)2i+1) + ρ2(

(2)2i+2) + ρ2(

(2)2i ).

Thus we prove the claims we made in Example 32.4.5.

Example 32.7.2. It is easier to verify the structure of Example 32.6.2, mainlyby checking its presentation in the table included, and to deduce that the algo-rithm performs the following computations:

t := X1 Xi3:

(1)3i+2 = ρ1(3i+1),

σ2((1)3i+2) = σ2ρ1(

(1)3i+1) =

(1)3i ,

(2)3i+2 = ρ2(

(≥2)3i ),

σ2(3i+2) = 3i ,

σ3((1)3i+2 +

(2)3i+2) = σ3ρ1(

(1)3i+1) + σ3ρ2(

(≥2)3i ) =

(1)3i +

(≥2)3i = 3i ,

3i+2 = ρ1((≥1)3i+1) + ρ2(

(≥2)3i );

t := X2 Xi3:

(1)3i+3 = ρ1(3i+2),

σ2((1)3i+3) = σ2ρ1(

(1)3i+2) =

(1)3i+1,

(2)3i+3 = ρ2(

(≥2)3i+1),

σ2(3i+3) = 3i+1,

σ3((1)3i+3 +

(2)3i+3) = σ3ρ1(

(1)3i+2) + σ3ρ2(

(≥2)3i+1) =

(1)3i +

(≥2)3i = 3i,

3i+3 = ρ1((≥1)3i+2) + ρ2(

(≥2)3i+1);

t := Xi+13 :

(1)3i+4 = ρ1(3i+3),

σ2((1)3i+4) = σ2ρ1(

(1)3i+3) =

(1)3i+2,

(2)3i+4 = ρ2(

(≥2)3i+2),

σ2(3i+4) = 3i+2,

σ3((1)3i+4 +

(2)3i+4) = σ3ρ1(

(1)3i+3) + σ3ρ2(

(≥2)3i+2) =

(1)3i+1 +

(2)3i+1,

(3)3i+4 = ρ3(

(3)3i+1),

σ3(3i+4) = 3i+1,

3i+4 = ρ1((≥1)3i+2) + ρ2(

(≥2)3i+1) + ρ3(

(3)3i+1).

546 Grobner III

Of course, for small examples, where space complexity is not a problem,most of the technology we introduced here – Horner representation, polyno-mial evaluation, efficient book-keeping of the forms σhρ j (

(≥ j)i ) – can be dis-

posed with and an easier paper-and-pencil computation can be performed.5 Wedescribe it by means of the following

Example 32.7.3. Let us compute the Macaulay basis of the m-primary idealI := f1, f2, f3, f4,

f1 := X32 − X1 X2

2, f2 := X21 X2, f3 := X3

1 − X22 + X1 X2, f4 := X4

2

(see Examples 28.2.6, 28.2.8, 29.2.2, 29.2.4 and 29.3.9), w.r.t. the inf-limitedordering < which is the reverse of the degree lexicographical ordering ≺ in-duced by X1 ≺ X2, that is the ordering

1 > X1 > X2 > X21 > X1 X2 > X2

2 > X31 > X2

1 X2 > X1 X22 > X3

2 > X41 > · · · .

The first definitions are trivial:

1 := Id,

2 := M(X1),

3 := M(X2),

4 := M(X21).

Then:

X1 X2: The elementary continuation γ1 := ρ1(3) := M(X1 X2) is not aninverse function since ev(γ1) = (0, 0, 1, 0).

X22: The same happens for the elementary continuation γ2 := ρ2(3) :=

M(X22) which satisfies ev(γ2) = (0, 0, −1, 0) = ev(γ1); therefore

we have

5 := γ2 + γ1 = M(X22) + M(X1 X2).

X31: The same happens also for the elementary continuation

γ3 := ρ1(4) := M(X31), ev(γ3) = (0, 0, 1, 0) = ev(γ1);

so that

6 := γ3 − γ1 = M(X31) − M(X1 X2).

X32: We begin by computing

ρ2(5) := M(X32) + M(X1 X2

2)

5 However, the computations performed on Example 32.4.2 and reported here are also obtainedvia a paper-and-pencil computation strongly supported by training and educated guesses.


and checking whether σ1ρ2(5) ∈ SpanK i , 1 ≤ i ≤ 6: from

σ1ρ2(5) = M(X22) =

(2)5

we obtain the elementary continuation

γ4 := M(X32) + M(X1 X2

2) + M(X21 X2), ev(γ4) = (0, 1, 0, 0).

X41: We produce the elementary continuation

γ5 := ρ1(6) := M(X41) − M(X2

1 X2), ev(γ5) = (0, −1, 0, 0),

and the inverse function

7 := γ5 + γ4 = M(X41) + M(X3

2) + M(X1 X22).

X51: We compute

ρ1(7) := M(X51) + M(X1 X3

2) + M(X21 X2

2),

σ2ρ1(7) = M(X1 X22) + M(X2

1 X2) = γ(1)4 ,

whence

γ6 := M(X51)+ M(X4

2)+ M(X1 X32)+ M(X2

1 X22), ev(γ6) = (0, 0, 0, 1).

Thus we obtain

the Macaulay basis Λ := i , 1 ≤ i ≤ 7,the order ideal T<(Λ) = N<(I) = 1, X1, X2, X2

1, X22, X3

1, X41,

the elementary continuations γ1, γ3, γ6 respectively associated to theelements of G<(I) = X1 X2, X3

2, X51.

Note also that Λ is ordered so as to satisfy Corollary 32.3.3.

Example 32.7.4. Example 32.4.5 gives a nice illustration of Macaulay’sDefinition 30.5.1.

Write, for each i ∈ N,

Λ2i := SpanK

( j , j ≤ 2i − 2 ∪ 2i )

Λ2i+1 := SpanK

( j , j ≤ 2i + 1)and qi := I(Λi ) and remark that, for each i ∈ N,

Λ2i ∩ Λ2i−1 = SpanK j , j ≤ 2i − 2 = M(I + mi−1),Λ2i + Λ2i−1 = SpanK j , j ≤ 2i = M(I + mi ),q2i ∩ q2i−1 = I + mi is a reduced decomposition,q2i+1 ⊂ q2i , q2i ⊂ q2i−1, q2i ⊂ q2i−2,each qi is a zero-dimensional principal system,

548 Grobner III

I is a principal system defined by any chain

qi1 ⊃ qi2 ⊃ · · · ⊃ qi j ⊃ qi j+1 ⊃ · · · ⊃ I

satisfying

i1 < i2 < · · · < i j < i j+1 < · · ·,i j+1 − i j ≥ 1 if i j+1 ≡ 1 (mod 2),i j+1 − i j ≥ 2 if i j+1 ≡ 2 (mod 2).

Example 32.7.5. We now have the technology to describe the inverse systemof the example we have discussed throughout Section 30.5, that is the ideal

I := (x1 + x3, x2 + x3) ⊂ k[x1, x2, x3],

for which, for any ordering such that x3 < x2 < x1 we have N(I) = xi3 : i ∈

N.Writing,

δi := M(xi3) +

i∑d=1

(−1)d∑

τ∈Wd

τ xi−d3

where W := xa11 xa2

2 : (a1, a2) ∈ N2, it is easy to verify that

δi ∈ M(I) andσ1(δi ) = σ2(δi ) = σ3(δi ) = δi−1.

33

Moller II

In connection with his solution of Problem 23.3.3, Macaulay gave an algo-rithm, which, given an order ideal

N ⊂ T := Xa11 . . . Xan

n : (a1, . . . , an) ∈ Nn,

produces:

a finite set of points

X := a1, . . . , as ⊂ kn, ai := (ai1, . . . , ain), #(N) = #(X);a bijection Φ : X → N;a set of polynomials

gτ ∈ P := k[X1, . . . , Xn], τ ∈ Xiω : ω ∈ N, 1 ≤ i ≤ nsuch that, writings

I := f : f (ai1, . . . , ain) = 0, 1 ≤ i ≤ sand, for each τ ∈ N, using the functional τ defined by

τ ( f ) = f (ai1, . . . , ain), f ∈ P, ai := Φ−1(τ )

we have:

N = N(I);gτ : τ ∈ G(I) is the reduced Grobner basis of I w.r.t. the lexicographical

ordering induced by X1 < · · · < Xn ;gτ : τ ∈ G(I) and τ : τ ∈ N are inverse.

After presenting a slight generalization of this construction by Macaulay(Section 33.1) I present some recent and interesting converse results:

549

550 Moller II

Lazard’s description of the structure of the lexicographical Grobner basis ofan ideal in two variables (Theorem 33.1.5);

an algorithm by Cerlienco and Mureddu which, given a finite set X ⊂ kn ofpoints computes, with the notation above, the order ideal N(I) and abijection Φ : X → N satisfying the properties granted by Macaulay’sresult (Section 33.2).

I merge them into a description of the Grobner structure of an intersection ofprimary ideals (Section 33.3); the tool to prove this structural theorem is adirect application of Moller’s algorithm (Section 33.6).

33.1 Macaulay’s Trick

In connection with his solution of Problem 23.3.3, Macaulay needed to show,for any function H(T ) : N → N satisfying the proved bound, the existenceof an ideal I ⊂ P satisfying H(T ; I) = H(T ), at least in the case of a zero-dimensional ideal; if the ideal is assumed to be homogeneous, the extremalmonomial ideal L, for which hH(T ; L) = hH(T ), is the required solution;but for the non-homogeneous case, Macaulay needed to produce an ideal Isatisfying both H(I) = H(L) and therefore also the relation T<(I) = L for anydegree-compatible term ordering <.

We discuss here a slightly extension of his trick, which allows us to solvethe following.

Problem 33.1.1. Given a finite set of terms m1, . . . , mr ∈ T and a term or-dering < on T , produce a set of elements g1, . . . , gr ∈ P such that;

• T(gi ) = mi , for each i;• G := g1, . . . , gr is a Grobner basis;

so that, denoting by I the ideal generated by G, we have:

• T(I) = T(G) = (m1, . . . , mr ).

LetM := n1, . . . , ns ⊂ T

be a finite sequence 1 such that:

for each i, 1 ≤ i ≤ r, there exists Ji ⊂ 1, . . . , s such that mi = ∏l∈Ji

nl;for each i, j, 1 ≤ i < j ≤ r, lcm(mi , m j ) = ∏

l∈Ji ∪J j

nl .

1 Caveat lector! A sequence and not just a set. If we have m1 := X2, m2 := XY , we must returnn1 := n2 := X, n3 := Y and J1 := 1, 2, J2 := 1, 3.

33.1 Macaulay’s Trick 551

Clearly such a list M can be easily obtained, by repeated gcds. Now let uschoose, for each l, 1 ≤ l ≤ s, an element hl ∈ P such that T(hl) < nl ; and letus define:

γl := nl − hl , for each l, 1 ≤ l ≤ s;gi := ∏

l∈Ji

γl , for each i, 1 ≤ i ≤ r.

With this notation, for each pair i, j, 1 ≤ i < j ≤ r, we have by constructionti j = ∏

l∈J j \Ji

nl , and t j i = ∏l∈Ji \J j

nl , where ti j , t j i are the elements satisfying

ti j T(gi ) = T(i, j) = lcm(T(gi ), T(g j )) = t j i T(g j ).

Proposition 33.1.2. The set G := g1, . . . , gr is a Grobner basis.

Proof. We have to prove, for each pair i, j, 1 ≤ i < j ≤ r, that the S-pairS(i, j) has a Grobner representation. To do so, let us define

φi j :=⎛⎝ ∏

l∈J j \Ji

γl

⎞⎠ − ti j and φ j i :=

⎛⎝ ∏

l∈Ji \J j

γl

⎞⎠ − t j i .

Clearly, since

ti j = T

⎛⎝ ∏

l∈J j \Ji

γl

⎞⎠ and t j i = T

⎛⎝ ∏

l∈Ji \J j

γl

⎞⎠ ,

we have T(φi j ) < ti j and T(φ j i ) < t j i . Therefore we can claim that

S(i, j) = −φi j gi + φ j i g j

is the required standard representation. In fact we have

0 = −∏

l∈Ji ∪J j

γl +∏

l∈J j ∪Ji

γl

= −⎛⎝ ∏

l∈J j \Ji

γl

⎞⎠ gi +

⎛⎝ ∏

l∈Ji \J j

γl

⎞⎠ g j

= −(φi j + ti j )gi + (φ j i + t j i )g j

= −φi j gi + φ j i g j − (ti j gi − t j i g j

)= −φi j gi + φ j i g j − S(i, j),

552 Moller II

so that, the claim holds, since

T(φi j gi ) < ti j T(gi ) = T(i, j) = t j i T(g j ) > T(φ j i g j ).

For any finite set X of points

X := a1, . . . , as ⊂ kn, ai := (ai1, . . . , ain)

let us denote:

for each i , by i the linear functional i ∈ P∗ defined by

i ( f ) = f (ai1, . . . , ain) for each f (X1, . . . , Xn) ∈ P;and writeL(X) := Spank(i , 1 ≤ i ≤ s) ⊂ P∗;I(X) := f ∈ P : f (ai ) = 0, for each i = P(L(X)).

With this notation we can now present Macaulay’s result: let N ⊂ T be afinite-order ideal of T , and let

G := m1, . . . , mr , ml = Xe1l1 · · · Xenl

n , for each l,

be the minimal basis of the monomial ideal T \ N.Since N is finite, for each i there exists di ∈ N such that

Xdii ∈ G and eil ≤ di , for each l.

Let us then choose, for each i, j, k, j = k, the elements

ai j ∈ k, 1 ≤ i ≤ n, 0 ≤ j < di : ai j = aik,

and define, for each l, 1 ≤ l ≤ r ,

gl :=n∏

i=1

eil−1∏j=0

(Xi − ai j ),

which satisfies T(gl) = ml .

Moreover, to each term t = Xe11 · · · Xen

n ∈ N let us associate the affine point

a(t) := (a1e1 , . . . anen ) ∈ kn,

and let X := a(t) : t ∈ N. Then:

Corollary 33.1.3 (Macaulay).With this notation, for any degree-compatible term ordering, we have:

(1) N = N(I(X)),

(2) G(I(X)) := g1, . . . , gr is the reduced Grobner basis of I(X).

33.1 Macaulay’s Trick 553

Since ei ≤ di , for each t = Xe11 . . . Xen

n ∈ X jτ : 1 ≤ j ≤ n, τ ∈ N andfor each i, it is natural to consider also the polynomials

gt :=n∏

i=1

ei −1∏j=0

(Xi − ai j ), t = Xe11 . . . Xen

n ∈ X jτ : 1 ≤ j ≤ n, τ ∈ N

and investigate their relation with the Lagrange Interpolation Formula (Corol-lary 28.2.2).

Let us order N := t1, . . . , ts in such a way that t1 < t2 < · · · < ts , where< is the lexicographical ordering induced by X1 < · · · < Xn ; and let us writeai := a(ti ) in order to fix a suitable enumeration of X and L(X). Moreover letus define qi := gti , for each i, 1 ≤ i ≤ s. Then:

Lemma 33.1.4. For any degree-compatible term ordering, we have:

(1) gt : t ∈ B(I(X)), is the border basis of I(X);(2) gt : t ∈ G(I(X)), is the reduced Grobner basis of I(X);(3) q := qi : 1 ≤ i ≤ s is a triangular set of L(X).

For n = 2, the structure of the Grobner basis constructed by Macaulay forthe ideal I(X) is an illustrative example of the Lazard Theorem which describesthe structure of the lexicographical Grobner basis for any ideal I ⊂ k[X1, X2]:

Theorem 33.1.5 (Lazard). Let P := k[X1, X2] and let < be the lexicograph-ical ordering induced by X1 < X2.

Let I ⊂ P be an ideal and let f0, f1, . . . , fk be a Grobner basis of Iordered so that

T( f0) < T( f1) < · · · < T( fk).

Then:

• f0 = PG1 · · · Gk+1;• f j = P Hj G j+1 · · · Gk+1, 1 ≤ j < k;• fk = P Hk Gk+1;

where:

P is the primitive part of f0 ∈ k[X1][X2];Gi ∈ k[X1], 1 ≤ i ≤ k + 1;Hi ∈ k[X1][X2] is a monic polynomial of degree d(i), for each i;d(1) < d(2) < · · · < d(k);

554 Moller II

Hi+1 ∈ (G1 . . . Gi , H1G2 . . . Gi , . . . , Hj G j+1 . . . Gi , . . . , Hi−1Gi , Hi ) forall i .

Proof. Let P and Gk+1 be, respectively, the primitive part and the content ofgcd( f0, . . . , fh) in k[X1][X2]; since a set g0, . . . , gh is a manimal Grobnerbasis if and only if gg0, . . . , ggh is, we can divide by PGk+1 and assumewlog that P = Gk+1 = 1 and gcd( f0, . . . , fh) = 1.

Since, for each i , T( fi ) < T( fi+1), we must have d(i) ≤ d(i + 1); butd(i) = d(i +1) would imply T( fi ) | T( fi+1) so that we have d(i) < d(i +1).

Setting gi := Lp( fi ) for each i , both Xd(i+1)−d(i)2 fi and fi+1 are in the ideal

and have degree d(i + 1) in X2; therefore successive Euclidean division of theleading polynomials leads to a polynomial f := Lp( f )Xd(i+1)

2 + · · · in theideal, where Lp( f ) = gcd(gi , gi+1).

Therefore T( f ) is a multiple of some T( f j ). If gi+1 = gcd(gi , gi+1), ne-cessarily j < i + 1 and T( f j ) divides T( fi+1), a contradiction. In conclusiongi+1 | gi and we can set Gi+1 := gi/gi+1.

Since Gi+1 fi+1−Xd(i+1)−d(i)2 fi is a polynomial of degree less than d(i +1)

in X2 which reduces to zero by the Grobner basis, it follows that Gi+1 fi+1 ∈( f0, . . . , fi ); therefore, inductively,

gi | f j for each j ≤ i ⇒ gi+1 | f j for each j ≤ i + 1.

Therefore, gcd( f0, . . . , fh) = 1 implies that gh = 1 and each gi divides fi .Setting Hi := fi/gi for all i , since Gi+1 fi+1 ∈ ( f0, . . . , fi ), dividing by

Gi+1gi+1 = gi = Gi+1 · · · Gh

we obtain the last claim.

33.2 The Cerlienco–Mureddu Correspondence

Cerlienco and Mureddu in 1990 proved a partial converse of Macaulay’s re-sult:

Problem 33.2.1. Given a finite set of points,

a1, . . . , as ⊂ kn, ai := (ai1, . . . , ain),

how do we compute N(I) w.r.t. the lexicographical ordering < induced by X1 <

· · · < Xn where

I := f ∈ P : f (ai ) = 0, 1 ≤ i ≤ s.

They later generalized it to a class (CeMu-ideals) of zero-dimensional ideals.Note that a zero-dimensional ideal I ⊂ P can be considered as given if we

know:

33.2 Cerlienco–Mureddu Correspondence 555

• the set Z(I);• for each a ∈ Z(I), a Macaulay basis of the corresponding primary compo-

nent of I.

Let us set the following notation:

• < is the lexicographical ordering induced by X1 < · · · < Xn ;• I ⊂ P is a zero dimensional ideal;• for each a ∈ Z := Z(I), a := (a1, . . . , an):

λa : P → P is the translation λa(Xi ) = Xi + ai , for each i ,ma = (X1 − a1, . . . , Xn − an),qa is the ma-primary component of I,Λa := M(λa(qa)) ⊂ SpanK (M),υa, for each υ ∈ N<(λa(qa)), the Macaulay equation υa := (υ) so

thatυa : υ ∈ N<(λa(qa)) is the Macaulay basis of Λa, enumerated in order

to satisfy the properties of Corollary 32.3.2;2

• s := ∑a∈Z deg(qa);

• L := λ1, . . . , λs := υaλa : υ ∈ N<(λa(qa)), a ∈ Z ordered as statedin Corollary 32.3.3;

• X := x1, . . . , xs := (a, υ) ∈ N<(λa(qa)), a ∈ Z enumerated so that

x j = (a, υ) ⇐⇒ λ j = υaλa;

• for each j, 1 ≤ j ≤ s, M(λ j ) := M(υ)λa where λ j = υaλa.

Note that Cerlienco and Mureddu stated their result under the followingequivalent assumptions:

• λ = M(λ) for each λ ∈ L;• υa = M(υ), for each λ = υaλa ∈ L;• each λa(qa) is a monomial ideal.

Definition 33.2.2. The ordered sets L(I) := L and X(I) := X are called, re-spectively, a Macaulay representation and a CeMu-skeleton of I := P(L);each λ = υaλa ∈ L is called a CeMu-functional and each x = (a, υ) ∈ X aCeMu-card.

If, moreover, for each λ = υaλa ∈ L, λ = M(λ) = M(υ)λa, thenI is called a CeMu-ideal, X its CeMu-scheme, and each x = (a, υ) ∈ X aCeMu-condition.

2 Note that in particular υ = T<(υa).

556 Moller II


(1) I = ⋂a∈Z qa = P(Spank(L));

(2) for each j, 1 ≤ j ≤ s, x j = (a, υ) and for each υ ′ | υ there is i < jsuch that xi = (a, υ ′);

(3) for each j, 1 ≤ j ≤ s, x j = (a, υ) ∈ X, and for each f ∈ P

M(λ j )( f ) = M(υ)(λa( f )) = (D(υ)( f ))(a) = c(υ, λa( f ));

(4) for each σ, 1 ≤ σ ≤ s, the sets Lσ := λ1, . . . , λσ and Xσ := xi , 1 ≤i ≤ σ are a Macaulay representation and a CeMu-skeleton of Iσ =P(Spank(Lσ ));

(5) I1 ⊂ · · · ⊂ Iσ ⊂ Iσ+1 ⊂ · · · ⊂ I;(6) I = √

I ⇐⇒ υ = 1 for each (a, υ) ∈ X ⇐⇒ #L = #Z.

The Cerlienco–Mureddu result proposes an algorithm which, to eachMacaulay representation and the corresponding CeMu-skeleton,

L := λ1, . . . , λs, X := x1, . . . , xs ⊂ kn × T ,

xi = (ai , υi ), ai := (ai1, . . . , ain), υi =n∏

l=1

Xαill ,

associates

• an order ideal N := N(L) and• a bijection Φ := Φ(L) : L → N,

which, as we will prove later, satisfies

Fact 33.2.4. N<(L) = N(P(Spank(L))) for the lexicographical ordering in-duced by X1 < · · · < Xn.

Since they do so by induction on s = #(L), let us consider the subset L′ :=

λ1, . . . , λs−1, and the corresponding3 order ideal N′ := N(L′) and bijectionΦ ′ := Φ(L′).

We need also to consider, for each m < n, the sets

T [1, m] := T ∩ k[X1, . . . , Xm]

= Xa11 . . . Xam

m : (a1, . . . , am) ∈ Nm,

M[1, m] := M(τ ) : τ ∈ T [1, m]and the projection

πm : kn → km, πm(x1, . . . , xn) = (x1, . . . , xm),

3 If s = 1 the only possible solution is N = 1, Φ(λ1) = 1.


which we freely use also to denote the projections

πm : T Nn → N

m T [1, m], πm(Xα11 . . . Xαn

n ) = Xα11 . . . Xαm

m ,

πm : M → M[1, m], πm(M(τ )) = M(πm(τ )),

and πm : kn × T → km × T [1, m], πm(a, τ ) = (πm(a), πm(τ )).

Recalling Macaulay’s notation (Definition 30.4.1) for Noether equations asmembers of k[X−1

1 , . . . , X−1n ], we note that for each Noetherian equation

(τ ) := M(τ ) +∑

t∈T(I)

γ (t, τ, N(I))M(t) = τ−1 +∑

t∈T(I)

γ (t, τ, N(I))t−1,

with τ = Xd11 . . . Xdn

n , there are unique polynomials

fi (X−11 , . . . , X−1

i ) ∈ k[X−11 , . . . , X−1

i ]

such that

(τ ) =((

· · ·((

1 + X−11 f1(X−1

1 ))X−d1

1 + X−12 f2(X−1

1 , X−12 )

)X−d2

2 + · · ·

+ fn−1(X−11 , . . . , X−1

n−1)

)X−dn−1

n−1 + X−1n fn(X−1

1 , . . . , X−1n )

)X−dn

n

and we set

πm((τ )) :=(

· · · (1 + X−11 f1(X−1

1 ))X−d1

1 + · · ·

+ fm−1(X−11 , . . . , X−1

m−1)

)X−dm−1

m−1 + X−1m fm(X−1

1 , . . . , X−1m )

= (σXdmm ...Xdn

n((τ )))(X−1

1 , . . . , X−1m , 0, . . . , 0)

∈ k[X−11 , . . . , X−1

m ].

Finally, for a CeMu-functional λ = υaλa we set

πm(λ) := πm(υaλa) := πm(υa)λπm (a).

Let us also write, for each ν, 1 ≤ ν < n, and each δ ∈ N,

Yνδ := Spankπν(λ) : λ ∈ L′, there exists ω ∈ T [1, ν] : Φ ′(λ) = ωX δ

ν+1.

With an abuse of notation, if P(Spank(L)) is radical, we simply identifyeach xi = (ai , 1) and the corresponding λi = λai with ai .

558 Moller II

With this notation, let us set

m := max(

j : π j (λs) ∈ Spank(π j (L′))

),

d := minδ : πm(λs) ∈ Ymδ,W := πm(λ) : Φ ′(λ) = ωXd

m+1, ω ∈ T [1, m] ∪ πm(λs)ω := Φ(W)(πm(λs)),

ts := ωXdm+1

where N(W) and Φ(W) are the result of the application of the present algo-rithm to W, which can be inductively applied since #(W) ≤ s − 1. We thendefine

N := N′ ∪ ts and Φ(λi ) :=

Φ ′(λi ), i < s,ts, i = s.

Example 33.2.5. Let us consider the set Y := ai , 1 ≤ i ≤ 6 where

a1 = (0, 0) a2 = (0, 1) a3 = (2, 0)

a4 = (0, 2) a5 = (1, 0) a6 = (1, 1);

the Cerlienco–Mureddu Algorithm returns:

(0,0) a1 := (0, 0), Φ(a1) := t1 := 1;(0,1) a2 := (0, 1), m = 1, d = #(0, 0) = 1, W = (0, 1),ω = 1, Φ(a2) := t2 := X2,

(2,0) a3 := (2, 0), m = 0, d = #(0, 0) = 1, W = (2, 0),ω = 1, Φ(a3) := t3 := X1,

(0,2) a4 := (0, 2), m = 1, d = #(0, 0), (0, 1) = 2, W = (0, 2),ω = 1, Φ(a4) := t4 := X2

2,(1,0) a5 := (1, 0), m = 0, d = #(0, 0), (2, 0) = 2, W = (1, 0),ω = 1, Φ(a5) := t5 := X2

1,(1,1) a6 := (1, 1), m = 1, d = #(1, 0) = 1, W = (0, 1), (1, 1),ω = X1, Φ(a6) := t6 := X1 X2.

Example 33.2.6. Let us consider the set X := bi , 1 ≤ i ≤ 9 where

b1 = (0, 0, 1), b2 = (0, 1, −2), b3 = (2, 0, 2),

b4 = (0, 2, −2), b5 = (1, 0, 3), b6 = (1, 1, 3),

b7 = (1, 1, 1), b8 = (2, 0, 1), b9 = (2, 0, 0)

and let us set ai := π2(bi ), for each i , so that π2(X) = Y, where Y is the setof points discussed in Example 33.2.5.


Clearly the Cerlienco–Mureddu correspondence returns Φ(bi ) = Φ(ai ) foreach i ≤ 6 and

t7 := X3, t8 := X1 X3, t9 := X23.

Example 33.2.7. With reference to Example 32.4.5, noting that each setSpanK i , 0 ≤ i ≤ 2ρ is a Macaulay representation of I + mρ , then foreach

• s = 2i we have

m = 0, Y1δ = M(1), δ < i ,

∅, δ ≥ i,ω = 1, t2i = Xi

1;

• s = 2i + 1 we have

m = 1, Y1δ = M(1), . . . M(X−i

1 δ = 0M(1), . . . M(X−i−1

1 δ = 1,ω = Xi

1, t2i = Xi1 X2.

Let

L := λ1, . . . , λs, X := x1, . . . , xs ⊂ kn × T ,


l=1

Xαill

be the Macaulay representation and the CeMu-skeleton of a zero-dimensionalideal I ⊂ P and let N := N(L) and Φ := Φ(L) the result of the Cerlienco–Mureddu correspondence. Then:

Lemma 33.2.8. If Y = λ1, . . . , λr ⊂ L is an initial segment of L then

• Y is a CeMu-skeleton,• N(Y) ⊂ N(L),

• for each j ≤ r < s, Φ(Y)(λ j ) = Φ(L)(λ j ).

Remark 33.2.9. We note that, by construction, we have

P(Spank(πν(L)′)) = Yν0 ⊃ Yν1 ⊃ · · · ⊃ Yνδ ⊃ Yνδ+1 ⊃ · · · ;I ∩ k[X1, . . . , Xν] = P(Spank(πν(L

′)))= P(Yν0) ⊂ · · · ⊂ P(Yνδ) ⊂ P(Yνδ+1) ⊂ · · · .

The result is essentially a special case of Theorem 26.2.6.

560 Moller II

33.3 Lazard Structural Theorem

Let I ⊂ P be a zero-dimensional ideal, and, using the same notation as above,let

L := λ1, . . . , λs, X := x1, . . . , xs ⊂ kn × T ,


l=1

Xαill

be a Macaulay representation and a CeMu-skeleton; let us now denote by N :=N(L) and Φ := Φ(L) the result of the Cerlienco–Mureddu correspondencewhich satisfies

Fact 33.3.1. We have

(A) N := N(I).

Since N is an order ideal, T := T \ N is a monomial ideal whose minimalbasis G := t1, . . . , tr will be ordered so that t1 < t2 < · · · < tr .

Writing further

B := (1 ∪ Xiτ : τ ∈ N) \ N

we obviously obtain the following.


(B) G(I) = G = t1, . . . , tr , t1 < t2 < · · · < tr ,(C) B(I) = B.

Let us extend the ordering of L to N = τ1, . . . , τs enumerating it so thatτσ = Φ(λσ ), for each σ and let us denote the ordering of L and N by ≺ so that

for each α, β, τα ≺ τβ, λα ≺ λβ ⇐⇒ α < β.

Write for each τ ∈ N

• L(τ ) := λ ∈ L : λ ≺ Φ−1(τ ) = λ ∈ L : Φ(λ) ≺ τ ,• X(τ ) := x j : λ j ∈ L(τ ),• I(L(τ )) := P(Spank(L(τ )))

and, for each τ ∈ N ∪ B,

• N(τ ) := ω ∈ N : ω ≺ τ ,so that we have

33.3 Lazard Structural Theorem 561

Corollary 33.3.3.

(D) For each τ ∈ N there is a unique polynomial

fτ := τ −∑

ω∈N(τ )

c( fτ , ω)ω

such that λ( fτ ) = 0, for each λ ∈ L(τ ).

(E) For each τ ∈ G there is a unique polynomial

fτ := τ −∑ω∈N

c( fτ , ω)ω

such that λ( fτ ) = 0, for each λ ∈ L.

Proof. Since #L(τ ) = #X(τ ) = #N(τ ) and #L = #X = #N, we can compute

fτ by interpolation.

In the same mood, though interpolation is not sufficient to prove it, we canstate

Fact 33.3.4.

(F) For each τ ∈ B there is a polynomial

fτ := τ −∑

ω∈N(τ )

c( fτ , ω)ω

such that λ( fτ ) = 0, for each λ ∈ L.

Corollary 33.3.5.

(G) The reduced Grobner basis of I is

G(I) := fτ : τ ∈ G;moreover, for each τ ∈ N, T( fτ ) = τ.

(H) The border basis of I is

B(I) := fτ : τ ∈ B.Proof. For each τ ∈ G ∪ B, the only term in fτ which is not a member of N isτ so that T( fτ ) = τ.

For any τ ∈ N, T( fτ ) = τ because the Cerlienco–Mureddu correspondencegives τ ∈ G(I(L(τ ))) and N(I(L(τ ))) = N(τ ).

Fact 33.3.6.

(I) For each ν, 1 ≤ ν < n:

562 Moller II

let jν be the value such that t jν < Xν+1 ≤ t jν+1; then ft1 , . . . , ft jν

is a minimal Grobner basis of both P(Spank(πν(L))) and ofI ∩ k[X1, . . . , Xν];

for each δ ∈ N, let j (νδ) be the value such that t j (νδ) < X δν+1 ≤

t j (νδ)+1; then Lp( ft1), . . . , Lp( ft j (νδ)) is a Grobner basis

of I(Yνδ).

(L) For each j, 1 ≤ j ≤ s, λ j ( fτ j ) = 0 so that L and λ j ( fτ j )−1 fτ j , 1 ≤

j ≤ s are triangular.

33.4 Some Factorization Results

Let us now restrict ourselves to a CeMu-ideal, assuming that

L := λ1, . . . , λs, X := x1, . . . , xs ⊂ kn × T ,


l=1

Xαill

are the Macaulay representation and the CeMu-scheme of a CeMu-ideal I, sothat, for each i ,

λi = M(λ) = M(υi )λai , for each i, 1 ≤ i ≤ s.

Under this assumption, for any term

τ := Xd11 . . . Xdn

n ∈ T \ N(L)

such that N ∪ τ is an order ideal, we define, for each m, 1 ≤ m ≤ n,

Nm(τ ) := Nm(L, τ ) := ω ∈ T [1, m] : τ > ωXdm+1m+1 . . . Xdn

n ∈ N,Am(τ ) := Am(L, τ ) := Φ−1(ωXdm+1

m+1 . . . Xdnn ) : ω ∈ Nm(τ ) ⊂ L,

Bm(τ ) := Bm(L, τ ) := πm(Am(τ )) ⊂ (k[X1, . . . , Xm])∗,Cm(τ ) := Cm(L, τ ) := πm(λ) ∈ Bm(τ ) : πm−1(λ) ∈ Bm−1(τ ),Lm(τ ) := Lm(L, τ ) := λ ∈ L : πm(λ) ∈ Cm(τ ) ⊂ L,Dm(τ ) := Dm(L, τ ) := xi ∈ X : πm(λi ) ∈ Cm(τ ) ⊂ km × T [1, m],Mm(τ ) := Mm(L, τ ) := ω ∈ T [1, m] : ω < Xdm

m , ωXdm+1m+1 . . . Xdn

n ∈ N,Mm(τ ) := ω ∈ Mm(τ ) : ω ≺ τ ,

where, with a slight abuse of notation, we have

Nn(τ ) := ω ∈ T : ω < τ , An(τ ) := λ : Φ(λ) < τ , C1(τ ) := B1(τ ).

33.4 Some Factorization Results 563

Example 33.4.1. With respect to Example 33.2.6, if we choose τ := X2 X3 wehave

N1 = A1 = B1 = C1 = D1 = M1 = ∅,

and 4

N2 = 1, X1, N3 = N \ X23,

A2 = b7, b8, A3 = bi , 1 ≤ i ≤ 8,B2 = (1, 1), (2, 0), B3 = bi , 1 ≤ i ≤ 8,C2 = (1, 1), (2, 0), C3 = b1, b2, b4, b5,D2 = b3, b6, b7, b8, b9, D3 = b1, b2, b4, b5,M2 = 1, X1, M3 = 1, X1, X2

1, X2, X1 X2, X22.

If we instead choose τ := X1 X23 we have

N1 = 1, N2 = 1, N3 = N,A1 = b9, A2 = b9, A3 = bi , 1 ≤ i ≤ 9,B1 = 2, B2 = (2, 0), B3 = bi , 1 ≤ i ≤ 9,C1 = 2, C2 = ∅, C3 = b1, b2, b4, b5, b6, b7,D1 = b3, b8, b9, D2 = ∅, D3 = b1, b2, b4, b5, b6, b7,M1 = 1, M2 = ∅, M3 = N \ X2

3.Lemma 33.4.2. With the notation above, we have:

(1) #(Bm(τ )) = #(Am(τ )) = #(Nm(τ ));(2) the Cerlienco–Mureddu correspondence associates to Bm(τ ) the order

ideal

N(Bm(τ )) = Nm(τ )

and the bijection Φ(Bm(τ )) defined by

Φ(Bm(τ ))(πm(x))Xdm+1m+1 . . . Xdn

n = Φ(x), for each x ∈ Am;

(3) #(Lm(τ )) = #(Cm(τ )) ≤ #(Mm(τ ));(4) under the Cerlienco–Mureddu correspondence one has

N(Cm(τ )) ⊂ ω ∈ T [1, m] : ω < Xdmm ;

(5) L = ⋃m Lm(τ ).

Proof.

(1) is trivial;

4 Recall that, with an abuse of notation, we are identifying each xi = (bi , 1) and the correspond-ing λi = λbi

with bi .

564 Moller II

(2) the Cerlienco–Mureddu algorithm when applied to the ordered set L

associates each element λ ∈ Am(τ ) to the term

Φ(λ) = Φ(πm(Am(τ )))(πm(λ))Xdm+1m+1 . . . Xdn

n ;(3) in order to obtain Mm(τ ) one has to remove from Nm(τ ) the subset

ωXdmm ∈ Nm(τ ) : ω ∈ T [1, m − 1] = ωXdm

m : ω ∈ Nm−1(τ )while for each ω ∈ Nm−1(τ ) there are dm + 1 CeMu-conditions y =(a, υ) ∈ km × T [1, m] for which

M(υ)λa ∈ Bm(τ ) and Φ(Bm−1(τ ))(πm−1(υaλa) = ω;(4) in order that there is ω ∈ N(Cm(τ )) such that ω ≥ Xdm

m , theCerlienco–Mureddu algorithm requires the existence of at least dm + 1CeMu-conditions x0, . . . , xdm , xi = (ai , υi ) such that

πm(x0) = · · · = πm(xi ) = · · · = πm(xdm ),

so that πm−1(M(υi )λai ) ∈ Bm−1(τ );(5) if λ ∈ L is such that Φ(λ) < τ , then there is a minimal value m ≤ n

for which λ ∈ Am(τ ), πm(λ) ∈ Bm(τ ), πm(λ) ∈ Cm(τ ), λ ∈ Lm(τ ).If λ ∈ L is such that Φ(λ) = Xe1

1 . . . Xenn > τ , there is m ≤ n such

that em > dm, while ei = di , for each i > m; this implies that there is ∈ Am(τ ) such that πm() = πm(λ) so that λ ∈ Lm(τ ).

As for (D)–(E), linear interpolation, is all one needs in order to prove

Proposition 33.4.3. With the same notation as in Lemma 33.4.2, we have:

(V) for each τ := Xd11 . . . Xdn

n ∈ G, and each m, 1 ≤ m ≤ n, there arepolynomials

gmτ := Xdmm +

∑ω∈Mm (τ )

c(gmτ , ω)ω

such that λ(gmτ ) = 0, for each λ ∈ Lm(τ );(T) for each τ := Xd1

1 . . . Xdnn ∈ N and each m, 1 ≤ m ≤ n, there are

polynomials

gmτ := Xdmm +

∑ω∈Mm (τ )

c(gmτ , ω)ω

such that λ(gmτ ) = 0, for each λ ∈ Lm(τ ), λ ≺ Φ−1(τ ).


Proof.

(V) Since #(Cm(τ )) ≤ #(Mm(τ )), we can evaluate each c(gmτ , ω) by inter-polation, so that (gmτ ) = 0, for each ∈ Cm(τ ) and λ(gmτ ) =πm(λ)(gmτ ), for each λ ∈ Lm(τ ).

(T) One has just to apply (V) to the set X(τ ).

For each τ := Xd11 . . . Xdn

n ∈ N, let us denote by ν := ν(τ) ≤ n the valuesuch that dν = 0 while dµ = 0 for each µ > ν so that τ ∈ T [1, ν], gmτ = 1for m > ν, and, writing

hτ :=n∏

m=1

gmτ ∈ k[X1, . . . , Xν−1][Xν],

lτ :=ν(τ)−1∏

m=1

gmτ ∈ k[X1, . . . , Xν−1],

pτ := gντ ∈ k[X1, . . . , Xν−1][Xν],

we havehτ = lτ pτ = lτ Xdν

ν + · · ·so that lτ ∈ k[X1, . . . , Xν−1] is the leading polynomial and the content of hτ ,while the monic polynomial pτ is the primitive component of hτ .

Therefore we have 5

Corollary 33.4.4. With the notation above, under the assumption that I is rad-ical, we have:

(W) for each τ = Xd11 . . . Xdν

ν ∈ N, there are

lτ ∈ k[X1, . . . , Xν−1]

and a monic polynomial

pτ = Xdνν +

∑ω∈Mν (τ )

c(pτ , ω)ω ∈ k[X1, . . . , Xν−1][Xν]

such that hτ := lτ pτ are such that

• T(hτ ) = τ,

• Lp(hτ ) = lτ ,• lτ (πν−1(a)) = 0, for all a ∈ X(τ ),

• pτ (a) = 0, for each a ∈ Dν(τ ),

• hτ (a) = 0, for each a ∈ X such that a ≺ Φ−1(τ );

5 This justifies why we need to require that I is radical: in this restricted setting, each functionalλi is an evaluation at a point and distributes with product.

566 Moller II

(X) for each i, 1 ≤ i ≤ r there are

li ∈ k[X1, . . . , Xν−1]


pi = Xdνν +

∑ω∈Mν (ti )

c(pi , ω)ω ∈ k[X1, . . . , Xν−1][Xν]

such that hi := li pi are such that

• T(hi ) = ti = Xd11 . . . Xdν

ν ∈ G ∩ T [1, ν],• Lp(hi ) = li ,• li (πν−1(a)) = 0, for each a ∈ ⋃ν−1

m=1 Dm(ti ),• pi (a) = 0, for each a ∈ Dν(ti ),• hi (a) = 0, for each a ∈ X.

While #(Cm(τ )) ≤ #(Mm(τ )), in general equality does not hold and thepolynomials gmτ are not unique. However, uniqueness can be forced via theCerlienco–Mureddu correspondence in such a way that the result does not re-quire us to assume that I is radical.

To start with, however, note that #(C1(τ )) = #(M1(τ )) so that g1τ is unique.We therefore begin our construction by setting γ1τ := g1τ and, inductively, form, 1 < m ≤ n:

• ζmτ := ∏m−1ν=1 γντ ;

• Qm(τ ) := M(ω)λa : ω ∈ T [1, m − 1], a ∈ Z := Z(I), M(ω)λa(ζmτ )

= 0;• Pm(τ ) := M

(πm

(υiω

))λai : M(υi )λai ∈ Lm(τ ), M(ω)λai ∈ Qm(τ );

• Rm(τ ) := (πm(ai ), πm(

υiω

)): M

(πm

(υiω

))λai ∈ Pm(τ );

• Em(τ ) := N(Rm(τ ));• Sm(τ ) := (πm(ai ), πm

(υiω

)) ∈ Rm(τ ) : (ai , υi ) ≺ Φ−1(τ );• Fm(τ ) := N(Sm(τ )).

Then:

Corollary 33.4.5. With the above notation we have

(n) for each τ := Xd11 . . . Xdn

n ∈ G, and each m, 1 ≤ m ≤ n, there are uniquepolynomials

γmτ := Xdmm +

∑ω∈Em (τ )

c(γmτ , ω)ω

such that πm(λ)(γmτ ) = 0, for each λ ∈ Lm(τ );


(m) for each τ := Xd11 . . . Xdn

n ∈ N, and each m, 1 ≤ m ≤ n there are uniquepolynomials

γmτ := Xdmm +

∑ω∈Fm (τ )

c(γmτ , ω)ω

such that πm(λ)(γmτ ) = 0, for each λ ∈ Lm(τ ), λ ≺ Φ−1(τ );(O) for each τ = Xd1

1 . . . Xdνν ∈ N, there are

Lτ ∈ k[X1, . . . , Xν−1]

and a unique monic polynomial

Pτ = Xdνν +

∑ω∈Fν (τ )

c(Pτ , ω)ω ∈ k[X1, . . . , Xν−1][Xν]

such that Hτ := Lτ Pτ are such that

• T(Hτ ) = τ, Lp(Hτ ) = Lτ ,

• πν−1(λ)(Lτ ) = 0, for each λ ∈ U ν−1m=1Lm(τ ),

• πν(λ)(Pτ ) = 0, for each λ ∈ Lν(τ ),

• πν(λ)(Hτ ) = 0, for each λ ∈ L : λ ≺ Φ−1(τ );(P) for each i, 1 ≤ i ≤ r there are

Li ∈ k[X1, . . . , Xν−1]


Pi = Xdνν +

∑ω∈Eν (ti )

c(Pi , ω)ω ∈ k[X1, . . . , Xν−1][Xν]

such that Hi := Li Pi are such that

• T(Hi ) = ti = Xd11 . . . Xdν

ν ∈ G ∩ T [1, ν], Lp(Hi ) = Li ,

• πν−1(λ)(Li ) = 0, for each λ ∈ ⋃ν−1m=1 Lm(ti ),

• πν(λ)(Pi ) = 0, for each λ ∈ Lν(ti ),• πν(λ)(Hi ) = 0, for each λ ∈ L.

Proof. The only non-trivial statements, i.e. the vanishing of πν−1(λ)(L) andπν(λ)(H) are an elementary consequence of the Leibniz Formula (Proposi-tion 31.4.1).

Fact 33.4.6. We have:

(Q) Li , Pi , Hi , 1 ≤ i ≤ r , satisfy

H1, . . . , Hr is a minimal Grobner basis of I,for each ν, 1 ≤ ν < n, H1, . . . , Hjν is a minimal Grobner basis

of I ∩ k[X1, . . . , Xν] and of I(πν(X));

568 Moller II

for each ν, 1 ≤ ν < n, L1, . . . , L j (νδ) is a Grobner basis of

I(Yνδ).

Clearly, if I is radical similar statements hold for

h1, . . . , hr , l1, . . . , l jνδ) and h1, . . . , h jν .The construction which led to Corollary 33.4.5 can be refined as follows:

for each τ := Xd11 . . . Xdn

n , for each ν ≤ n, iteratively for decreasing δ ≤ dν ,with initial value Pνdn+1(τ ) := Pν−1 := Pν−12, we compute

Yνδ(τ ) := πν(x) : ∃ω ∈ T [1, ν] : Φ(x) = ωX δν+1, x ∈ Pνδ+1(τ ),

Eνδ(τ ) := N(Yνδ(τ )),

Pνδ(τ ) :=

M(πν

(υi

ω

))λai : M(υi ))λai ∈ Lν(τ ), M(ω)λai ∈ Yνδ(τ )

,

Sνδ(τ ) := πν(x) ∈ Yνδ(τ ) : x ≺ Φ−1(τ ),Fm(τ ) := N(Sm(τ )),

so that:

Corollary 33.4.7.

(N) For each τ := Xd11 . . . Xdn

n ∈ G, and each m, 1 ≤ m ≤ n, there areunique polynomials

γmτ := Xdmm +

∑ω∈Em (τ )

c(γmτ , ω)ω

and

γmδτ := Xm +∑

ω∈Emδ(τ )

c(γmδτ , ω)ω, 1 ≤ δ ≤ dm,

such that

• πm(λ)(γmδτ ) = 0, for each λ ∈ Ymδ(τ ),

• πm(λ)(γmτ ) = 0, for each λ ∈ Lm(τ ),• γmτ = ∏

δ γmδτ .

(M) For each τ := Xd11 . . . Xdn

n ∈ N, and each m, 1 ≤ m ≤ n, there areunique polynomials

γmτ := Xdmm +

∑ω∈Fm (τ )

c(γmτ , ω)ω

and

γmδτ := Xm +∑

ω∈Fmδ(τ )

c(γmδτ , ω)ω, 1 ≤ δ ≤ dm

such that

• πm(λ)(γmδτ ) = 0, for each λ ∈ Ymδ(τ ), λ ≺ Φ−1(τ );• πm(λ)(γmτ ) = 0, for each λ ∈ Lm(τ ), λ ≺ Φ−1(τ );,

33.5 Some Examples 569

• γmτ = ∏δ γmδτ .

Remark 33.4.8. The only difference between the three bases

f1, . . . , fr , h1, . . . , hr and H1, . . . , Hr is that, unlike the others, the first is reduced. On the other side, for each i , wehave

T( fi ) = T(hi ) = T(Hi ) = ti .

Therefore we have

• f1 = h1 = H1 and• fi − hi ∈ (h1, . . . , hi−1), fi − Hi ∈ (H1, . . . , Hi−1) for each i, 1 < i ≤ r,

whence

• fi ∈ (h1, . . . , hi ), fi ∈ (H1, . . . , Hi ) for each i, 1 ≤ i ≤ r.

Fact 33.4.9. We have:

(R) For each i, 2 ≤ i ≤ r , Pi ∈ (Hj , j < i

): Li .

(S) For each j, 1 ≤ j ≤ s, λ j (Hτ j ) = 0; L and λ j (Hτ j )−1 Hτ j , 1 ≤ j ≤ s

are triangular.

Corollary 33.4.10. If I is radical, moreover,

(Z) li , pi , hi , 1 ≤ i ≤ r , satisfy

h1, . . . , hr is a minimal Grobner basis of I,for each ν, 1 ≤ ν < n, h1, . . . , h jν is a minimal Grobner basis of

I ∩ k[X1, . . . , Xν] and of P(Spank(πν(L))),for each ν, 1 ≤ ν < n, l1, . . . , l j (νδ) is a Grobner basis of I(Yνδ);for each i, 2 ≤ i ≤ r , pi ∈ (

h j , j < i)

: li ,for each j, 1 ≤ j ≤ s, λ j (hτ j ) = 0,L is triangular to λ j (hτ j )

−1hτ j , 1 ≤ j ≤ s.

33.5 Some Examples

Example 33.5.1. Let us consider the set Y introduced in Example 33.2.5.A direct application of the algorithm of Figure 28.1 returns:

(0,0) t1 := 1,G1 := X1, X2;

570 Moller II

(0,1) t2 = X2,G2 = X1, X2

2 − X2;(2,0) t3 := X1,

G3 = X21 − 2X1, X1 X2, X2

2 − X2;(0,2) t4 = X2

2,G4 = X2

1 − 2X1, X1 X2, X32 − 3X2

2 + 2X2;(1,0) t5 = X2

1,G5 = X3

1 − 3X21 + 2X1, X1 X2, X3

2 − 3X22 + 2X2;

(1,1) t6 = X1 X2,G6 = X3

1 − 3X21 + 2X1, X2

1 X2 − X1 X2, X1 X22 − X1 X2, X3

2 − 3X22 + 2X2.

Note that we have

X31 − 3X2

1 + 2X1 = (X1 − 2)(X1 − 1)X1,

X21 X2 − X1 X2 = X2(X1 − 1)X1,

X1 X22 − X1 X2 = X2(X2 − 1)X1,

X32 − 3X2

2 + 2X2 = X2(X2 − 1)(X2 − 2),

illustrating Lazard’s Theorem and Corollary 33.4.7. The fact that Moller’sAlgorithm returns the Cerlienco–Mureddu correspondence is not a coinci-dence.

Example 33.5.2. The result of the application of the algorithm of Figure 28.1to the set X of Example 33.2.6 returns, again, the Cerlienco–Mureddu corre-spondence and the Grobner basis G6 ∪ f1, f2, f3, f4 where

f1 := X3 X21 − 3X3 X1 + 2X3 − 3X2

2 − 6X2 X1 + 9X2 − X21 + 3X1 − 2,

f2 := X3 X2 + X3 X1 − 2X3 + 3X22 + X2 X1 − 7X2 − 2X2

1 + 3X1 + 2,

f3 := X23 X1 − 2X2

3 − 4X3 X1 + 8X3 − 15X22 − 30X2 X1 + 45X2 + 3X1 − 6,

f4 := X33 − 3X2

3 + 3X3 X1 − 4X3 − 3X22 − 6X2 X1 + 9X2 − 3X1 + 6,

and, modulo I(Y),

f1 = (X1 − 2)(X1 − 1)(X3 − 32 X2

2 + 92 X2 − 1),

f2 = (X2 + X1 − 2)(X3 + 3X2 − 2X1 − 1),

f3 = (X1 − 2)(X3 − 1)(X3 − 5X1 + 2),

f4 = (X3 − 1)X3(X3 + 3X21 − 8X1 + 2),


where

• (X21 −3X1 +2, X2 + X1 −2, X3 −1) is the Grobner basis of the ideal whose

roots are π2(b7), π2(b8),• b ∈ X : (X2

1 − 3X1 + 2)(b) = 0 = b1, b2, b4 to which the Cerlienco–Mureddu correspondence associates 1, X2, X2

2,• b ∈ X : (X2 + X1 − 2)(b) = 0 = b1, b2, b5 to which the Cerlienco–

Mureddu correspondence associates 1, X1, X2,• b ∈ X : (X1 − 2)(X3 − 1)(b) = 0 = b2, b4, b5, b6 to which the Cer-

lienco–Mureddu correspondence associates 1, X1, X2, X1 X2,• b ∈ X : (X2

3 − X3))(b) = 0 = b2, b3, b4, b5, b6 to which the Cerlien-co–Mureddu correspondence associates 1, X1, X2

1, X2, X1 X2.

Example 33.5.3. Let us set a := (0, 0, 0), b := (1, 0, 1), c := (0, −1, −1),

λa(qa) := (X41, X1 X2

2, X21 X2, X1 X3, X2 X3, X2

3)

λb(qb) := (X1, X32, X1 X3, X2

3)

λc(qc) := (X1, X22, X2

3),

I := qa ∩ qb ∩ qc.

so that s := deg(I) = 8 + 4 + 4 = 16.In the table below we list the sets X, L and the result N(L) of the Cerlienco–

Mureddu correspondence.

i 1 2 3 4 5 6 7 8ai a a a a a a a aυi 1 X1 X2 X3 X2

1 X1 X2 X22 X3

1Φ(λi ) 1 X1 X2 X3 X2

1 X1 X2 X22 X3

1i 9 10 11 12 13 14 15 16ai b b b b c c c cυi 1 X2 X3 X2

2 1 X2 X3 X2 X3

Φ(λi ) X41 X2

1 X2 X1 X3 X1 X22 X3

2 X42 X2 X3 X2

2 X3

The lex reduced Grobner basis of I is G(I) = fi , 1 ≤ i ≤ 9 where

f1 := X51 − X4

1,

f2 := X31 X2 − X2

1 X2,

f3 := X21 X2

2 − X1 X22,

f4 := X1 X32,

f5 := X52 + 2X4

2 + X32,

f6 := X21 X3 − X1 X3,

f7 := X1 X2 X3 − X21 X2,

572 Moller II

f8 := X32 X3 + 2X2

2 X3 + X2 X3 − 2X1 X22 − X2

1 X2,

f9 := X23 − 2X2

2 X3 − 4X2 X3 − 2X1 X3 − 3X42 + 2X1 X2

2 + 4X21 X2 + X4

1

and we have the following factorization of each fi modulo ( f1, . . . , fi−1):

f1 = X41(X1 − 1),

f2 = X21(X1 − 1)X2,

f3 = X1(X1 − 1)X22,

f4 = X1 X32,

f5 = X32(X2 + 1)2,

f6 = X1(X1 − 1)X3,

f7 = X1 X2(X3 − X2),

f8 ≡ X2(X2 + 1)2(X3 − X21),

f9 ≡ (X3 − X21 − 2X2 − X2

2)(X3 + 3X22 + 2X3

2 − X21).

Note that for

f2 Q2(t2) = M(X21)λa, M(X1)λb, M(X2

1)λc,L2(t2) = λ5, λ8,P2(t2) = λ1, λ2,E2(t2) = 1, X1;

f3 Q2(t3) = M(X1)λa, M(X1)λb, M(X1)λc,L2(t3) = λ2, λ5, λ8,P2(t3) = λ1, λ2, λ5, λ3,E2(t3) = 1, X1, X2

1, X2;f4 Q2(t4) = M(X1)λa, M(1)λb, M(X1)λc,

L2(t4) = λ2, λ5, λ8, λ9,P2(t4) = λ1, λ2, λ5, λ3, λ9, λ10, λ12,E2(t4) = 1, X1, X2

1, X31, X2, X1 X2, X2

2;f5 R2(t5) = λ1, λ3, λ7, λ13, λ15;f6 Q3(t6) = M(X1)λa, M(X1)λb, M(X1)λc,

L3(t6) = λ2, λ5, λ6, λ8,P3(t6) = λ1, λ2, λ5, λ3,E3(t6) = 1, X1, X2

1, X2;f7 Q2(t7) = M(X1)λa, M(1)λb, M(X1)λc,

L2(t7) = λ2, λ5, λ8, λ9,P2(t7) = λ1, λ2, λ5,E2(t7) = 1, X1, X2

1;Q3(t7) = M(X1 X2)λa, M(X2)λb, M(X1)λc,L3(t7) = λ6, λ10, λ12,P3(t7) = λ1, λ9, λ10,E3(t7) = 1, X1, X2;


f8 Q2(t8) = M(1)λa, M(1)λb, M(1)λc,L2(t8) = λ1, λ13, λ14,P2(t8) = λ1, λ13, λ14, ,E2(t8) = 1, X2, X2

2;Q3(t8) = M(X2)λa, M(X2)λb, M(X2

2)λc,L3(t8) = λ2, λ3, λ5, λ6, λ7, λ8, λ9, λ10, λ12,P3(t8) = λ1, λ2, λ3, λ9, λ10,E3(t8) = 1, X1, X2, X2

1, X1 X2;f9 P3(t9) = λi , i ≤ 16,

Y32(t9) = λ1, λ2, λ9, λ13, λ14,E32(t9) = 1, X1, X2

1, X2, X22,

γ32t9 = X3 − X21 − 2X2 − X2

2,

P32(t9) = λi , i ∈ 1, 2, 3, 5, 9, 10, 13, 14,Y31(t9) = λi , i ∈ 1, 2, 3, 5, 9, 10, 13, 14,E31(t9) = 1, X1, X2, X2

1, X1 X2, X22, X3

1, X32,

γ31t9 = X3 − X21 + 3X2

2 + 2X32,

and that each factor is obtained by interpolation as stated in Corollary 33.4.5.

Example 33.5.4. If, in the example above, we now add, where d = (1, 0, 0),

λ17 := M(X23)λa, Φ(λ17) = X3,

λ18 := M(1)λd, Φ(λ18) = X1 X3,

the corresponding lex reduced Grobner basis is

fi , 1 ≤ i ≤ 8 ∪ f10, f11where

f10 := X2 X23 + 2X2 X3 + 2X4

2 + 3X32 − 3X2

1 X2

≡ X2(X3 − 1 − 4X2 − 2X22)(X3 − X2

1 + 3X22 + 2X3

2);f11 := X3

3 − 2X1 X23 + 3X2

2 X3 + 6X2 X3 + X1 X3

≡ X3(X3 − X1 − 2X2 − X22)(X3 − X2

1 + 3X22 + 2X3

2).

The factorization is justified by

f10 Q2(t10) = M(1)λa, M(1)λb, M(1)λc, M(1)λd,L2(t10) = λ1,P2(t10) = λ1,E2(t10) = 1,γ2t10 = X2,

Q3(t10) = M(X2)λa, M(1)λb, M(1)λc, M(1)λd,L3(t10) = λi , i ≤ 18, 1 = i = 4,P3(t10) = λi , i /∈ 4, 5, 6, 7, 8, 18,

574 Moller II

Y32(t10) = λ9, λ13, λ14, λ18,E32(t10) = 1, X1, X2, X2

2,γ32t10 = X3 − 1 − 4X2 − 2X2

2,

P32(t10) = λi , i ∈ 1, 2, 3, 9, 10, 13, 14,Y31(t10) = λi , i ∈ 1, 2, 3, 9, 10, 13, 14,E31(t10) = 1, X1, X2, X2

1, X1 X2, X22, X3

2,γ31t10 = X3 − X2

1 + 3X22 + 2X3

2;

f11 P3(t11) = λi , i ≤ 18,Y33(t11) = λ1, λ18,E33(t11) = 1, X1, ,γ33t11 = X3,

P33(t11) = λi , i /∈ 6, 7, 8,Y32(t11) = λ1, λ9, λ13, λ14,E32(t11) = 1, X1, X2, X2

2,γ32t11 = X3 − X1 − 2X2 − X2

2,

P32(t11) = λi , i ∈ 1, 2, 3, 9, 10, 13, 14,Y31(t11) = λi , i ∈ 1, 2, 3, 9, 10, 13, 14,E31(t11) = 1, X1, X2, X2

1, X1 X2, X22, X3

2,γ31t11 = X3 − X2

1 + 3X22 + 2X3

2.

33.6 An Algorithmic Proof

The fact that Moller’s algorithm returns the Cerlienco–Mureddu correspon-dence suggests that a proof can be obtained by a direct application of it. 6

The proof being by induction, we begin with

Lemma 33.6.1. If #L = 1 conditions (A), (F), (I), (L), (Q), (R), (S) hold.

Proof. When we have a single point (a1, . . . , an) ∈ kn , we have

• N = 1,• B = G = X1, . . . , Xn,• f1 = 1,• fXi = Xi − ai , for each i,

and the properties are obviously satisfied.

6 Of which a simplified version in this setting is presented in Figure 33.1.

33.6 An Algorithmic Proof 575

Fig. 33.1. Moller’s algorithm for Macaulay representation

r := 1, B := ∅t1 := 1, N := t1, q1 := t1, q := q1,For h = 1..n do

t := Xh , bt := Xh − ah1, B := B ∪ tWhile r ≤ s do

Let t := min<t ∈ B : λr+1(bt ) = 0r := r + 1, B := B \ t,tr := t, N := N ∪ tr , qr := λr (bt )

−1bt , q := q ∪ qr ,For each τ ∈ B do bτ := bτ − λr (bτ )qr ,For h = 1..n do

If Xhtr ∈ B thent := Xhtr ,f := Xhbtr − ∑

τ∈NXh τ∈B

c(btr , τ )bXhτ

bt := f − λr ( f )qrB := B ∪ Xhtr , h = 1..n

N, q, bτ : τ ∈ B

This gives a starting point for induction: let us assume we have a Macaulayrepresentation and the corresponding CeMu-skeleton

L := λ1, . . . , λs, X := x1, . . . , xs ⊂ kn × T ,


l=1

Xαill

of a zero-dimensional I, and let us denote

X′ := x1, . . . , xs−1, L′ := λ1, . . . , λs−1 and I′ := P(Spank(L

′),

for which we assume conditions (A)–(L) hold. If moreover I (and so also I′) isa CeMu-ideal, we also assume that conditions (M)–(S) hold for I′.

In particular:

Φ ′ := N′ → L′ is the Cerlienco–Mureddu correspondence,

G′ := G(I′) = ω1, . . . , ωr , ω1 < ω2 < · · · < ωr ,

B′ := B(I′),f ′ω, ω ∈ B′, are the polynomials whose existence is implied by (F),

Fi := f ′ωi

are the polynomials whose existence is implied by (E), so thatFi : 1 ≤ i ≤ r is the reduced Grobner basis of I′,L ′

i , P ′i , H ′

i are the polynomials whose existence is implied by (P).

Setting

I := min<

j, 1 ≤ j ≤ r : λs(Fj ) = 0we then have

576 Moller II

Lemma 33.6.2. If L′ satisfies conditions (A–L) then

Φ(L)(λs) = ωI .

Proof. Let ωI = Xd11 . . . Xdn

n and let m + 1 := max(i : di = 0), so that

FI ∈ k[X1, . . . , Xm+1].

Since, by (I), for each ν,

I′ ∩ k[X1, . . . , Xν] = P(Spank(πν(L′))),

and

Fj ∈ k[X1, . . . , Xν], ν ≤ m ⇒ j < I

we deduce that

πν(λs)(Fj ) = λs(Fj ) = 0, for each Fj ∈ k[X1, . . . , Xν], ν ≤ m, whileπm+1(λs)(FI ) = λs(FI ) = 0.

This allows us to deduce that

m := max(

j : π j (λs) ∈ Spank(π j (L′)).

Therefore πm+1(λs) ∈ Spank(πm+1(L′); also

dm = minδ : πm(λs) ∈ Ymδ;in fact, for each δ < dm , since

T(Fj ) = ω j < X δm < Xdm

m ⇒ j < I,

and πm(λs)(Fj ) = 0, (I) lets us deduce that πm(λs) ∈ Ymδ and πm(λs) /∈Ymdm .

As a consequence we consider

W := πm(λ) : Φ ′(λ) = ωXdmm+1, ω ∈ T [1, ν] ∪ πm(λs);

in this setting the Cerlienco–Mureddu correspondence gives a relation betweeneach point πm(xi ) and the corresponding term τi .

Moreover, since the argument is on the cardinality of the Macaulay repre-sentation and #(W) < #(L), we directly deduce that the ideal P(πm(W)) hasthe Grobner basis Lp( ft1), . . . , Lp( ft j (mdm )

). Also

πm(λs)(Lp( ft j )) = 0, for each j < I while πm(λs)(Lp( ftI ) = 0.

so that the same argument gives that the Cerlienco–Mureddu correspondencereturns Φ(πm(λs)) = Xd1

1 . . . Xdmm .


As a consequence, the application of Moller’s algorithm to L = L′ ∪ λs

produces:

qs := c−1 FI , with c = λs(FI );N := N′ ∪ ωI ;B := B′ \ ωI ∪ XiωI , 1 ≤ i ≤ n;fτ := f ′

τ − λs( f ′τ )qs for each τ ∈ B′ \ ωI , τ > ωI , and

fτ := f ′τ , for each τ ∈ B′ \ ωI , τ < ωI , since λs( f ′

τ ) = 0;for each τ := XiωI ∈ B′

fτ := (Xi − ais)FI −∑

Xi ω∈B′c(FI , ω) fXi ω

where

FI = ωI +∑ω∈N′

c(FI , ω)ω.

Corollary 33.6.3. If L′ satisfies conditions (A)–(L) then L satisfies conditions

(A), (F), (I), (L).If moreover I is a Ce-Mu-ideal and L

′ satisfies conditions (M)–(S) then L

satisfies conditions (Q), (R), (S).

Proof.

(A) and (F) are obvious;(I) and (Q) are a direct consequence of the application of the Cerlienco–

Mureddu algorithm to P(πm(W));(L) λs( fωI ) = 0 by construction;(R) On the basis of Remark 33.4.8 we know that FI ∈ (H ′

1, . . . , H ′I ); also all

we need to prove is that, for each i ,

Hi ∈ (H1, . . . , Hi−1) = Hj , T(Hj ) < T(Hi );therefore

• if T(Hi ) = ti ∈ G′, i I , we have

Hi = H ′i − aFI ∈ (H ′

1, . . . , H ′i−1) = (H1, . . . , Hi−1)

so that, also (H ′1, . . . , H ′

i ) = (H1, . . . , Hi );• finally, for τ = Xi tI we have Lτ = L ′

I , and

Lτ Pτ = Hτ ≡ fτ ≡ (Xi − ais)FI ≡ (Xi − ais)L ′I P ′

I ≡ 0

modulo (H ′1, . . . , H ′

I ) = (H1, . . . , HI ).

578 Moller II

The same argument proves the claim for h1, . . . , hr .(S) λs(HωI ) = 0 and λs(hωI ) = 0 because both HωI − fωI and hωI − fωI

have a representation in terms of Fi , i < I and λs(Fi ) = 0, for eachi < I.

In conclusion we have:

Theorem 33.6.4. For a zero-dimensional ideal I, given by a Macaulay repre-sentation L, using the same notation as above, we have:

(A) N := N(I);(B) G(I) = G = t1, . . . , tr , t1 < t2 < · · · < tr ;(C) B(I) = B;(D) for each τ ∈ N there is a unique polynomial

fτ := τ −∑

ω∈N(τ )

c( fτ , ω)ω

such that λ( fτ ) = 0, for each λ ∈ L(τ );(E) for each τ ∈ G there is a unique polynomial

fτ := τ −∑ω∈N

c( fτ , ω)ω

such that λ( fτ ) = 0, for each λ ∈ L;(F) for each τ ∈ B there is a polynomial

fτ := τ −∑

ω∈N(τ )

c( fτ , ω)ω

such that λ( fτ ) = 0, for each λ ∈ L;(G) the reduced Grobner basis of I is

G(I) := fτ : τ ∈ G;moreover, for each τ ∈ N, T( fτ ) = τ ;

(H) the border basis of I is

B(I) := fτ : τ ∈ B;(I) for each ν, 1 ≤ ν < n:

let jν be the value such that t jν < Xν+1 ≤ t jν+1; then ft1 , . . . , ft jν

is a minimal Grobner basis both of P(Spank(πν(L))) and ofI ∩ k[X1, . . . , Xν];

for each δ ∈ N, let j (νδ) be the value such that t j (νδ) < X δν+1 ≤

t j (νδ)+1; then Lp( ft1), . . . , Lp( ft j (νδ)) is a Grobner basis

of I(Yνδ);


(L) for each j, 1 ≤ j ≤ s, λ j ( fτ j ) = 0 so that L and λ j ( fτ j )−1 fτ j , 1 ≤ j ≤

s are triangular.

If I is a CeMu-ideal:

(M) for each τ := Xd11 . . . Xdn

n ∈ N, and each m, 1 ≤ m ≤ n, there areunique polynomials

γmτ := Xdmm +

∑ω∈Fm (τ )

c(γmτ , ω)ω

and

γmδτ := Xm +∑

ω∈Fmδ(τ )


such that

• πm(λ)(γmδτ ) = 0, for each λ ∈ Ymδ(τ ), λ ≺ Φ−1(τ ),

• πm(λ)(γmτ ) = 0, for each λ ∈ Lm(τ ), λ ≺ Φ−1(τ ),

• γmτ = ∏δ γmδτ ;

(N) for each τ := Xd11 . . . Xdn

n ∈ G, and each m, 1 ≤ m ≤ n, there are uniquepolynomials

γmτ := Xdmm +

∑ω∈Em (τ )

c(γmτ , ω)ω

and

γmδτ := Xm +∑

ω∈Emδ(τ )


such that

• πm(λ)(γmδτ ) = 0, for each λ ∈ Ymδ(τ ),

• πm(λ)(γmτ ) = 0, for each λ ∈ Lm(τ ),• γmτ = ∏

δ γmδτ ;

(O) for each τ = Xd11 . . . Xdν

ν ∈ N, there are

Lτ ∈ k[X1, . . . , Xν−1]


Pτ = Xdνν +

∑ω∈Fν (τ )

c(Pτ , ω)ω ∈ k[X1, . . . , Xν−1][Xν]

such that Hτ := Lτ Pτ are such that

• T(Hτ ) = τ, Lp(Hτ ) = Lτ ,

• πν−1(λ)(Lτ ) = 0, for each λ ∈ U ν−1m=1Lm(τ ),

580 Moller II

• πν(λ)(Pτ ) = 0, for each λ ∈ Lν(τ ),

• πν(λ)(Hτ ) = 0, for each λ ∈ L : λ ≺ Φ−1(τ );

(P) for each i, 1 ≤ i ≤ r , there are

Li ∈ k[X1, . . . , Xν−1]


Pi = Xdνν +

∑ω∈Eν (ti )

c(Pi , ω)ω ∈ k[X1, . . . , Xν−1][Xν]

such that Hi := Li Pi are such that

• T(Hi ) = ti = Xd11 . . . Xdν

ν ∈ G ∩ T [1, ν], Lp(Hi ) = Li ,

• πν−1(λ)(Li ) = 0, for each λ ∈ ∪ν−1m=1Lm(ti ),

• πν(λ)(Pi ) = 0, for each λ ∈ Lν(ti ),• πν(λ)(Hi ) = 0, for each λ ∈ L;

(Q) Li , Pi , Hi , 1 ≤ i ≤ r , satisfy

H1, . . . , Hr is a minimal Grobner basis of I,for each ν, 1 ≤ ν < n, H1, . . . , Hjν is a minimal Grobner basis

of I ∩ k[X1, . . . , Xν] and of I(πν(X)),for each ν, 1 ≤ ν < n, L1, . . . , L j (νδ) is a Grobner basis of

I(Yνδ).

(R) for each i, 2 ≤ i ≤ r , Pi ∈ (Hj , j < i

): Li .

(S) for each j, 1 ≤ j ≤ s, λ j (Hτ j ) = 0; L and λ j (Hτ j )−1 Hτ j , 1 ≤ j ≤ s

are triangular;(T) for each τ := Xd1

1 . . . Xdnn ∈ N and each m, 1 ≤ m ≤ n, there are

polynomials

gmτ := Xdmm +

∑ω∈Mm (τ )

c(gmτ , ω)ω

such that λ(gmτ ) = 0, for each λ ∈ Lm(τ ), λ ≺ Φ−1(τ );(V) for each τ := Xd1

1 . . . Xdnn ∈ G, and each m, 1 ≤ m ≤ n, there are

polynomials

gmτ := Xdmm +

∑ω∈Mm (τ )

c(gmτ , ω)ω

such that λ(gmτ ) = 0, for each λ ∈ Lm(τ ).

If moreover I is radical:

(W) for each τ = Xd11 . . . Xdν

ν ∈ N, there are

lτ ∈ k[X1, . . . , Xν−1]



pτ = Xdνν +

∑ω∈Mν (τ )

c(pτ , ω)ω ∈ k[X1, . . . , Xν−1][Xν]

such that hτ := lτ pτ are such that

• T(hτ ) = τ,

• Lp(hτ ) = lτ ,• lτ (πν−1(a)) = 0, for all a ∈ X(τ ),

• pτ (a) = 0, for each a ∈ Dν(τ ),

• hτ (a) = 0, for each a ∈ X such that a ≺ Φ−1(τ );

(X) for each i, 1 ≤ i ≤ r , there are

li ∈ k[X1, . . . , Xν−1]


pi = Xdνν +

∑ω∈Mν (ti )

c(pi , ω)ω ∈ k[X1, . . . , Xν−1][Xν]

such that hi := li pi are such that

• T(hi ) = ti = Xd11 . . . Xdν

ν ∈ G ∩ T [1, ν],• Lp(hi ) = li ,• li (πν−1(a)) = 0, for each a ∈ ⋃ν−1

m=1 Dm(ti ),• pi (a) = 0, for each a ∈ Dν(ti ),• hi (a) = 0, for each a ∈ X;

(Z) li , pi , hi , 1 ≤ i ≤ r , satisfy

h1, . . . , hr is a minimal Grobner basis of I,for each ν, 1 ≤ ν < n, h1, . . . , h jν is a minimal Grobner basis of

I ∩ k[X1, . . . , Xν] and of P(Spank(πν(L))),for each ν, 1 ≤ ν < n, l1, . . . , l j (νδ) is a Grobner basis of I(Yνδ),for each i, 2 ≤ i ≤ r , pi ∈ (

h j , j < i)

: li ,for each j, 1 ≤ j ≤ s, λ j (hτ j ) = 0,L is triangular to λ j (hτ j )

−1hτ j , 1 ≤ j ≤ s.

Part five

Beyond Dimension Zero

And when he had opened the fifth seal, I saw under the altar the souls of them that wereslain for the word of God, and for the testimony which they held.

And they cried with a loud voice, saying, How long, O Lord, holy and true, dost thounot judge and avenge our blood on them that dwell on the earth?Revelation (Authorised Version)

The things depending from Mercury: animality, quicksilver, agate, marjoram, monkey,blackbird, mullet.E.C. Agrippa, De occulta phylosophia

Revelliez-vous a notre voixEt sortez de la nuit profonde,Peuples, ressaisissez vos droits,Le soleil luit pour tout le monde.Sylvain Marechal, Chanson des Egaux

34

Grobner IV

In the introduction to Chapter 27 I connected the notion of ‘solving’ to boththe Lasker–Noether Theorem and the Kronecker Model, thus suggesting that‘solving’ an ideal I ⊂ k[X1, . . . , Xn] ⊂ P consists of returning, for eachassociated prime p of I, an admissible sequence ( f1, . . . , fr ) for the quotientfield of the integral domain P/p.

A careful analysis of such an admissible sequence led Grobner to describea ‘good’ basis, Primbasis, for each prime p ⊂ P. Grobner was probably mo-tivated in this discussion by the fact that a Primbasis is essentially a completeintersection. What led computer algebra to reconsider Grobner’s approach isthe fact that his Primbasis is naturally a Grobner basis under a lexicographicalordering.1 This led computer algebra to generalize Grobner results thus givingdifferent Basissatze:

• we first consider the case of a zero-dimensional ideal (Section 34.1),where Grobner’s result can be extended from primes to primary ideals,while Grobner’s structural results do not necessarily hold for a radicalideal;

• Grobner improved his results (Section 34.2) by considering the effect onKronecker’s Model of the Primitive Element Theorem, thus describing thestructure of the basis of a zero-dimensional radical ideal I ⊂ k[X1, . . . , Xn]in allgemeine, that is generic, position; the result is what Grobner called amonoidale Primbasis:

I = (g(Y ), X2 − g2(Y ), . . . , Xn − gn(Y ))

where Y := ∑ni=1 ai Xi is ‘generic’, deg(gi ) < deg(g), g is squarefree and

is irreducible if and only if I is prime;

1 This has implicitly already been used in the discussion on representation and arithmetics of afield in Kronecker’s Model (Section 8.3).

585

586 Grobner IV

• such results were then extended by Grobner (Section 34.3) to a prime idealI, dim(I) = d > 0, by simply connecting the basis of I with that ofIk(X1, . . . , Xd)[Xd+1, . . . , Xn], where X1, . . . , Xd is a minimal set of in-dependent variables.

The strength of Grobner’s Allgemeine Nulldimensionale Basissatz (Theo-rem 34.2.1) suggests (Section 34.4) specializing the notion of Noether posi-tion (Section 27.9) to that of allgemeine position and studying the structureof the lexicographical Grobner basis of an ideal I when it is projected onto anallgemeine coordinate Y := ∑n

i=1 ai Xi .In Section 34.5 the notion of ‘solving’ which is implicit throughout this book

is discussed.Finally, in Section 34.6 the Gianni–Kalkbrener Theorem, which is a strong

and powerful structural description of the Grobner basis of a polynomial idealw.r.t. the lexicographical ordering, is presented.

34.1 Nulldimensionale Basissatze

The discussion in Section 27.12 of the structure of zero-dimensional idealsJ ⊂ k[X1, . . . , Xn] in a polynomial ring over the algebraic closure field ksuggests that study of the more general case of a zero-dimensional ideal I ⊂k[X1, . . . , Xn] will require a reconsideration of Kronecker theory (Chapter 8)starting from the easy

Remark 34.1.1. Let P := k[X1, . . . , Xn] and let I ⊂ P be an ideal. Then thefollowing conditions are equivalent:

• I is a maximal ideal,• K := P/I ⊃ k is a finite algebraic extension.

We therefore fix a zero-dimensional ideal I – not necessarily a maximal one –and we define,2 for j, 0 ≤ j ≤ n:

• I j := I ∩ k[X1, . . . , X j ],• L j := k[X1, . . . , X j ]/I j ,• π j to be both the canonical projection

π j : k[X1, . . . , X j ] → L j


π j : k[X1, . . . , Xn] → L j [X j+1, . . . , Xn].

2 Where, with some abuse of notation, we set I0 := (0), L0 := k, π0( f ) = f for any f ∈k[X1, . . . , Xn ].

34.1 Nulldimensionale Basissatze 587

Then we assume that we have the reduced Grobner basis G of I underthe lexicographical ordering induced by X1 < · · · < Xn . Since I is zero-dimensional, we know from Theorem 27.12.3 that, for each j, there are

• a minimal d j ∈ N such that Xd jj ∈ T(G); and

• a monic polynomial

f j ∈ k[X1, . . . , X j−1][X j ] \ k[X1, . . . , X j−1],

such that

• T( f j ) = Xd jj ,

• each other element h ∈ G \ f j must be a combination of terms not

divisible by Xd jj or, equivalently, satisfies deg j (h) < d j .

We will therefore write G := f1, . . . , fn ∪ hi j with

• f j ∈ k[X1, . . . , X j−1][X j ] \ k[X1, . . . , X j−1] monic, T( f j ) = Xd jj and

degl( f j ) < dl , for each l = j ,• hi j ∈ k[X1, . . . , X j ] \ k[X1, . . . , X j−1] such that degl(hi j ) < dl , for

each l.

Moreover, we note that, under these assumptions, H := f1, . . . , fn is aGrobner basis itself 3 generating an ideal H ⊂ I ⊂ k[X1, . . . , Xn] such thatdeg(I) ≤ deg(H) = ∏

l dl .

Having set the notation we will use throughout this and the next section, wecan state the first result:

Theorem 34.1.2 (Grobner; Nulldimensionaler Primbasissatz). The follow-ing conditions are equivalent:

(1) I is prime;(2) for each j, 1 ≤ j ≤ n, there exists f j ∈ k[X1, . . . , X j ] \

k[X1, . . . , X j−1] such that

(a) for each j , I j = ( f1, . . . , f j ),(b) I = H = ( f1, . . . , fn),(c) for each j, f j is monic in X j ,(d) for each j, π j−1( f j ) ∈ L j−1[X j ] is irreducible, of degree d j , over

the field L j−1.

Moreover the conditions above imply:

3 Each S-pair satisfies Buchberger’s First Criterion, since T( fi ) = Xdii and T( f j ) = X

d jj are

relatively prime for each i, j, i = j .

588 Grobner IV

(i) for each j, I j is maximal and L j is a field;(ii) for each j, ( f1, . . . , f j ) is the reduced Grobner basis of I j w.r.t. the

lexicographical ordering induced by X1 < · · · < X j ;(iii) [Ln : k] = ∏

l dl = deg(I); [L j : k] = ∏l≤ j dl = deg(I j );

(iv) for each j the ideal IEj := I jP is prime and the chain

(0) ⊂ IE1 ⊂ IE

2 ⊂ · · · ⊂ IEn−1 ⊂ I

cannot be further refined;(v) for each j, dim(IE

j ) = n − j, r(IEj ) = j.

Proof. (See Section 8.2.)

(2) ⇒ (1) By construction, inductively, each L j is a simple algebraic field ex-tension of L j−1 of degree d j . Therefore each I j (and so also I) ismaximal and so prime.

(i) is a direct consequence of the argument above.(1) ⇒ (2) Because I is prime and 0-dimensional, I1 = I ∩ k[X1] = (0) is

prime, therefore it is generated by a monic irreducible polynomial f1.So inductively, we can assume that we have found f1, . . . , f j−1

satisfying (c) and (d) and generating the prime ideal I j−1. SinceI ∩ k[X j ] = (0), π j−1(I j ) = (0) and is prime, so there is a monicpolynomial

f j ∈ k[X1, . . . , X j ] \ K [X1, . . . , X j−1]

such that π j−1( f j ) is a generator of π j−1(I j ) and so it is irreducible.Also I j = ( f1, . . . , f j ).

(ii) Is obvious.(iii) we have a tower of finite algebraic simple extensions

k = L0 ⊂ L1 ⊂ · · · ⊂ Ln = P/I

each having degree d j .(iv) and (v) From the chain

(0) ⊂ IE1 ⊂ IE

2 ⊂ · · · ⊂ IEn−1 ⊂ I

and Lemma 27.9.3 we obtain the formula

n = dim(0) > dim(IE1 ) > · · · > dim(IE

n−1) > dim(I) = 0,

whence (v) and the impossibility of refining the chain.

Corollary 34.1.3. The ideal I is prime if and only if

• its reduced Grobner basis w.r.t. the lexicographical ordering induced byX1 < · · · < Xn is G = f1, · · · , fn, and

• for each j, π j−1( f j ) ∈ L j−1[X j ] is irreducible.


Definition 34.1.4 (Grobner). Under the assumptions above, the basis G = f1, . . . , fn is called the Primbasis of I = H.

Theorem 34.1.5 (Gianni; Nulldimensionale Primarbasissatz). The follow-ing conditions are equivalent:

(1) I is primary;(2) for each j, 1 ≤ j ≤ n exist f j , g j , hi j ∈ k[X1, . . . , X j ] \

k[X1, . . . , X j−1] such that writing,4 for each j, 0 ≤ j ≤ n,

• J j := (g1, . . . , g j ) ⊂ K [X1, . . . , X j ],• M j := k[X1, . . . , X j ]/J j ,• ρ j for both the canonical projection

ρ j : k[X1, . . . , X j ] → M j


ρ j : k[X1, . . . , Xn] → M j [X j+1, . . . , Xn],

the following hold:

(a) for each j, I j = (f1, . . . , f j

) + (hil : l ≤ j);(b) for each j, (g1, . . . , g j ) is the Primbasis of the prime ideal J j ;(c) I = ( f1, . . . , fn) + (hil : l ≤ n);(d) for each j, f j and g j are monic in X j ;(e) for each j, ρ j−1( f j ) is a power of the irreducible polynomial

ρ j−1(g j );(f) deg j (hi j ) < deg j ( f j ), ρ j−1(hi j ) = 0.

Moreover, the conditions above imply, for each j:

(i) h ∈ I j , deg j (h) < deg j ( f j ) ⇒ ρ j−1(h) = 0;

(ii) J j = √I j ;

(iii) Hj := f1, . . . , f j generates a J j -primary ideal H j ⊂ I j ⊂k[X1, . . . , X j ], deg(I j ) ≤ deg(H j ) = ∏ j

l=1 dl .

Proof.

(2) ⇒ (1) Since an ideal is primary if and only if its radical is prime, we havejust to prove that each prime ideal J j is the radical of the ideal I j .One has I1 = ( f1), J1 = (g1), and there is r ∈ N such that f1 = gr

1so J1 = √

I1.

4 Again, with some abuse of notation, we set J0 := (0), M0 := k, ρ0( f ) = f for any f ∈k[X1, . . . , Xn ].

590 Grobner IV

Then, by induction on j , (e) implies that, for a suitable r j ∈ N:

p := gr jj − f j ∈ J j−1k[X1, . . . , X j ]

and so, for some s, ps ∈ I j−1k[X1, . . . , X j ]; therefore

gr j sj = (p + f j )

s = ps +(

sps−1 + · · · + f s−1j

)f j ∈ I j .

So J j ⊂ √I j and, by maximality, J j = √

I j .(ii) Is part of the argument above.(i) Let h ∈ I j , deg j (h) < deg j ( f j ); according to (a), we can express it as

h = p f j +∑

i

pi hi j + u with u ∈ I j−1k[X1, . . . , X j ].

Then, by (f),

ρ j−1(h) = ρ j−1(p)ρ j−1( f j ) +∑

i

ρ j−1(pi )ρ j−1(hi j )

= ρ j−1(p)ρ j−1( f j )

giving a contradiction on degrees unless ρ j−1(h) = 0.(iii) It is sufficient to note that each Hj is a Grobner basis.(1) ⇒ (2) Since I is primary, so is each I j .

For j = 1, the claim states the existence of polynomials f1 and g1,with g1 irreducible and f1 a power of it, such that I1 = ( f1), which isclearly true.So assume that we have proved the claim for j − 1. Then ρ j−1(I j ) ⊂M j−1[X j ] and is generated by a power of an irreducible monic poly-nomial; therefore there are f j , g j ∈ k[X1, . . . , X j ] satisfying (d)and (e).Then (b) holds, since J j = (g1, . . . , g j ) is prime by the Primbasis-satz.There are now polynomials

h1 j , . . . , hs j ∈ k[X1, . . . , X j ] \ k[X1, . . . , X j−1]

such that I j = I j−1 + ( f j , h1 j , . . . , hs j ). By pseudodivision by f j wecan assume deg j (hi j ) < deg j ( f j ), so that, by the argument on degreesketched above, ρ j−1(hi j ) = 0 and (a), (c), (f) hold.

Corollary 34.1.6. The ideal I is primary if and only if its reduced Grobnerbasis w.r.t. the lexicographical ordering induced by X1 < · · · < Xn can beexpressed as

G := f1, . . . , fn ∪ hi j ,


where

• for each j there is g j ∈ k[X1, . . . , X j ] \ k[X1, . . . , X j−1] monic such thatρ j−1(g j ) is irreducible and ρ j−1( f j ) is a power of it, and

• for each i, j, ρ j−1(hi j ) = 0.

Under these assumptions, H := f1, . . . , fn is the reduced Grobner basisw.r.t. the lexicographical ordering induced by X1 < · · · < Xn of a primaryideal H such that√

H = √I,

H ⊂ I, anddeg(I) ≤ deg(H) = ∏ j

l=1 dl .

Definition 34.1.7. Under the assumptions above, the basis

H = f1, . . . , fnis called the Primarbasis of I.

As Kronecker’s Model gives a characterization of the structure of nulldimen-sional prime ideals, one can expect that, in the same way, Duval’s Model willallow us to characterize nulldimensional radical ideals. But the situation ismore complex. We can in fact only state the following

Theorem 34.1.8 (Nulldimensionaler Radikalbasissatz). Among the follow-ing conditions

(1) I is radical;(2) for each j, 1 ≤ j ≤ n exists

f j ∈ k[X1, . . . , X j−1][X j ] \ k[X1, . . . , X j−1]

such that

(a) for each j, I j = ( f1, . . . , f j );(b) I = ( f1, . . . , fn);(c) for each j, f j is monic in X j , d j := deg j ( f j );(d) for each j, L j is a direct sum of fields, L j = ⊕

i Li j , whose canon-ical projections, and their field extensions, will be denoted,5

πi j : L j [X j+1, . . . , Xn] → Li j [X j+1, . . . , Xn],

(e) for each j, πi j−1π j−1( f j ) ∈ Li j−1[X j ] is squarefree,

the implication (2) ⇒ (1) holds. Moreover, condition (2) above implies:

5 Where, with the customary abuse of notation, we have I0 := (0), L0 := k, π0( f ) = f for anyf ∈ k[X1, . . . , Xn ].

592 Grobner IV

(i) for each j, I j is radical and L j is a Duval field;(ii) for each j, ( f1, . . . , f j ) is the reduced Grobner basis of I j w.r.t. the

lexicographical ordering induced by X1 < · · · < X j ;(iii)

∏l dl = deg(I); ∏

l≤ j dl = deg(I j ).

Proof. (See Section 11.4)

(i) Each L j is a Duval field and a direct sum of fields, L j = ⊕i Li j ; therefore,for each j , there is a prime pi ∈ k[X1, . . . , X j ] such that

Li j = k[X1, . . . , X j ]/pi and pi = ker(πi j−1π j−1)

so that I j = ∩i pi .(2) ⇒ (1) is a special case of (i).(ii) and (iii) are trivial,

but there is no converse implication (1) ⇒ (2), as the example below shows.

Example 34.1.9. In k[X1, X2], the ideal

I := (X21 − X1, X1 X2, X2

2 − X2) = (X1 − 1, X2) ∩ (X1, X2) ∩ (X1, X2 − 1)

is radical but does not satisfy condition (2) of the theorem above.On the other hand we have the decomposition

I = I1∩I2, I1 := (X1, X22−X2) = (X1, X2)∩(X1, X2−1), I2 := (X1−1, X2)

where each component satisfies condition (2).Such components are naturally obtained a la Duval splitting the ideal I ac-

cording to whether the element x1 ∈ k[x1, x2] = k[X1, X2]/I is zero or invert-ible in the components of the direct sum of fields

k[x1, x2] = k[X1, X2]/(X1 − 1, X2) ⊕ k[X1, X2]/(X1, X2)

⊕ k[X1, X2]/(X1, X2 − 1).

Whether x1 is zero or invertible is a natural question in the Duval Model:once the first generator X2

1 − X1 of I has been tested to find if it is monic andsquarefree, thus producing the Duval field

D1 := k[x1] := k[X1]/(X21 − X1),

one needs to investigate the second generator x1 X2 ∈ D1[X2] which,

• if x1 = 0, gives the second monic and irreducible polynomial X2, thus pro-ducing the Duval sequence (X1 − 1, X2), while,

• if x1 = 0, it vanishes, thus producing the Duval sequence (X1, X22 − X2).

34.2 Primitive Elements and Allgemeine Basissatz 593

Definition 34.1.10. Under the assumptions above, the basis G = f1, . . . , fnwhose elements satisfy condition (2) is called the Radikalbasis of I.

34.2 Primitive Elements and Allgemeine Basissatz

The success of the reinterpretation of Kronecker’s Model in terms of ‘good’bases of null-dimensional ideals leads to an investigation of what would be theeffect of the Primitive Element Theorem (Theorem 8.4.5) on the representationof such ideals; this shows that the Primitive Element Theorem allows us to nat-urally extend this interpretation to the Duval Model and that the corresponding‘good’ lexicographical Grobner basis has a very nice shape.

Let us first recall that the construction of a primitive element y from a givenset x1, . . . , xn of algebraic (separable) elements consists of repeatedly defin-ing

y2 := x1 + c2x2, y3 := y2 + c3x3, . . . , yn := yn−1 + cn xn,

where ci = 0 for each i , and recall that for almost all choices of (c2, . . . , cn) ∈C(n − 1, k), the resulting

y := yn = x1 +n∑

i=2

ci xi

is primitive.For technical reasons 6 we will fix an infinite subfield k′ ⊂ k and consider

only choices (c2, . . . , cn) ∈ C(n − 1, k′).Let us therefore consider the polynomial ring k[X1, . . . , Xn], a zero-

dimensional ideal I ⊂ k[X1, . . . , Xn] and an infinite subfield k′ ⊂ k.For any c := (c2, . . . , cn) ∈ C(n − 1, k′) we set

Yc := X1 +n∑

i=2

ci Xi ,

and we consider the linear change of coordinates (see Example 27.8.2)

k[Yc, X2, . . . , Xn] = k[X1, . . . , Xn]

defined by X1 = Yc − ∑ni=2 ci Xi .

Since I is zero-dimensional, there is a polynomial gc(Yc) ∈ k[Yc] such thatI ∩ k[Yc] = (gc).

6 Essentially, we will need to apply the result in the non-zero-dimensional case, where we willconsider integral elements xd+1, . . . , xn over the field k(X1, . . . , Xd ), and we will need wlogto deal only with combinations y := xd+1 + ∑n−d

i=2 ci xd+i with ci ∈ k.

594 Grobner IV

Let us write Z(I) = a1, . . . , as, where a j = (a j1, . . . , a jn) and note thatfor all j, l, 1 ≤ j, l ≤ s:

Yc(a j ) = Yc(al) ⇐⇒ a j1 +n∑

i=2

ci a ji = al1 +n∑

i=2

ci ali

⇐⇒ (a j1 − al1

) +n∑

i=2

ci(a ji − ali

) = 0,

so that there is a non-empty Zariski open set U ⊂ C(n − 1, k′) such that

Yc(a j ) = Yc(al) for each c ∈ U and j, l, 1 ≤ j, l ≤ s.

As a consequence:

Theorem 34.2.1 (Grobner; Allgemeine Nulldim. Basissatz). With the no-tation above, if I is radical, then there is a non-empty Zariski open set U ⊂C(n − 1, k′) such that for each c ∈ U exist g0, g2, . . . , gn ∈ k[Yc] so that,writing

f1(Yc) := g0(Yc), and fi := Xi − gi (Yc), 2 ≤ i ≤ n,

g1(Yc) := Yc −n∑

i=2

ci gi (Yc),

the polynomials f1, . . . , fn satisfy condition (2) of Theorem 34.1.8 for I ⊂k[Yc, X2, . . . , Xn]. In particular:

(a) g0(Yc) is squarefree and monic, degYc(g0) =: δ;

(b) (g0(Yc)) = I ∩ k[Yc];(c) degYc

(gi ) < degYc(g0) = δ, for each i;

(d) (g0(Yc), X2 − g2(Yc), . . . , Xn − gn(Yc)) is the reduced Grobner basis ofI w.r.t. the lexicographical ordering induced by Yc < X2 < · · · < Xn;

(e) for each j, (g0(Yc), X2 −g2(Yc), . . . , X j −g j (Yc)) is the reduced Grobnerbasis of I ∩ k[Yc, X2, . . . , X j ] w.r.t. the lexicographical ordering inducedby Yc < X2 < · · · < X j ;

(f) k[X1, . . . , Xn]/I ∼= k[Yc]/g0(Yc);(g) δ = degYc

(g0) = deg(I);(h) writing R := α ∈ k : g0(α) = 0, one has

Z(I) = (g1(α), g2(α), . . . , gn(α)) : α ∈ R;(i) I is prime iff g0(Yc) is irreducible.

Proof. For each c := (c2, . . . , cn) ∈ C(n − 1, k′), since I is radical and zero-dimensional, I ∩ k[Yc] is radical and is generated by a monic and squarefree


polynomial g0, thus implying (a) and (b). Moreover, if I is prime, g0 is irre-ducible.

By the discussion above we deduce the existence of the non-empty Zariskiopen set U ⊂ C(n − 1, k′) such that

Yc(a j ) = Yc(al) for each c ∈ U and j, l, 1 ≤ j, l ≤ s;

therefore we have

δ = degYc(g0) = #R = #Z(I) = deg(I)

and (g) holds.Then, for each i, 1 ≤ i ≤ n, there exists a unique polynomial gi (Yc) ∈

k[Yc], degYc(gi ) < δ, such that a ji = gi (Yc(a j )) for each j .

Therefore fi := Xi − gi (Yc) ∈ I for each i ≥ 2 and(g0(Yc), X2 − g2(Yc), . . . , Xn − gn(Yc)

)is the reduced Grobner basis of I w.r.t. the lexicographical ordering induced byYc < X2 < · · · < Xn, thus proving (c), (d), (e) and (f). Also, (i) holds sinceeach fi is linear and (h) holds because, for each j,

a j1 = Yc(a j ) −n∑

i=2

ci a ji = Yc(a j ) −n∑

i=2

ci gi (Yc(a j )) = g1(Yc(a j )).

Definition 34.2.2. Under the assumptions above, the basis

(g0(Yc), X2 − g2(Yc), . . . , Xn − gn(Yc))

is called the allgemeine basis of I.

Example 34.2.3. To illustrate the Nulldimensionalen Basissatze theorems, letus begin by considering the maximal ideal m and the primary ideal q inQ[X, Y ] where

• q is such that a := (√

2√

3,√

2 + √3) ∈ Z(q),

• writing

q = (Y − √2 − √

3)2, (X − √2√

3)2, (Y − √2 − √

3)(X − √2√

3),

q has q as its primary component at a,• m = √

q,

• m = √q = (X − √

2√

3, Y − √2 − √

3).

596 Grobner IV

Since K = Q[√

3,√

2] = Q[T, U ]/m where m = (T 2 − 3, U 2 − 2), byLemma 27.12.11 in order to obtain m we need to compute J ∩ k[X, Y ] where

J = (T 2 − 3, U 2 − 2, X − U T, Y − U − T );the computation of the Grobner basis of J under the lexicographical orderinginduced by X < Y < T < U , which is

G := X2 − 6, Y 2 − 2X − 5, T + XY − 3Y, U − XY + 2Y gives us the Grobner basis of m under the lexicographical ordering induced byX < Y , which is

G ∩ Q[X, Y ] = X2 − 6, Y 2 − 2X − 5.It is easy to verify that the Primbasis has the structure described by the

Primbasissatz (Theorem 34.1.2) and that the roots of the ideal are all and onlythe four conjugates (±√

2√

3, ±√2 ± √

3) of a; in particular f1 = X2 − 6 ∈Q[X ] and f2 = Y 2 − 2

√6 − 5 ∈ Q[

√6][Y ], where Q[

√6] = Q[X ]/ f1(X),

are irreducible.As for the computation of q, since

q = ((Y − √

2 − √3)2, (X − √

2√

3)2, (Y − √2 − √

3)(X − √2√

3)= (2

√6 − 2

√3Y − 2

√2Y + Y 2 + 5, 6 − 2

√6X + X2,

2√

3 + 3√

2 − √3X − √

2X − √6Y + XY

)we have

J = U 2 − 2, T 2 − 3, 2T U − 2T Y − 2UY + Y 2 + 5,

6 − 2T U X + X2, 2T + 3U − T X − U X − T UY + XY whose Grobner basis under the lexicographical ordering induced by X < Y <

T < U is

G := X4 − 12X2 + 36,

X2Y 2 − 6Y 2 − 2X3 − 5X2 + 12X + 30,

Y 4 − 4XY 2 − 10Y 2 + 4X2 + 20X + 25,

T − 12 (11Y 3 X + 27Y 3 + 13

3 Y X3 + 11Y X2 − 23Y X − 75Y ),

U + 92 XY 3 − 11Y 3 − 7

4 X3Y − 92 X2Y + 9XY + 30Y ,

so that

q = (X4 − 12X2 + 36,

X2Y 2 − 6Y 2 − 2X3 − 5X2 + 12X + 30,

Y 4 − 4XY 2 − 10Y 2 + 4X2 + 20X + 25).


Denoting by ρ1 : Q[X, Y ] → Q[√

6][Y ] the morphism such that ρ1(X) =√6, one has

X4 − 12X2 + 36 = (X2 − 6)2,

Y 4 − 4XY 2 − 10Y 2 + 4X2 + 20X + 25 = (Y 2 − 2X − 5)2,

ρ1(X2Y 2 − 6Y 2 − 2X3 − 5X2 + 12X + 30) = 0,

so that√

q = m and the given basis satisfies the Primarbasissatz (Theo-rem 34.1.5).

In order to verify the Allgemeine Nulldimensionalen Basissatz (Theo-rem 34.2.1) we have just to remark that the four roots of m are

(√6, +√2+√

3), (−√6, −√

2+√3), (−√

6, +√2−√

3), (√

6, −√2−√

3)

and there is no real need to perform a ‘generic’ change of coordinates, sincethe four roots are distinguished by their Y coordinates, so that it is sufficientto compute the Grobner basis of m under the lexicographical ordering inducedby Y < X , which is 7

X − 12 Y 2 + 5

2 , Y 4 − 10Y 2 + 1.

Note also that the Grobner basis of q under the lexicographical orderinginduced by Y < X is f1, f2, f3 where

f1 := Y 8 − 20Y 6 + 102Y 4 − 20Y 2 + 1

= (Y 4 − 10Y 2 + 1)2,

f2 := XY 4 − 10XY 2 + X − 12 Y 6 + 15

2 Y 4 − 512 Y 2 + 5

2

= (Y 4 − 10Y 2 + 1)(X − 12 Y 2 + 5

2 ),

f3 := X2 − XY 2 + 5X + 14 Y 4 − 5

2 Y 2 + 254

= (X − 12 Y 2 + 5

2 )2.

7 Note that

Y 4 − 10Y 2 + 1 =(

Y 2 − 2√

2Y − 1) (

Y 2 + 2√

2Y − 1)

=(

Y 2 − 2√

3Y + 1) (

Y 2 + 2√

3Y + 1)

(±√2 ± √

3)2 = 5 + 2√

6,

(±√2 ∓ √

3)2 = 5 − 2√

6.

598 Grobner IV

34.3 Higher-Dimension Primbasissatz

Let us now consider a prime ideal p ⊂ k[X1, . . . , Xn] =: P such thatdim(p) =: d = 0. Then, up to a renumbering of the variables, we have

p ∩ k[X1, . . . , Xd ] = (0).

Let us then define

K := k(X1, . . . , Xd),Q := k[X1, . . . , Xd ],

and let us consider the polynomial ring

K [Xd+1, . . . , Xn] = k(X1, . . . , Xd)[Xd+1, . . . , Xn]

and the prime

p := pk(X1, . . . , Xd)[Xd+1, . . . , Xn].

Clearly dim(p) = 0 since, for each i > d there is a non-zero polynomial

f (X1, . . . , Xd , Xi ) ∈ p ∩ k[X1, . . . , Xd , Xi ] ⊂ p ∩ K [Xi ].

Lemma 34.3.1. With the notation above, we have p ∩ P = p.

Proof. Let p/q ∈ p, p ∈ p, q ∈ Q \ 0; note that q ∈ p, since p ∩ Q = (0).

Therefore, if p/q = p′ ∈ P , so that p′q = p ∈ p, then p′ ∈ p.

Since p is maximal we can apply Theorem 34.1.2 in order to deduce

Theorem 34.3.2 (Grobner; Hoherdimensional Basissatz). Let p ⊂ k[X1, . . . , Xn] be a prime ideal,

dim(p) = d, p ∩ k[X1, . . . , Xd ] = (0),

and set r := n − d.Then we have:

(1) There are polynomials p1, . . . , pr ∈ P and F ∈ Q such that

(a) pi ∈ k[X1, . . . , Xd+i ] \ k[X1, . . . , Xd+i−1],(b) pi , as an element of K [Xd+1, . . . , Xd+i ], is monic in Xd+i ,(c) pi is irreducible in k[X1, . . . , Xd+i ],(d) pi ∈ p,(e) p = (p1, . . . , pr ) : F.

(2) For each q ∈ P \ p, there exist g ∈ P \ p and p ∈ p such thatqg − p ∈ Q \ 0.

(3) The ideal p is an isolated primary component of (p1, . . . , pr ) in P .

34.3 Higher-Dimension Primbasissatz 599

Proof.

(1) Applying Theorem 34.1.2 to p := pK [Xd+1, . . . , Xn] we obtain a se-quence of polynomials

fi ∈ K [Xd+1, . . . , Xd+i ] \ K [Xd+1, . . . , Xd+i−1]

satisfying the conditions listed there. Multiplying fi by the lcm qi ∈ Qof the denominators of the coefficients, we get rid of denominators andwe obtain the required pi := fi qi ∈ p ∩ P = p, which obviouslysatisfies (a), (b), (c) and (d).Let g1, . . . , gs ⊂ P be a basis of p; from

g j =∑

i

ai j

bi jfi with ai j ∈ P, bi j ∈ Q,

getting rid of denominators we obtain

Fj g j =∑

i

ci j pi , with ci j ∈ P, Fj ∈ Q.

Therefore, if we set F := ∏j Fj ∈ Q, since F ∈ Q \ 0 and p ∩Q =

(0), we have F /∈ p and, by Proposition 27.2.11, p : F = p, so that

p ⊆ (p1, . . . , pr ) : F ⊆ p : F = p

and (e) follows.(2) Let p := (p1, . . . , pr ) and π : K [Xd+1, . . . , Xn] → K [Xd+1, . . . ,

Xn]/p be the projection. Since q ∈ P \ p ⊂ K [Xd+1, . . . , Xn] \ p,

π(q) = 0 and is an invertible element of the field K [Xd+1, . . . , Xn]/p.This means that there exists g′ ∈ K [Xd+1, . . . , Xn] \ p : π(q)π(g′) =1, that is qg′ − 1 ∈ p; more precisely there are g ∈ P and h ∈ Q \ 0such that g′ = g/h; if we define p := qg − h we have

p = qg − h = h(qg′ − 1) ∈ p ∩ P = p and qg − p = h ∈ Q \ 0.(3) Let p = (p1, . . . , pr ) = ⋂r

i=1 qi , be an irredundant primary represen-tation in P and for each i, let pi be the associated prime. Therefore

p = p : F =r⋂

i=1

(qi : F);

by Proposition 27.2.11 we know that, for each i ,

• F ∈ qi ⇒ qi : F = P and• F /∈ qi ⇒ qi : F is a pi -primary.

Therefore there is one component, say q1, for which p = q1 : F = q1

while F ∈ qi if i > 1.

600 Grobner IV

If, for some i = 1, pi ⊂ p we would have the contradiction F ∈ qi ⊂pi ⊂ p with F ∈ p ∩ Q = (0).

Definition 34.3.3 (Grobner). Under the assumptions above, p1, . . . , pr iscalled the Primbasis of p.

Let now consider a radical unmixed ideal f ⊂ k[X1, . . . , Xn] =: P ,dim(f) =: d = 0 and let us again write K := k(X1, . . . , Xd) and considerthe polynomial ring K [Xd+1, . . . , Xn] = k(X1, . . . , Xd)[Xd+1, . . . , Xn] andthe extension ideal fe = fK [Xd+1, . . . , Xn].

Let us also assume that X1, . . . , Xd is a maximal set of independent vari-ables for each associated prime of f so that

f = fec = fk(X1, . . . , Xd)[Xd+1, . . . , Xn] ∩ P.

Let us also set r := n − d and write Yc := Xd+1 + ∑ri=2 ci Xd+i for each

c := (c2, . . . , cr ) ∈ C(r − 1, k). Then:

Corollary 34.3.4 (Allgemeiner Hoherdimensionaler Basissatz). With theassumptions above, there is a non-empty Zariski open set U ⊂ C(r − 1, k)

such that for each c ∈ U, there exist h0, h2, . . . , hr ∈ K [Yc] such that, denot-ing

g1(Yc) := h0(Yc), gi := Xd+i − hi (Yc), 2 ≤ i ≤ r,

h1(Yc) := Yc −r∑

i=2

ci hi (Yc),

the following hold:

(a) g1(Yc) is squarefree and monic in k[X1, . . . , Xd ][Yc], degYc(g1) =: δ;

(b) (g1(Yc)) = fe ∩ K [Yc];(c) degYc

(hi ) < degYc(h0) = δ, for each i;

(d)(

g1(Yc), Xd+2 − h2(Yc), . . . , Xn − hr (Yc))

is the reduced Grobner

basis of fe w.r.t. the lexicographical ordering induced by Yc < Xd+2 <

· · · < Xn;(e) Xd+1 − h1(Yc) ∈ fe;(f) k[X1, . . . , Xn]/f ∼= k[X1, . . . , Xd , Yc]/g1(Yc);(e) δ = degYc

(g1) = deg(I);(f) f is prime iff g1(Yc) is irreducible.

Corollary 34.3.5. With the assumptions above and denoting by < the lexico-graphical ordering induced by X1 < · · · < Xd < Yc < Xd+2 < · · · < Xn,there is a non-empty Zariski open set U ⊂ C(r − 1, k) such that for each

34.4 Ideals in Allgemeine Positions 601

c ∈ U, there exist δ ∈ N, q2, . . . , qr ∈ k[X1, . . . , Xd ] and p0, p2, . . . , pr ∈k[X1, . . . , Xd , Yc], degYc

(pi ) < δ, such that, denoting

g1 := Y δc + p0, and gi := qi Xd+i − pi , 2 ≤ i ≤ r

we have

(g1, . . . , gr ) ⊂ k[X1, . . . , Xd , Yc, Xd+2, . . . , Xn] is a basis of f;T<(g1) = Y δ

c ;T<(gi ) = T<(qi )Xd+i for each i ≥ 2.

Definition 34.3.6. Under the assumptions above, the basis (g1, . . . , gr ) iscalled the Allgemeine Basis of f.

34.4 Ideals in Allgemeine Positions

Let us now extend the notion of Noether position and improve Corollary 27.9.6by considering the linear transformations

Lc : k[X1, . . . , Xn] → k[X1, . . . , Xn]

defined by

Lc(Xi ) :=

X j + ∑ni= j+1 ci Xi if i = j ,

Xi if i = j,

where c := (c j+1, . . . , cn) ∈ C(n − j, k), and stating

Lemma 34.4.1. Let R = k[x1, . . . , xn] be an integral domain, d the transcen-dence degree of k(x1, . . . , xn) over k and assume x1, . . . , xd is a transcen-dental basis of R over k.

There is a non-empty Zariski open set U ⊂ C(n − j, k) such that for eachc := (c j+1, . . . , cn) ∈ U, setting

y j := Lc(x j ) = x j +n∑

i= j+1

ci xi

we have:

(1) if j ≤ d, x1, . . . , x j−1, y j , x j+1, . . . , xd is a transcendental basis ofR;

(2) if j = d + 1, y j is a primitive element for R, integral overk[x1, . . . , xd ];

(3) if j > d + 1, and xd+1 is a primitive element for R, integral overk[x1, . . . , xd ], there is g ∈ k[X1, . . . , Xd , T ] such that

y j = g(x1, . . . , xd , xd+1).

602 Grobner IV

Proof. (1) holds trivially for each c ∈ C(n − j, k) and the same is truefor (3): one has just to define g := g j + ∑n

i= j+1 ci gi where each gi ∈k[X1, . . . , Xd , T ], j ≤ i ≤ n, is a polynomial such that xi = gi (x1, . . . ,

xd , xd+1).

The central point is (2) which holds as a direct consequence of the PrimitiveElement Theorem (Lemma 8.4.2, Theorem 34.2.1).

Corollary 34.4.2. Let P := k[X1, . . . , Xn] and let f ⊂ P be an ideal.There is a Zariski open set N ⊂ GL(n, k) (respectively B(n, k), N (n, k))

such that for each M := (ci j

) ∈ N, and each associated prime p ∈ P of f,writing

• P/p =: k[x1, . . . , xn] =: R,• d := dim(p),• yi := M(xi ) = ∑

j ci j x j , for each i ,

we have

• y1, . . . , yd is a transcendental basis of R,• yi is integral over k[y1, . . . , yd ] for each i > d,• yd+1 is a primitive element for R,• for 0 ≤ i ≤ n − d − 1 there are polynomials

hi (Y1, . . . , Yd , T ) ∈ k[Y1, . . . , Yd ][T ],

h0 monic, such that, writing gi (T ) := hi (y1, . . . , yd , T ) we have

• k[x1, . . . , xn] = k[y1, . . . , yn] = k[Y1, . . . , Yd ][T ]/h0(T ),• g0(yd+1) = 0,• yd+1+i = gi (yd+1) for each i ,

• there exist

• δ ∈ N,• q2, . . . , qn−d ∈ k[Y1, . . . , Yd ] and• p0, p2, . . . , pn−d ∈ k[Y1, . . . , Yd , Yd+1], degYd+1

(pi ) < δ,

such that, denoting

g1 := Y δd+1 + p0, and gi := qi Yd+i + pi , 2 ≤ i ≤ n − d

we have

• (g1, . . . , gr ) ⊂ k[Y1, . . . , Yd , Yd+1, . . . , Yn] is the allgemeine basisof p,

• g1 is irreducible in k(Y1, . . . , Yd)[Yd+1],• g1(y1, . . . , yd , yd+1) = 0,• yd+i = pi (y1, . . . , yd , yd+1)q

−1i (y1, . . . , yd), 2 ≤ i ≤ r .

34.4 Ideals in Allgemeine Positions 603

Definition 34.4.3. Let P := k[X1, . . . , Xn], f ⊂ P be an ideal, Y1, . . . , Ynbe a system of coordinates of P and let M ∈ GL(n, k) be such that Yi =M(Xi ), for each i.

The ideal f is said to be in allgemeine position w.r.t. Y1, . . . , Yn – orY1, . . . , Yn to be an allgemeine position for f – if M ∈ N where N is theZariski open set N ⊂ GL(n, k) whose existence is implied by Corollary 34.4.2.

Let

P := k[X1, . . . , Xn],f ⊂ P be an ideal,G ⊂ P be the reduced Grobner basis of f w.r.t. the lexicographical ordering

induced by X1 < · · · < Xn

and let us wlog 8 assume that X1, . . . , Xn is in allgemeine position for f.Then:

Corollary 34.4.4 (Gianni). If f is radical and unmixed, d := dim(f), then

(1) there is a squarefree p ∈ G such that T(p) = Xed+1,

(2) for each j > d + 1 there is an irreducible p j ∈ G such that T(p j ) =m j X j , m j ∈ T [1, d],

(3) for each j > d there is q j ∈ G such that T(q j ) = Xe jj .

Moreover p is irreducible iff f is prime.

Proof. The existence of p and each p j follows from Corollary 34.3.5.The existence of the q j s follows from the Noether Normalization Lemma

(Theorem 27.9.1) if f is prime.In the general case, let f = ⋂r

i=1 pi be the irredundant primary representa-

tion of f where each pi is prime, and let each q(i)j be the minimal polynomial

over k of x j ∈ k[x1, . . . , xn] := P/pi .

Then Q j := ∏ri=1 q(i)

j ∈ f and T(Q j ) is a power of X j . This implies theexistence of q j ∈ G such that T(q j ) | T(Q j ) and the claim.

8 Up to the ‘generic’ linear transformation

M : k[X1, . . . , Xn ] → k[X1, . . . , Xn ]

defined by

M(Xi ) =∑

jci j X j for each i

where M := (ci j

) ∈ N and N is the Zariski open set N ⊂ GL(n, k) whose existence is impliedby Corollary 34.4.2.

604 Grobner IV

Corollary 34.4.5. If f := ⋂dl=1 ul is an irredundant equidimensional repre-

sentation and, for each l, Gl denotes the reduced Grobner basis of√

ul w.r.t.the lexicographical ordering induced by X1 < · · · < Xn, then, for each l,

(1) there is a squarefree p ∈ Gl such that T(p) = Xel+1,

(2) for each j > l + 1 there is an irreducible p j ∈ Gl such that T(p j ) =m j X j , m j ∈ T [1, l],

(3) for each j > l there is q j ∈ Gl such that T(q j ) = Xe jj .

The restrictions of Corollary 34.4.2 and 34.4.4 to a zero-dimensional idealgive

Corollary 34.4.6. Let P := k[X1, . . . , Xn] and let f ⊂ P be a zero-dimensional ideal.

There is a non-empty Zariski open set U ⊂ C(n − 1, k) such that for eachassociated prime p ∈ P of f and each c := (c2, . . . , cn) ∈ U, setting

Yc := X1 +n∑

i=2

ci Xi ,

and writing

• P/p =: k[x1, . . . , xn] =: R,• yc := x1 + ∑n

i=2 ci xi ,• Y for the linear form Y := X1 + ∑n

i=2 ci Xi ,• G ⊂ k[Y, X1, X2, . . . , Xn] for the reduced Grobner basis of

f + (Y − X1 −n∑

i=2

ci Xi )

w.r.t. the lexicographical ordering induced by Y < X1 < · · · < Xn

we have

• yc is a primitive element for R,• for 0 ≤ i ≤ n, i = 1 there are polynomials

gi (Y ) ∈ k[Y ],

g0 monic, such that, writing g1(Y ) := Y − ∑ni=2 ci gi we have

• R = k[Y ]/g0(Y ),• g0(yc) = 0,• xi = gi (yc) for each i ,

• G = (g0(Y ), X1 − g1(Y ), . . . , Xn − gn(Y )

).

34.5 Solving 605

Definition 34.4.7. With the notation above the linear form Y := X1 +∑ni=2 ci Xi is said to be an allgemeine coordinate for the zero-dimensional

ideal f if c := (c2, . . . , cn) ∈ U where U is the Zariski open set U ⊂ C(n−1, k)

whose existence is implied by Corollary 34.4.6.

34.5 Solving

Let us consider P := k[X1, . . . , Xn] and let Ω(k) be the universal field (Defi-nition 9.4.1) of k.

For any n-tuple β := (β1, . . . , βn) ∈ Ω(k)n we can consider the morphismΨβ : P → Ω(k) defined by Ψβ( f ) = f (β1, . . . , βn) for each f ∈ P.

Then clearly ker(Ψβ) = f ∈ P : f (β1, . . . , βn) = 0 =: p is a prime 9 andIm(Ψβ) = k[β1, . . . , βn] ∼= P/p is an integral domain whose quotient field isthe extension field k ⊂ K := k(β1, . . . , βn) ⊂ Ω(k).

Conversely, for any prime p ⊂ P , we can consider

the integral domain R := P/p,its quotient field K , which is a field extension of k and a subfield of Ω(k),

the images βi ∈ R ⊂ K ⊂ Ω(k) of each Xi modulo p

and we have

p = f ∈ P : f (β1, . . . , βn) = 0,R = k[β1, . . . , βn],

K = k(β1, . . . , βn).

In this setting d := dim(p) is the transcendental degree of K (Definition27.9.2) and, up to a suitable renumbering and relabelling the variables andthe βs, we have P = k[X1, . . . , Xn] = k[Y1, . . . , Yd , Z1, . . . , Zr ] and (Sec-tion 8.2)

K = k(β1, . . . , βd)(βd+1, . . . , βn)

∼= k(Y1, . . . , Yd)(βd+1, . . . , βn)

=: k(Y1, . . . , Yd)[α1, . . . , αr ]∼= k(Y1, . . . , Yd)[Z1, . . . , Zr ]/( f1, . . . , fr )

where r = n − d =: r(p) and ( f1, . . . , fr ) ⊂ k(Y1, . . . , Yd)[Z1, . . . , Zr ]is a suitable admissible sequence 10 or, equivalently, a Primbasis (Defini-tions 34.1.4 and 34.3.3) .

9 Being a field, Ω(k) has no zero-divisor.10 We do not care about minimality; we consider admissible that the root

(X1,√

X1,√

X1,√

X1) ∈ Ω(k)4

606 Grobner IV

Moreover, if Y1, . . . , Yd , Z1, . . . , Zr is in allgemeine position for p, then

f1 is a monic element in k[Y1, . . . , Yd ][Z1],for i ≥ 2, fi = qi Zi + pi for suitable qi ∈ k[Y1, . . . , Yd ] and pi ∈

k[Y1, . . . , Yd , Z1],R = k[Y1, . . . , Yd , α1] ∼= k[Y1, . . . , Yd , Z1]/( f1),

p = ( f1, . . . , fr ).

We can therefore consider p to be ‘given’ if we are given

• the integral domain R by means of

• the values d := dim(p) and r = n − d =: r(p),• a system of coordinates Y1, . . . , Yd , Z1, . . . , Zr of P where

Y1, . . . , Yd is a maximal set of independent variables for p,• and an admissible sequence ( f1, . . . , fr ) ⊂ k[Y1, . . . , Yd ][Z1, . . . , Zr ]

such that

R ∼= k[Y1, . . . , Yd , Z1, . . . , Zr ]/( f1, . . . , fr ), p = ( f1, . . . , fr )

• and the r elements αi ∈ R integral over k[Y1, . . . , Yd , α1, . . . , αi−1] andsatisfying fi (Y1, . . . , Yd , α1, . . . , αi−1, αi ) = 0,

so that

p = f ∈ k[Y1, . . . , Yd , Z1, . . . , Zr ] : 0 = f (Y1, . . . , Yd , α1, . . . , αr ) ∈ R.If space-time considerations do not forbid us, then we can perform a

‘generic’ change of coordinates so that

the system of coordinates Y1, . . . , Yd , Z1, . . . , Zr is in allgemeine positionfor p,

( f1, . . . , fr ) is an allgemeine basis

and we can consider p as ‘given’ by giving

• the values d := dim(p) and r = n − d =: r(p),• a system of coordinates Y1, . . . , Yd , Z1, . . . , Zr of P in allgemeine

position for p,• δ ∈ N,• polynomials q2, . . . , qr ∈ k[Y1, . . . , Yd ] and• p0, p2, . . . , pr ∈ k[Y1, . . . , Yd , Z1], degZ1

(pi ) < δ,

such that, writing F1 := Z δ1 + p0, we have

is associated to the prime p = (X21 − X2, X2 − X3, X3 − X4) and the field K = k(X1,

√X1)

is represented as

K = k(Y1)[Z1, Z2, Z3]/(Z1 − Y 21 , Z2 − Z1, Z3 − Z1).

34.5 Solving 607

R = k[β1, . . . , βn] ∼= k[Y1, . . . , Yd ][βd+1] ∼= k[Y1, . . . , Yd ][Z1]/(F1),F1(β1, . . . , βd , βd+1) = 0,βd+i = pi (β1, . . . , βd , βd+1)/qi (β1, . . . , βd).

These considerations allow us to explain in which sense we considered inSection 20.4 as ‘computed’ the set Z(I) of the roots of an ideal I ⊂ P .

In fact, let

I = ⋂ri=1 qi , be the irredundant primary representation of I,

for each i, pi be the associated prime of qi , anddi := dim(pi ),

and let us assume that each pi is ‘given’ in the sense above.For any element β := (β1, . . . , βn) ∈ Ω(k)n satisfying f (β1, . . . , βn) = 0

for each f ∈ I, writing p := ker(Ψβ) we have

p = f ∈ P : f (β1, . . . , βn) = 0 ⊃ I

so that there is at least one i for which p ⊃ pi .Let us write p := pi and let us wlog assume that both X1, . . . , Xdeg(p) and

X1, . . . , Xdeg(p) are a maximal set of independent variables for, respectively,p and p, and that p is given by means of

d := di = dim(p), r = n − d =: r(p),( f1, . . . , fr ) ⊂ k(X1, . . . , Xd)[Xd+1, . . . , Xn],α1, . . . , αr ∈ Rr

so that

P/p ∼= k[X1, . . . , Xn]/( f1, . . . , fr ),( f1, . . . , fr ) is an admissible sequence,each fi is a monic polynomial in k[X1, . . . , Xd , α1, . . . , αi−1][Xi ],each αi is integral over k[X1, . . . , Xd , α1, . . . , αi−1] andsatisfies fi (αi ) = 0,p = ( f1, . . . , fr ) = f ∈ P : 0 = f (X1, . . . , Xd , α1, . . . , αr ) ∈ R.

Then, since p ⊃ pi , we obtain the ring projection

Ψ : Ri := P/pi P/p = k[β1, . . . , βn]

defined by Ψ (Xi ) = βi and Ψ (α j ) = βd+ j for each i, j .Conversely, for any i and any ring homomorphism Ψ : Ri = P/pi → Ω(k),

if we write βi := Ψ (Xi ) and βd+ j := Ψ (α j ) for each i, j , we have, for eachf ∈ p,

f (β1, . . . , βn) = f (Ψ (X1), . . . , Ψ (Xd), Ψ (α1), . . . , Ψ (αr ))

= Ψ ( f (X1, . . . , Xd , α1, . . . , αr ))

= Ψ (0) = 0

so that (β1, . . . , βn) is a root of IΩ(k)[X1, . . . , Xn].

608 Grobner IV

34.6 Gianni–Kalkbrener Theorem

Adapting the notation of Section 26.2, we consider here the polynomial rings

k[Y] := k[Y1, . . . , Yd ],

k[Y][Z] := k[Y, Z] := k[Y1, . . . , Yd , Z1, . . . , Zr ]∼= k[X1, . . . , Xn] =: P

and the monomial semigroups

Y := Y a11 · · · Y ad

d : (a1, . . . , ad) ∈ Nd,

Z := Zb11 · · · Zbr

r : (b1, . . . , br ) ∈ Nr ,

T := Xc11 · · · Xcn

n : (c1, . . . , cn) ∈ Nn

= tY tZ : tY ∈ Y, tZ ∈ Z,

where n = d + r and we identify P and k[Y, Z] by

Xi :=

Yi if i ≤ d,Zi−d if i > d;

a term ordering <Z on Z, a term ordering <Y on Y, and the block ordering< on T inducing Y < Z, that is the one which, for each t (1)t (2) ∈ T , t (i) :=t (i)Y t (i)Z , t (i)Y ∈ Y, t (i)Z ∈ Z, i = 1, 2, is defined by

t (1) < t (2) ⇐⇒ t (1)Z <Z t (2)

Z or t (1)Z = t (2)

Z and t (1)Y <Y t (2)

Y ;the algebraic closure k of k; and, for any α = (b1, . . . , bd) ∈ kd , the projection

Φα : P ∼= k[Y][Z] → k[Z]

defined by

Φα( f ) = f (b1, . . . , bd , Z1, . . . , Zr ] for each f ∈ k[X1, . . . , Xn].

Lemma 34.6.1 (Gianni–Kalkbrener). Let I ⊂ k[Y][Z] be an ideal and G itsGrobner basis w.r.t. <. Then

(1) Φα(M<Z (I)) ⊆ M(Φα(I));(2) Φα(G) is a Grobner basis of Φα(I) in k[Z] if M(Φα(I)) =

Φα(M<Z (I)).

Proof.

(1) For f = lc( f )T<Z ( f ) + · · · ∈ k[Y][Z] we have

Φα(M<Z ( f )) = lc( f )(α)T<Z ( f ) = M(Φα( f ))

unless lc( f )(α)T<Z ( f ) = 0.

34.6 Gianni–Kalkbrener Theorem 609

(2) We have

M(Φα(I)) = Φα(M<Z (I))

= Φα(M<Z (G)) ⊆ M(Φα(G)) ⊆ M(Φα(I))

so that M(Φα(G)) = M(Φα(I)).

Let us now assume that

< is the lexicographical ordering < on T induced by X1 < X2 < · · · < Xn

and its restriction to each subset T [1, i] ⊂ k[X1, . . . , Xi ],I ⊂ k[Y][Z] is a zero-dimensional ideal andG is its Grobner basis w.r.t. <.

Lemma 34.6.2 (Gianni–Kalkbrener). Writing

J := I ∩ k[X1, . . . , Xd+1] ∼= I ∩ k[Y1, . . . , Yd , Z1],

and H := G ∩ k[X1, . . . , Xd+1] ∼= G ∩ k[Y1, . . . , Yd , Z1] we have

(1) there exists a polynomial g ∈ J such that Φα(g) generates Φα(J), anddegd+1(g) = deg(Φα(g)),

(2) Φα(H) ⊂ k[Xd+1] ∼= k[Z1] is a Grobner basis of Φα(J).

Proof.

(1) First let us prove the claim under the assumption that I is primary,thus applying the Nulldimensionale Primarbasissatz (Theorem 34.1.5)which gives that G = f1, . . . , fn ∪ hi j satisfies

H = f1, . . . , fd+1 ∪ hi j , j ≤ d + 1,H ′ := G ∩ k[Y1, . . . , Yd ] = f1, . . . , fd ∪ hi j , j ≤ d,fd+1 ∈ k(Y)[Z1] is monic,hid+1 ∈ √

I ∩ k[Y1, . . . , Yd ].

Then,

• either there is g ∈ G ∩ k[Y1, . . . , Yd ] such that Φα(g) = 0 so that

Φα(J) = (1) = Φα(g) and degd+1(g) = 0 = deg(Φα(g));• or H ′ ⊂ ker(Φα), Φα(hid+1) = 0 for each i , Φα(J) is generated by

Φα(H) = Φα( fd+1) ∪ Φα(hid+1) = Φα( fd+1)and degd+1( fd+1) = deg(Φα( fd+1)), because fd+1 is monic ink(Y)[Z1].

610 Grobner IV

In general, let us consider the irredundant primary decomposition I =⋂rl=1 ql = ∏r

l=1 ql . Our argument proves the existence, for each l, ofa polynomial gl ∈ ql ∩ k[X1, . . . , Xd+1] such that Φα(gl) generatesΦα(ql) and degd+1(gl) = deg(Φα(gl).Then clearly g := ∏r

l=1 gl satisfies the claim.(2) We know, from Lemma 34.6.1(1), that Φα(M<Z (J)) ⊆ M(Φα(J)),

and, from Theorem 26.2.2, that H is the Grobner basis of J, so in or-der to deduce the claim from Lemma 34.6.1(2) it is sufficient to proveΦα(M<Z (J)) ⊇ M(Φα(J)).

But, since k[Xd+1] is a principal ideal domain, this is a direct conse-quence of the result above. We have

g = Lp(g)X δd+1 + · · · ∈ k[Y][Xd+1]

with δ = degd+1(g) = deg(Φα(g)) and Lp(g)(b1, . . . , bd) = 0; there-fore

M(Φα(J)) = (X δd+1) = (Φα(g)) ⊆ Φα(M<Z (J)).

Let us write, for each d, 1 ≤ d ≤ n, δ ∈ N,

Gd := G ∩ k[X1, . . . , Xd ] andGdδ := g ∈ G, g ∈ k[X1, . . . , Xd ], degi (g) ≤ δ

and we recall (Theorem 26.2.2 and Theorem 26.2.6) that each Gd andLpdδ(G) := Lp(g), g ∈ Gdδ are Grobner bases w.r.t. < of, respectively,Id := I ∩ k[X1, . . . , Xd ] and Lpdδ(I).

We moreover enumerate G := g1, . . . , gs in such a way that

T(g1) < T(g2) < · · · < T(gs−1) < T(gs);therefore we have

G11 ⊆ G12 ⊆ · · · ⊆ G1 ⊆ · · · ⊆ Gd−1 ⊆ · · · ⊆ Gdδ ⊆ Gdδ+1 ⊆ · · · ⊆ Gd ⊆ · · ·

and each Gdδ is a section of both Gdδ+1 and Gd .We thus obtain the following immediate improvement of Trink’s Algorithm

for solving polynomial equations:

Theorem 34.6.3 (Gianni–Kalkbrener). Let

α := (b1, . . . , bd) ∈ Z(Id),σ be the minimal value such that Φα(Lp(gσ )) = 0,


j, δ the values such that

gσ = Lp(gσ )X δ+1j + · · · ∈ k[X1, . . . , X j ] \ k[X1, . . . , X j−1].

Then

• j = d + 1,• for each g ∈ Gd, Φα(g) = 0,• for each g ∈ Gd+1δ , Φα(g) = 0,• Φα(gσ ) = gcd (Φα(g) : g ∈ Gd+1) ∈ k[Xd+1],• for each b ∈ k, (b1, . . . , bd , b) ∈ Z(Id+1) ⇐⇒ Φα(gσ )(b) = 0.

Example 34.6.4. To illustrate Gianni–Kalkbrener’s Theorem, we consider theideal I ⊂ Z2[T1, T2, T3, T4] generated by ( f1, f2, f3, f4) where

f1 := T4 + T3 + T2, f2 := T 34 + T 3

3 + T1, f3 := T 163 + T3, f4 := T 16

4 + T4,

whose Grobner basis, under the lex ordering induced by T1 < T2 < T3 < T4

is h1, h2, h3, h4, h5, h6, h7, h8 where

h1 := 1T 161 + T1,

h2 := (T111 + T6

1 + T1)T 32 + T 12

1 + T 71 + T 2

1 ,

h3 := T1T 92 + T 2

1 T 62 + T 8

1 T 32 + T 4

1 ,

h4 := 1T 162 + T2,

h5 := T1T 23 + (T1T2)T3

+ T 82 T 9

1 + T 82 T 4

1 + T 52 T 10

1 + T 52 T 5

1 + T 22 T 6

1 + T 22 T1,

h6 := T2T 23 + T 2

2 T3 + T 32 + T1,

h7 := 1T 163 + T3,

h8 := 1T4 + T3 + T2

and the bold term of each hi denotes its leading polynomials Lp(hi ). Note thatan (incomplete) factorization of h1 = T 16

1 + T1 is

h1 = T1(T 101 − T 5

1 + 1)(T 51 + 1).

Among the 16 roots of h1:

τ1 = 0 is such that

Lp(h4)(0) = 0,hi (0, T2) = 0, for each i ≤ 3;

each root τ1 of T 101 + T 5

1 + 1 = Lp(h2)/Lp(h3) is such that

Lp(h3)(τ1) = 0,h1(τ1, T2) = h2(τ1, T2) = 0,h4(T2) = τ−1

1 h3(τ1, T2)(T 72 + τ1T 4

2 + τ 121 T2);

612 Grobner IV

each root τ1 of T 51 + 1 = h1/Lp(h2) is such that

Lp(h2)(τ1) = 0,h1(τ1, T2) = 0,h3(τ1, T2) = h2(τ1, T2)(T 6

2 + τ 21 ),

h4(T2) = τ−11 h2(τ1, T2)(T 13

2 + τ1T 102 + τ 2

1 T 72 + τ 3

1 T 42 + τ 4

1 T2).

Therefore the roots (τ1, τ2) of I ∩ K [T1, T2] are

(0, β) : β16 + β = 0,(ζ, δ) : ζ 10 + ζ 5 + 1 = δ9 + ζ δ6 + ζ 7δ3 + ζ 3 = 0,(ε, η) : ε5 + 1 = η3 + ε = 0.Among these roots:

(τ1, τ2) := (0, 0) is such that

Lp(h7)(0, 0) = 0,hi (0, 0, T3) = 0, for each i ≤ 6;

each root (τ1, τ2) := (0, β) : β15 + 1 = 0 is such that

Lp(h6)(0, β) = 0,hi (0, β, T3) = 0, ∀i ≤ 5;β−1h7(T3) = (β−1T3)

16 − (β−1T3) is obviously a multiple of

(β−1T3)2 + (β−1T3) + 1 = β−2h6(0, β, T3);

while for the other roots (τ1, τ2)

Lp(h5)(τ1, τ2) = 0,δh5(ζ, δ) = ζh6(ζ, δ),

there exists h(ζ, δ, T3) : ζh7(T3) = h5(ζ, δ)h(ζ, δ, T3),

ηh5(ε, η) = εh6(ε, η),

there exists h(ε, η, T3) : εh7(T3) = h5(ε, η)h(ε, η, T3).

Example 34.6.5. A more elementary example which explains better the rela-tion between the different polynomials in Gi∂ is the ideal I ⊂ Z2[T1, T2, T3]generated by the Grobner basis ( f1, f2, f3, f4, f5, f6) where

f1 := T 21 − T1,

f2 := T1T2,

f3 := T 22 − T2,

f4 := T1T3,

f5 := T2T3 − T2,

f6 := T 23 − T3

in which I ∩ Z2[T1] = (T 21 − T1) whose roots are 0, 1:


• for τ1 = 0 we have Φτ1 (I ∩ Z2[T1, T2]) = (0, 0, T 22 − T2) = (T 2

2 − T2)

whose roots are 0, 1:– for α := (τ1, τ2) = (0, 0) we have

Φα(I ∩ Z2[T1, T2, T3]) = (0, 0, 0, 0, 0, T 23 − T3) = (T 2

3 − T3)

whose roots are 0, 1;– for α := (τ1, τ2) = (0, 1) we have

Φα(I ∩ Z2[T1, T2, T3]) = (0, 0, 0, 0, T3 − 1, T 23 − T3) = (T3 − 1)

whose root is 1;• for τ1 = 1 we have Φτ1(I ∩ Z2[T1, T2]) = (0, T2, T 2

2 − T2) = (T2), whoseroot is 0 so that for α := (τ1, τ2) = (1, 0) we have– Φα(I ∩ Z2[T1, T2, T3]) = (0, 0, 0, 0, T3, 0, T 2

3 − T3) = (T3) whose rootis 0.

35

Gianni–Trager–Zacharias

The Primbasissatze presented in the previous chapter give the tools needed inorder to devise algorithms for computing Lasker–Noether decompositions andrelated concepts.

In Section 35.1 I introduce the computational problems related to Lasker–Noether decomposition which will be discussed throughout the chapter.

Section 35.2 presents the Gianni–Trager–Zacharias solution (mainly a directapplication of the Primbasissatze) for a zero-dimensional ideal.

Section 35.3 contains the result (Theorem 35.3.4) allowing us to reduce thegeneral case to the zero-dimensional one, and presents both their approach(GTZ-scheme) and a suggested improvement (ARGH-scheme): the GTZ-scheme has the disadvantage that the algorithms produce many redundantspurious components which must be tested and removed; the ARGH-schemeavoids such production of spurious components but at the price of also remov-ing embedded components which must be recovered later.

Section 35.4 gives the solution, by means of the GTZ- and ARGH-schemes,of the decomposition problems.

If the ideal to be decomposed is in allgemeine position, the best shape of thebases allows us to strongly improve the algorithms (Section 35.5); however, theprice of being in allgemeine position is full-density of all the data; this requirestechniques allowing the computation of an allgemeine coordinate preservingsparsity as much as possible (Section 35.6); in connection with this problem,there have also been proposals to apply Moller’s algorithm (Section 35.7) inorder to avoid density when performing the ARGH-scheme.

Section 35.8 is devoted to a presentation 1 of the proposal by Eisenbud,Huneke and Vasconcelos of applying ‘direct methods’ for decomposition as

1 However, limited by the fact that their theoretical tools are outside the scope of the book.

614

35.1 Decomposition Algorithms 615

an alternative to the Gianni–Trager–Zacharias approach by reduction to thezero-dimensional case.

Section 35.9 is devoted to an adaptation, by Caboara, Conti and Traverso, ofthe ARGH-scheme which is able to completely avoid a change of coordinates;Section 35.10 shows the application, proposed by Heiß–Oberst–Pauer, of in-verse systems in order to produce a squarefree decomposition of a primary.

35.1 Decomposition Algorithms

Let P := k[X1, . . . , Xn], f ⊂ P be an ideal and

f =r⋂

i=1

qi

be an irredundant primary representation.For each i , let pi := √

qi be the associated prime and δ(i) := dim(qi ) be thedimension of the primary qi . Let d := max(δ(i)) = dim(f) and

M := i : pi is isolated.We will discuss throughout this chapter the following problems:2

primality test: given f ⊂ P decide whether f is prime;primarity test: given f ⊂ P decide whether f is primary and return the prime√

f;radicality test: given f ⊂ P decide whether f is radical;equidimensionality test: given f ⊂ P decide whether f is unmixed;primary decomposition: given f ⊂ P return an irredundant primary repre-

sentation f = ⋂ri=1 qi of f;

prime decomposition: given f ⊂ P return the set of all the associated primesof f;

equidimensional decomposition: given f ⊂ P return an irredundant equidi-mensional representation f = ⋂d

i=1 ui ;top-dimensional component: given f ⊂ P return its top-dimensional com-

ponent Top(f);radical computation: given f ⊂ P return its radical

√f;

minimal prime decomposition: given f ⊂ P return the irredundant primerepresentation

√f = ⋂

i∈M pi of√

f;equidimensional radical decomposition: given f ⊂ P return the irredundant

equidimensional representation√

f = ⋂di=1 vi of its radical.

2 Remember that, all through this book, we assume that the fields are infinite and perfect and that,if their characteristic is p = 0, it is possible to extract pth roots.

616 Gianni–Trager–Zacharias

35.2 Zero-dimensional Decomposition Algorithms

Recalling that we have tools which relate ideals – and their decompositions –in k[X1, . . . , Xn] with their extensions in k(X1, . . . , Xd)[Xd+1, . . . , Xn](Section 27.5), we begin our discussion by showing how Grobner’s Basissatzeallow us to solve them for a zero-dimensional ideal.

Let us therefore first assume that f ⊂ P is a zero-dimensional ideal and letus compute its Grobner basis G w.r.t. the lexicographial ordering induced byX1 < · · · < Xn .

As a consequence of Theorem 27.12.3 we know that

G = f1, . . . , fn ∪ hi j where

• f j ∈ k[X1, . . . , X j−1][X j ] \ k[X1, . . . , X j−1],

• T( f j ) = Xd jj , for some d j ∈ N,

• deg j ( f j ) = d j , for each j,• degl( f j ) < dl , for each l = j, for each j,• hi j ∈ k[X1, . . . , X j ] \ k[X1, . . . , X j−1],• degl(hi j ) < dl , for each l, i, j.

Then:

primality test: by the Nulldimensionale Primbasissatz (Theorem 34.1.2), f isprime iff

• G = f1, . . . , fn,• f1 is irreducible in k[X1],• π j−1( f j ) is irreducible in L j−1[X j ] where L j−1 is the field

L j−1 := k[X1, . . . , X j−1] \ ( f1, . . . , f j−1)

and π j−1 : k[X1, . . . , X j ] → L j−1[X j ] is the canonical projection;

primarity test: by the Nulldimensionale Primarbasissatz (Theorem 34.1.5), fis primary iff

• g1 := SQFR( f1) = √f1, the squarefree associate of f1, is irre-

ducible in k[X1],• g j := SQFR(ρ j−1( f j )) = √

ρ j−1( f j ) is irreducible in M j−1[X j ]where M j−1 is the field

M j−1 := k[X1, . . . , X j−1] \ (g1, . . . , g j−1),

and ρ j−1 : k[X1, . . . , X j ] → M j−1[X j ] is the canonical projection,• hi j ∈ (g1, . . . , g j−1), for each i, j ,

in which case√

f = (g1, . . . , gn);

35.2 Zero-dimensional Decomposition Algorithms 617

radicality test: a radicality test follows directly from Seidenberg’s Lemma.

Lemma 35.2.1. Let f ⊂ P be a zero-dimensional ideal. Let m ∈f ∩ k[X1] be a squarefree polynomial and let m = ∏

i mi be its fac-torization. Then

f = f + (m) =⋂

i

(f + (mi )) .

Proof. Clearly f ⊂ ⋂i (f + (mi )) .

Conversely, by Lagrange’s Chinese Remainder Theorem (Theo-rem 2.7.1) there are γi such that

1 =∑

i

γi

∏j =i

m j ;

therefore, for any f ∈ P , if f ∈ ⋂i (f + (mi )) then exist, for each i ,

fi ∈ f, ai ∈ P such that f = fi + ai mi and we have

f =∑

i

γi f∏j =i

m j =∑

i

fi

(γi

∏j =i

m j

)+

(∑i

γi ai

)m ∈ f.

Lemma 35.2.2 (Seidenberg Lemma). Let f ⊂ P be a zero-dimensional ideal and assume that, for each j , there is a squarefreepolynomial g j in f ∩ k[X j ]. Then f is squarefree.

Proof. The proof is by induction on n, the statement being trivial forn = 1.Let g1 ∈ f ∩ k[X1] be the squarefree polynomial whose existence isimplied by the assumption and let g1 = ∏

i hi be its factorization ink[X1].Then we have

f = f + (g1) =⋂

i

(f + (hi ))

and the claim is proved if we prove that each factor f + (hi ) is anintersection of primes. We can therefore assume wlog that g1 is irre-ducible.Let us then consider the field L1 := k[X1]/g1 and the projection

π1 : k[X1, X2, . . . , Xn] → L1[X2, . . . , Xn];since ker(π1) = (g1) ⊂ f we know (Lemma 27.5.4(12)) that f = fec.

Clearly, for each j , π1(g j ) is a squarefree polynomial in π1(f) ∩L1[X j ] so that, by induction, there is a decomposition π1(f) = fe =⋂

l pl into prime components.


For each l, let us consider a subset k1, . . . , kr ⊂ f such thatpl = (π1(k1), . . . , π1(kr )) and let pl := (k1, . . . , kr , g1) so that

• pel = π1(pl) = pl ,

• pcl = π−1

1 (pl) = pl ,• P/pl ∼= L1[X2, . . . , Xn]/pl (Lemma 27.5.4(11)) so that• pl is prime

and

f + (g1) = f = fec =⋂

l

pcl =

⋂l

pl .

Corollary 35.2.3 (Seidenberg). Let f ⊂ P be a zero-dimensionalideal and, for each i, let fi (Xi ) be the generator of the principal idealf ∩ k[Xi ] and gi be the squarefree associate of fi . Then√

f = f + (g1, . . . , gn).

Example 35.2.4. It is useful to note that the statement does not implythat (g1, . . . , gn) is

√f.

A trivial example can explain the more subtle relation among thetwo ideals. Let us consider

f = (X3 − X, (X − Y )2) ∈ k[X, Y ]

whose roots are

Z(f) = (1, 1), (0, 0), (−1,−1);the required polynomials are

f1 := X3− X, f2 := Y 6−2Y 4+Y 2, and g1 = X3− X, g2 = Y 3−Y.

Therefore (g1, g2) has the nine roots

(±1, ±1), (0,±1), (±1, 0), (0, 0)which are obtained by combining in all ways the coordinates of thethree elements of Z(f); each of these nine roots is double in ( f1, f2)

as are the roots of f.In conclusion the ideal (g1, . . . , gn) has spurious but single roots;

joining f and (g1, . . . , gn) has the effect of removing both the spuriousroots of (g1, . . . , gn) and the multiplicity of the roots of f.


Corollary 35.2.5. Let f ⊂ P be a zero-dimensional ideal and, foreach i, let fi (Xi ) be the generator of the principal ideal f ∩ k[Xi ].

Then, the following conditions are equivalent:

• f is squarefree,• for each i, fi is squarefree.

The interesting aspect of Seidenberg’s approach is that the computa-tion of the fi s just requires elementary linear algebra computation.This will be discussed in Section 35.7 and applied in Algo-rithm 35.7.2;

primary decomposition: an easy approach to obtaining the primary decom-position of f := ⋂r

i=1 qi is to compute the prime decomposition and,for each associate prime pi = √

qi , to deduce qi by means of repeat-edly computing the decreasing chain

a1 ⊇ a2 ⊇ · · · ⊇ ai ⊇ · · · ⊇ f

defined by

a1 := f + p j = p j and al+1 := f + p jal , for each l ≤ 1

until aρ+1 = aρ and then setting qi := aρ; this algorithm is justifiedby the following:

Proposition 35.2.6. Let f ⊂ P be a (not necessarily zero-dimensional) ideal and let f = ⋂r

i=1 qi be its irredundant primaryrepresentation. Let q j be a p j -primary isolated component of f andlet ρ j be the characteristic number of q j . If p j is maximal then, usingthe notation above, we have:

(1) for each σ ≥ ρ,√

f + pσj = p j ;

(2) writing b := ∏i = j qi , we have b ⊆ p j ;

(3) for each σ ≥ ρ, f + pσj = q j ;

(4) for each σ, aσ = f + pσj .

Proof.

(1) Clearly√

f + pσj ⊆ p j ; assume that for some prime p we have√

f + pσj ⊆ p; since pσ

j ⊆ f + pσj this implies

p j =√

pσj ⊆

√f + pσ

j ⊆ p;the maximality of p j allows us to conclude that p j = p andestablish the claim.


(2) Since q j is isolated, qi ⊆ pi ⊆ p j for all i = j , whence theclaim.

(3) The inclusion q j ⊇ f + pσj being trivial, let us prove the con-

verse: we have

bq j =∏

i

qi ⊆r⋂

i=1

qi ⊆ f + pσj ;

since b ⊆ p j , the required inclusion q j ⊆ f+pσj follows from

the fact that f + pσj is p j -primary.

(4) Since the claim holds for σ = 1, we can argue by induction:

aσ = f + p jaσ−1 = f + p j f + p jpσ−1j = f + pσ

j ;

prime decomposition: the prime decomposition can be performed by itera-tively decomposing the radical of each ideal

f j := f ∩ k[X1, . . . , X j ].

We first need the following

Lemma 35.2.7. Let a, b ⊂ P be ideals. Then:

(1)√

a + √b = √

a + b;

(2)√√

a + √b = √

a + b;

(3)√

a ∩ b = √a ∩ √

b = √ab.

Proof.

(1) Let d ∈√

a + √b; then there exist a ∈ a, b ∈ P, ρ, σ ∈ N

such that

dρ = a + b, bσ ∈ b;this implies 3

dρσ = (a + b)σ = a(

aσ−1 + · · · + σbσ−1)

+ bσ ∈ a + b.

The other inclusion follows from a + b ⊆ a + √b.

(2) Apply the formula above twice.(3) If c ∈ √

a ∩ b, then there exists ρ ∈ N such that cρ ∈ a ∩ b;therefore c ∈ √

a ∩ √b.

3 If 0 = p := char(k) | σ , more simply

dρσ = (a + b)σ = aσ + bσ ∈ a + b.


If c ∈ √a ∩ √

b then there exist ρ, σ ∈ N such that cρ ∈a, cσ ∈ b; therefore cρ+σ = cρcσ ∈ ab and c ∈ √

ab.

Since ab ⊆ a ∩ b we have√

ab ⊆ √a ∩ b.

Corollary 35.2.8. Let a, b, c ⊂ P be ideals. Then:√(a ∩ b) + c =

√(a + c) ∩ (b + c).

Proof. One has

√(a ∩ b) + c =

√√a ∩ b + √

c

=√√

ab + √c

= √ab + c

=√

(a + c) (b + c)

=√

(a + c) ∩ (b + c).

If we have already produced the irredundant maximal representation√f j = ⋂

l ml j , then√f j+1 =

√f j+1 + f j =

√f j+1 +

⋂l

ml j =⋂

l

√f j+1 + ml j ,

and our aim is to decompose the radical of each f j+1 + ml j .

We can in particular assume that ml j is generated by a Kroneckeradmissible sequence k1, . . . , k j which is, by the NulldimensionalePrimbasissatz (Theorem 34.1.2) its Grobner basis w.r.t. the lexico-graphical term ordering induced by X1 < · · · < X j ; therefore, theGrobner basis 4 Gl j of f j+1 + ml j w.r.t. the lexicographical termordering induced by X1 < · · · < X j < X j+1 has the shape 5

4 In fact, Gl j could be directly obtained from G by performing complete reduction on the setG ∪ k1, . . . , k j .

Since we are working by iteration, we can even assume that we have already computed theGrobner basis G′ of G∪k1, . . . , k j−1 and all we need to do is perform reduction of G′ modulok j and then add k j .

5 Since(f j+1 + ml j

) ∩ k[X1, . . . , X j ] = ml j we have

Gl j ∩ k[X1, . . . , X j ] = k1, . . . , k j .Let us now consider in Gl j a polynomial

g =D∑

t=0

ht (X1, . . . , X j )Xtj+1 ∈ k[X1, . . . , X j ][X j+1] \ k[X1, . . . , X j ]


(k1, . . . , k j , g) with

g ∈ k[X1, . . . , X j ][X j+1] \ k[X1, . . . , X j ]

and monic.Writing L j := k[X1, . . . , X j ]/ml j and

π j : k[X1, . . . , X j ][X j+1] → L j [X j+1]

for the canonical projection, we can effectively compute irreduciblemonic polynomials

gh ∈ k[X1, . . . , X j ][X j+1] \ k[X1, . . . , X j ]

such that π j (g) = ∏h π j (gh)eh ; if we write

mh := ml j + (gh) = (k1, . . . , k j , gh)

we clearly have the decomposition√

f j+1 + ml j = ⋂h mh into prime

components.radical computation: the radicality test based on Seidenberg’s Lemma has

the advantage that in case of failure,√

f is already directly availablesince, according to Corollary 35.2.3, one has√

f = f + (g1, . . . , gs).

35.3 The GTZ Scheme

Gianni, Trager and Zacharias proposed an effective scheme (the GTZ-scheme)which allows us to reduce the computation of the decomposition algorithmsfrom the generic case to the zero-dimensional case, whose solution we havealready discussed.

Let us assume that we are given an ideal f ⊂ P := k[X1, . . . , Xn].Corollary 27.11.9 allows us, from our knowledge of the Grobner basis of

f w.r.t. any term ordering, to compute the dimension d := dim(f) of f and amaximal set of independent variables for it. Up to a renumbering we can wlogassume that such a maximal set of independent variables is X1, . . . , Xd.

of minimal degree D := deg j+1(g).

Then such a polynomial is unique and monic: in fact, since ml j is maximal, if Lp(g) :=h D = 1 we would have a contradiction, since Lp(g) is invertible modulo ml j and there ish′ ∈ k[X1, . . . , X j ] such that Lp(g)h′ = 1 mod ml j so that

g′ := Can(h′g, ml j , <) = X Dj+1 +

D−1∑t=0

h′t (X1, . . . , X j )Xt

j+1 ∈ f j+1 + ml j

and T(g′) | T(g).

35.3 The GTZ Scheme 623

Our aim is to compute at least a partial primary decomposition of f. In orderto simplify the discussion let us fix a yet-unknown-to-us irredundant primaryrepresentation

f :=r⋂

i=1

qi

of f and let us assume, wlog, that the primaries are ordered so that, for a suitablevalue 1 ≤ s ≤ r,

X1, . . . , Xd is a maximal set of independent variables for qi ⇐⇒ i ≤ s.

If we therefore consider the ring k(X1, . . . , Xd)[Xd+1, . . . , Xn], which is thequotient ring of k[X1, . . . , Xn] w.r.t. the multiplicative system

k[X1, . . . , Xd ] \ 0and the canonical homomorphism

φ : R : = k[X1, . . . , Xd ][Xd+1, . . . , Xn]

→ k(X1, . . . , Xd)[Xd+1, . . . , Xn] =: S,

all the notations and results of Section 27.5 are available. In particular, fromCorollary 27.5.19 we obtain

Corollary 35.3.1. With the notation above, we have

• fe = ⋂si=1 qe

i is an irredundant primary representation;• fec = ⋂s

i=1 qi is an irredundant primary representation;• fe is zero-dimensional;• fec is unmixed.

Proof. We have qi ∩ k[X1, . . . , Xd ] \ 0 = ∅ iff dim(qi ) ≥ d, andX1, . . . , Xd is contained in a maximal set of independent variables for qi .

Since dim(qi ) ≤ dim(f) = d we have

qi ∩ k[X1, . . . , Xd ] \ 0 = ∅ ⇐⇒ i ≤ s.

If f is prime the relation between f = fec and fe is already discussed inTheorem 34.3.2.

From there we learn that, in this case, if we take

• the reduced Grobner basis

f1, . . . , fn−d ⊂ k(X1, . . . , Xd)[Xd+1, . . . , Xn]

of fe w.r.t. the lexicographical ordering induced by Xd+1 < · · · < Xn ,• for each i , the lcm qi ∈ k[X1, . . . , Xd ] of the denominators of the coeffi-

cients of fi ,


• pi := qi fi ∈ k[X1, . . . , Xn], and• F := ∏

i qi ,

then fec = (p1, . . . , pn−d) : F. We show now that this result can even bestrengthened.

Proposition 35.3.2. Let G be the reduced Grobner basis of f with any blockordering inducing X1, . . . , Xd < Xd+1, . . . , Xn, and let 6

s := SQFR

(∏g∈G

lc(Prim(g))

)∈ k[X1, . . . , Xd ].

Then:

(1) G is a (not necessarily reduced) Grobner basis of fe;(2) for any s ∈ k[X1, . . . , Xd ] which is a multiple of s we have

fec = f : s∞.

Proof.

(1) This is the statement of Corollary 26.2.3.(2) The inclusion f : s∞ ⊆ fec holds for any value s ∈ k[X1, . . . , Xd ]

since, if h ∈ f : s∞, there exists ρ ∈ N such that sρh ∈ f so that

h = (sρh)/sρ ∈ fe ∩ k[X1, . . . , Xn] = fec.

To prove the converse inclusion, let us remark that if h ∈ k[X1, . . . ,

Xn] is such that h ∈ fec, then its normal form in k(X1, . . . ,

Xd)[Xd+1, . . . , Xn] w.r.t. G is zero.Let us consider a rewriting step h → h′; there are a polynomial p ∈k[X1, . . . , Xd ], a term t ∈ k[Xd+1, . . . , Xn] and an element g ∈ Gsuch that

h − h′ = lc(g)−1 ptg and T(h′) < T(h).

Since h′ ∈ fec we can by induction assume that there exists ρ ∈ N suchthat sρh′ ∈ f; this allows us to deduce that

sρ+1h = s(sρh′) + sρ

(lc(g)−1 pts

)g ∈ f,

and h ∈ f : s∞.

6 Each polynomial Prim(g) is considered to be an element of

k[X1, . . . , Xd ][Xd+1, . . . , Xn ].


Lemma 35.3.3. Let s ∈ k[X1, . . . , Xd ] be such that fec = f : s∞ and letσ ∈ N be such that f : sσ = f : s∞.

Then f = fec ∩ (f + (sσ )) .

Proof. One inclusion being trivial, let us consider a polynomial f ∈ fec ∩(f + (sσ )) . Then there are g ∈ f and r ∈ k[X1, . . . , Xn] such that f = g +rsσ ∈ fec. Therefore rsσ ∈ fec = f : sσ , rs2σ ∈ f, r ∈ f : s∞ = f : sσ , so thatrsσ ∈ f and f ∈ f.

Theorem 35.3.4 (Gianni–Trager–Zacharias). Given a d-dimensional idealf ⊂ k[X1, . . . , Xn] it is possible to explicitly compute

• a set of d variables, which, up to a renumbering, we can assume to beX1, . . . , Xd,

• a polynomial t ∈ k[X1, . . . , Xd ],

such that if f := ⋂ri=1 qi is an irredundant primary representation of f and,

wlog, the primaries are ordered so that

• X1, . . . , Xd is a maximal set of independent variables for qi ⇐⇒ i ≤ s

the following hold:

(1) X1, . . . , Xd is a maximal set of independent variables for f;(2) fe is zero-dimensional;(3) fe = ⋂s

i=1 qei is an irredundant primary representation;

(4) fec is unmixed;(5) if fe = ⋂s

i=1 Qi is an irredundant primary representation, then fec =⋂si=1 Qc

i is an irredundant primary representation;(6) up to a renumbering, for each i ≤ s we have qi = Qc

i ;(7) fec = ⋂s

i=1 qi is an irredundant primary representation;(8) fec = f : t;(9) f = fec ∩ (f + (t));

(10) f + (t) f.

Corollary 35.3.5. Given a d-dimensional ideal f ⊂ k[X1, . . . , Xn] and as-suming wlog that the variables are ordered so that X1, . . . , Xd is a maxi-mal set of independent variables for f, it is possible to explicitly compute twoideals

f0 ⊂ k(X1, . . . , Xd)[Xd+1, . . . , Xn] and f+ ⊂ k[X1, . . . , Xn]


such that

(1) f0 is zero-dimensional,(2) f0 = fe,

(3) f = fc0 ∩ f+,

(4) there exists t ∈ k[X1, . . . , Xd ] such that f+ = f + (t).

Definition 35.3.6. Let f ⊂ k[X1, . . . , Xn] be a d-dimensional ideal.A GTZ-decomposition of f is the assignment of two ideals

f0 ⊂ k(X1, . . . , Xd)[Xd+1, . . . , Xn] and f+ ⊂ k[X1, . . . , Xn]

satisfying the conditions above.

The effect of the GTZ-decomposition is therefore the production of a partialdecomposition

f = fc0 ∩ f+ = fec ∩ (f + (t)) ,

where

• since f0 = fe is zero-dimensional the techniques discussed in Section 35.2can be applied to give the decomposition of f0, from which that of fec canbe obtained simply by contraction;

• either

• f+ = (1) and we are through, or• f+ is zero-dimensional and its decomposition can be directly computed,

or• the decomposition of f+ can be computed by iteratively computing a

GTZ-decomposition of it.

The GTZ-scheme consists of iteratively computing GTZ-decompositions

a(i) = a(i)c0 ∩ a(i)+, where a(i) :=

f if i = 1,a(i − 1)+ if i > 1,

until a(i)+ is either (1) or zero-dimensional, thus reducing the decompositionalgorithm computation to the zero-dimensional case.

Termination of the GTZ-scheme is granted by the following argument: sincet ∈ k[X1, . . . , Xd ] and f ∩ k[X1, . . . , Xd ] = ∅, then t /∈ f and f + (t) f; as aconsequence, any chain

a(1) ⊂ a(2) ⊂ · · · ⊂ a(i) ⊂ · · ·where

• a(1) = f,• for each i , a(i) is neither (1) nor zero-dimensional, and


• for each i , a(i) has a GTZ-decomposition

a(i) = a(i)c0 ∩ a(i)+, where a(i)+ = a(i + 1),

is necessarily finite.

Remark 35.3.7. It is best to point out immediately a weakness of this approach.If

• f := ⋂ri=1 qi is an irredundant primary representation,

• the primaries are ordered so that X1, . . . , Xd is a maximal set of indepen-dent variables for qi iff i ≤ s,

• f0 ⊂ k(X1, . . . , Xd)[Xd+1, . . . , Xn] and f+ ⊂ k[X1, . . . , Xn] are a GTZ-decomposition, so that

• f0 = fe is zero-dimensional,• f = fc

0 ∩ f+,

• there exists t ∈ k[X1, . . . , Xd ] such that f+ = f + (t),

then f+ = f + (t) has the decomposition

f+ = f + (t) =r⋂

i=1

(qi + (t)) .

In such a decomposition, for each i ≤ s, we have

t ∈ √qi

since X1, . . . , Xd is a maximal set of independent variables for both qi and√qi , while t ∈ k[X1, . . . , Xd ].As a consequence

√qi + (t)

√qi , for each i ≤ s, and each primary

component q of (qi + (t)) is embedded into√

qi .Of course it could happen that each component qi in the decomposition f :=⋂ri=1 qi already contains an embedded primary q j such that

√q j = √

qi + (t),in which case the primary decomposition of f is preserved. But in general, thiswill not happen and any application of the GTZ-decomposition algorithm hasthe negative effect of introducing spurious components belonging to primeswhich are not associated to f.

Such spurious primaries of course must be removed; the only way I knowof doing so is to test, for any instance in which

√q j ⊂ √

qi , whether alsoq j ⊂ qi .

Example 35.3.8. To illustrate both the GTZ-scheme and its weakness let usconsider the ideal

f = a(1) = (XY Z , Y Z2, T ) = (Z , T ) ∩ (Y, T ) ∩ (X, Z2, T ) ⊂ k[X, Y, Z , T ]


which has dimension 2 and two maximal sets of independent variables, namelyX, Y and X, Z.

Choosing X, Z we obtain

a(1)0 = (Y, Y, T ) = (Y, T ), t = X Z2, a(1)+ = (XY Z , Y Z2, T, X Z2);note that we have the decomposition

a(1)+ = (Z , T, X Z2) ∩ (Y, T, X Z2) ∩ (X, Z2, T, X Z2)

= (Z , T ) ∩ (X, Y, T ) ∩ (Y, Z2, T ) ∩ (X, Z2, T )

with the spurious components (X, Y, T ) and (Y, Z2, T ).The ideal a(2) := a(1)+ = (XY Z , X Z2, Y Z2, T ) has dimension 2 and

a single maximal set of independent variables, namely X, Y . We thereforeobtain

a(2)0 = (Z , Z2, Z2, T ) = (Z , T ),

t = XY,

a(2)+ = (XY Z , X Z2, Y Z2, T, XY ) = (XY, X Z2, Y Z2, T );a(2)+ has the decomposition

a(2)+ = (Z , T, XY ) ∩ (X, Y, T, XY ) ∩ (Y, Z2, T, XY ) ∩ (X, Z2, T, XY )

= (X, Z , T ) ∩ (Y, Z , T ) ∩ (X, Y, T ) ∩ (Y, Z2, T ) ∩ (X, Z2, T )

= (X, Y, T ) ∩ (Y, Z2, T ) ∩ (X, Z2, T ).

The ideal a(3) := a(2)+ = (XY, X Z2, Y Z2, T ) has dimension 1 andthree maximal sets of independent variables, each corresponding to each1-dimensional component, namely

• X, related to (Y, Z2, T ),• Y , related to (X, Z2, T ),• Z, related to (X, Y, T ).

Choosing X we obtain

a(3)0 = (Y, Z2, Y Z2, T ) = (Y, Z2, T ),

t = X,

a(3)+ = (XY, X Z2, Y Z2, T, X) = (X, Y Z2, T ),

and

a(3)+ = (X, Y, T, X) ∩ (Y, Z2, T, X) ∩ (X, Z2, T, X)

= (X, Y, T ) ∩ (X, Y, Z2, T ) ∩ (X, Z2, T )

= (X, Y, T ) ∩ (X, Z2, T ).


For the 1-dimensional ideal a(4) := a(3)+ = (X, Y Z2, T ) we choose theset Y and we obtain

a(4)0 = (X, Z2, T ), t = Y, a(4)+ = (X, Y Z2, T, Y ) = (X, Y, T ).

Since both ideals are zero-dimensional we therefore obtain the primarydecompositions

a(4) = (X, Z2, T ) ∩ (X, Y, T ),

a(3) = (X, Z2, T ) ∩ (X, Y, T ) ∩ (Y, Z2, T ),

a(2) = (X, Z2, T ) ∩ (X, Y, T ) ∩ (Y, Z2, T ) ∩ (Z , T ),

f = a(1) = (X, Z2, T ) ∩ (X, Y, T) ∩ (Y, Z2, T) ∩ (Z , T ) ∩ (Y, T ).

where we have marked in bold the spurious components.

Remark 35.3.9. If one is not interested in multiplicity and even embeddedcomponents (as we already noted in Remark 27.13.5) and intends to study onlythe decomposition of

√f one can easily get rid of spurious components, but at

the price of also removing the true embedded ones, using a slight modificationof the GTZ-scheme, which was proposed by Alonso and Raimondo, adaptinga construction by Giusti and Heintz.

In this modification each GTZ-decomposition f = fc0 ∩ f+ is replaced by

a decomposition (ARGH-decomposition) of the radical of√

f through the as-signment together of

f0 ∈ k(X1, . . . , Xd)[Xd+1, . . . , Xn],

and another ideal f√ ∈ k[X1, . . . , Xn] which satisfies the formula

√f =

√f0

c ∩√

f√.

and which can be computed via

f√ := f :(fc0

)∞,

as we will prove in the proposition below. A more efficient way of computingf√ by means of Moller’s algorithm is discussed in Algorithm 35.7.1.

The GTZ-scheme iteratively computes GTZ-decompositions

a(i) = a(i)c0 ∩ a(i)+ = a(i)ec ∩ a(i)+

where

a(i) :=

f if i = 1,a(i − 1)+ if i > 1


until a(i)+ is either (1) or zero-dimensional; similarly, the ARGH-schemeiteratively computes the ARGH-decompositions√

a(i) =√

a(i)c0 ∩

√a(i)√ =

√a(i)ec ∩

√a(i)√

where

a(i) :=

f if i = 1a(i − 1)√ if i > 1

until a(i)√ is (1).

Proposition 35.3.10. Let us, as usual, assume that we have a d-dimensionalideal f ⊂ k[X1, . . . , Xn], that the variables are ordered so that X1, . . . , Xdare a maximal set of independent variables for f, and that the irredundantprimary representation f := ⋂r

i=1 qi is wlog ordered so that

• X1, . . . , Xd is a maximal set of independent variables for qi ⇐⇒ i ≤ s• there exists j ≤ s :

√qi ⊃ √

q j ⇐⇒ s < i ≤ u.

Then, writing

f0 := fe ⊂ k(X1, . . . , Xd)[Xd+1, . . . , Xn]

f√ := f :(fc0

)∞ ⊂ k[X1, . . . , Xn],

we have

(1)√

f0 = ⋂si=1

√qe

i is an irredundant primary representation,(2)

√fec = √

fc0 = ⋂s

i=1√

qi is an irredundant primary representation,

(3)√

f√ = ⋂ri=u+1

√qi is an irredundant primary representation,

(4)√

f = √fc0 ∩

√f√ = ⋂s

i=1√

qi ∩ ⋂ri=u+1

√qi .

Proof. The equalities

√f =

s⋂i=1

√qi ∩

r⋂i=u+1

√qi , f0 =

s⋂i=1

qei ,

fec =s⋂

i=1qi ,

√fc0 =

s⋂i=1

√qi

being an obvious consequence of the previous results, in order to complete the

proof we just need to prove that√

f√ = ⋂ri=u+1

√qi : we have that

f√ = f :(fc0

)∞ =r⋂

i=1

qi :(fec)∞

,

35.4 Higher-dimensional Decomposition Algorithms 631

and, by Corollary 27.2.12

qi :(fec)∞ :=

qi ⇐⇒ fec ⊆ √

qi ⇐⇒ u < i ≤ r(1) ⇐⇒ fec ⊆ √

qi ⇐⇒ 1 ≤ i ≤ u.

35.4 Higher-dimensional Decomposition Algorithms

We can now discuss how the decomposition algorithms can be performed bymeans of GTZ-decompositions;

primality test: Primality/primariety tests of a higher-dimensional ideal canbe reduced to the zero-dimensional case.In fact, if we are given a d-dimensional ideal f ⊂ k[X1, . . . , Xn],we can consider a maximal set of independent variables, sayX1, . . . , Xd, the ring k(X1, . . . , Xd)[Xd+1, . . . , Xn] and the canon-ical homomorphism

φ : R := k[X1, . . . , Xd ][Xd+1, . . . , Xn]

→ k(X1, . . . , Xd)[Xd+1, . . . , Xn];

using the notations of Section 27.5, the assumptionf∩k[X1, . . . , Xd ] = 0 implies that the results of Remark 27.5.18(3)and (4) hold so that:

Corollary 35.4.1. Under these assumptions we have:

• f is prime iff fe is prime and f = fec;• f is primary iff fe is primary and f = fec.

So given a d-dimensional ideal f ⊂ k[X1, . . . , Xn] one computesa maximal set of independent variables, say X1, . . . , Xd, and thereduced Grobner basis G of f with any term ordering < suchthat

Xi > t, for each term t ∈ k[X1, . . . , Xi−1], and each i > d;

our knowledge of G allows us to deduce both (see Proposi-tion 35.3.2) the Grobner basis G ′ of fe w.r.t. the lexicographical


ordering such that Xd+1 < · · · < Xn , and the polynomial s :=SQFR

(∏g∈G lc(Prim(g))

)which satisfies the formula fec = f : s∞.

Therefore f is prime iff

• G ′ satisfies the Nulldimensionale Primbasissatz (Theorem 34.1.2),giving the primality of fe, and

• f = f : s∞ granting the relation f = fec.

primariety test: The computation outlined above allows us also to performa primariety test; we just use the Grobner basis G ′ to test whether itsatisfies the Nulldimensional Primarbasissatz (Theorem 34.1.5), giv-ing primariety of fe. At the same time the primbasis of

√fe is also

obtained; therefore, if, equivalently, f = f : s∞ and f is primary, itsassociated prime is obtained by computing

√fec.

radicality test: If we perform here the GTZ-scheme computing the GTZ-decomposition f = fc

0 ∩ f+, we can iteratively reduce the problem tothe radicality of fc

0 and f+. But essentially we are unable to do betterthan computing

√f and checking whether f = √

f. We have in fact:

• if f0 = fe is not radical, then f also is not;• if f0 = fe is radical and f = fec, then f is radical;• if f0 = fe is radical but f = fec, then f is radical iff f = fec ∩ √

f+.

There is no improvement if we use the ARGH-scheme, where thesame test applies verbatim if we replace f+ with f√.

equidimensionality test: We can perform here the GTZ-scheme computingiteratively the GTZ-decompositions

a(i) = a(i)c0 ∩ a(i)+, where a(i) :=

f if i = 1,a(i − 1)+ if i > 1,

until either a(i)+ = (1) or dim(a(i)+) < dim(a(i)).We have therefore obtained a decomposition

f =(

ι⋂i=1

a(i)c0

) ⋂a(ι)+

where, for each i ≤ ι, dim(f) = dim(a(i)c0) > dim(a(ι)+). Therefore

f is equidimensional ⇐⇒ a(ι)+ = (1).primary decomposition: Again, we reduce this algorithm to prime decom-

position and to the application of Proposition 35.2.6.prime decomposition: Let us perform again the GTZ-scheme computing

iteratively the GTZ-decompositions

a(i) = a(i)c0 ∩ a(i)+, where a(i) :=

f if i = 1,a(i − 1)+ if i > 1,

35.4 Higher-dimensional Decomposition Algorithms 633

until either a(i)+ = (1) or dim(a(i)+) = 0, obtaining the decompo-sition

f =(

ι⋂i=1

a(i)c0

) ⋂a(ι)+.

The prime decomposition is then obtained by performing it on a(ι)+and on each component a(i)0; for each prime p associated to a(i)0,the algorithm will then return pc.

equidimensional decomposition: What apparently is the obvious solution,that is

• performing the GTZ-scheme, until either a(i)+ = (1) ordim(a(i)+) = 0, thus obtaining a decomposition f =(⋂ι

i=1 a(i)c0

) ∩ a(ι)+ where each a(i)c0 is unmixed;

• setting u0 := a(ι)+ and, for each δ, 0 < δ ≤ d, uδ :=⋂i :dim(a(i))=δ a(i)c

0• and checking whether uδ ⊇ ⋂

i =δ ui ,

fails because of spurious components.For instance, in the example discussed above, this approach wouldgive us the wrong component

u1 = (X, Z2, T ) ∩ (X, Y, T) ∩ (Y, Z2, T)

while the correct answer is u1 = (X, Z2, T ).

The algorithm described, therefore, gives only a not necessarily irre-dundant decomposition.

top-dimensional component: In this case, it is instead sufficient to per-form the GTZ-scheme until either a(i)+ = (1) or dim(a(i)+) <

dim(f), thus obtaining a decomposition f = (⋂ιi=1 a(i)c

0

) ⋂a(ι)+;

then

Top(f) =ι⋂

i=1

a(i)c0.

radical computation: We perform here the ARGH-scheme computing itera-tively the ARGH-decompositions

√a(i) =

√a(i)c

0 ∩√

a(i)√, where a(i) :=

f if i = 1,a(i − 1)√ if i > 1,

until a(i)√ is (1).


For each component a(i)c0, the radical

√a(i)0 is computed using Sei-

denberg’s Algorithm (Corollary 35.2.3) and its contraction returns√a(i)c

0 = √a(i)0

c.

minimal prime decomposition: Again we perform here the ARGH-scheme,computing iteratively the ARGH-decompositions

√a(i) =

√a(i)c

0 ∩√

a(i)√, where a(i) :=

f if i = 1,a(i − 1)√ if i > 1,

until a(i)√ is (1) and we perform primary decomposition on eacha(i)c

0.equidimensional radical decomposition: When the ARGH-scheme has

been performed, one has just to combine those components a(i)c0

which have the same dimension.

35.5 Decomposition Algorithms for Allgemeine Ideals

Let us now consider how the decomposition algorithms can be improved if theideal f ⊂ P is in allgemeine position.

35.5.1 Zero-dimensional Allgemeine Ideals

We will first discuss the case in which f is zero-dimensional; we again assumethat f is given by means of its Grobner basis G w.r.t. the lexicographical order-ing induced by X1 < · · · < Xn , where

G := f1, . . . , fn ∪ hi j and

• f j ∈ k[X1, . . . , X j−1][X j ] \ k[X1, . . . , X j−1],


• deg j ( f j ) = d j , for each j,• degl( f j ) < dl , for each l = j and each j,• hi j ∈ k[X1, . . . , X j ] \ k[X1, . . . , X j−1].• degl(hi j ) < dl , for each l, i, j .

primality test: The ideal f is prime and in allgemeine position iff

• G = f1, . . . , fn,• f1 is irreducible in k[X1],• for each j > 1, f j = X j − g j (X1) for a suitable polynomial

g j (X1) ∈ k[X1].

35.5 Decomposition Algorithms for Allgemeine Ideals 635

primariety test: The ideal f is primary and in allgemeine position iff

• g1(X1) := SQFR( f1) = √f1, the squarefree associate of f1(X1),

is irreducible in k[X1],• for each j > 2, setting,7 in k[X1, X j ],

f j (X1, g2(X1), . . . , g j−1(X1), X j) =: Xd jj −d j g j (X1)Xdi −1

j +· · · ,one has

f j (X1, g2(X1), . . . , g j−1(X1), X j )

≡ (X j − g j (X1)

)d j (mod g1),

• for each i, j, hi j (X1, g2(X1), . . . , g j (X1)) is multiple of g1(X1),

in which case√f = (g1, X2 − g2(X1), . . . , Xn − gn(Xn)).

Example 35.5.1. Let us consider the ideal f ⊂ k[X1, X2, X3] whoseGrobner basis w.r.t. the lexicographical ordering induced by X1 <

X2 < X3 is

G := f1, f2, f3, h13where

f1 := X61 + 3X4

1 + 3X21 + 1,

f2 := X32 − 3X5

1 X22 − 30X4

1 X2

− 45X21 X2 − 18X2 + 21X5

1 + 35X31 + 15X1,

f3 := X23 − 30X4

1 X3 − 48X21 X3 − 20X3

+ 84X41 X2

2 + 140X21 X2

2 + 60X22 − 288X5

1 X2

− 504X31 X2 − 224X1 X2 − 198X4

1 − 360X21 − 165,

h13 := X2 X3 − X51 X3 + 6X5

1 X22 + 6X3

1 X22 + 45X4

1 X2 + 2X1 X22

+ 72X21 X2 − 28X5

1 − 48X31 − 21X1 + 30X2;

7 This formulation applies unless 0 = p := char(k) | d j .It is easy to reformulate it in the case char(k) = p = 0; in this setting there is e j ≥ 0 such

that

f j (X1, g2(X1), . . . , g j−1(X1), X j ) =:

(X

d jj − d j g j (X1)X

di −1j + · · ·

)pe j

,

with gcd(p, d j ) = 1, and one must require

f j (X1, g2(X1), . . . , g j−1(X1), X j ) ≡ (X j − g j (X1)

)d j pe j

(mod g1).


then we have

g1 = X21 + 1,

f2(X1, X2) ≡ X32 − 3X1 X2

2 − 3X2 + X1 (mod g1)

g2 = X1,

f3(X1, X1, X3) ≡ X23 − 2X3 + 1 (mod g1)

g3 = 1,

h13(X1, X1, 1) ≡ 0 (mod g1),

thus giving√

f = (X21 + 1, X2 − X1, X3 − 1).

radicality test: The ideal f is radical and in allgemeine position iff

• G = f1, . . . , fn,• f1 is squarefree in k[X1],• for each j > 1, f j = X j − g j (X1) for a suitable polynomial

g j (X1) ∈ k[X1].

primary decomposition: If f is in allgemeine position, then writing

Z(f) = a1, . . . , ar , where a j = (a j1, . . . , a jn),

we have a j1 = al1, for each j = l, so that one obtains the primarydecomposition of f by computing the factorization f1 = ∑r

i=1 geii of

the generator f1 of f ∩ k[X1] and setting f = ⋂ri=1(f + (gei

i )).prime decomposition: In order to obtain the prime decomposition, one apply

directly, for each i , the radical computation algorithm below to theGrobner basis

G ′ := Rem(g, gi ) : g ∈ G ∪ gi

of f + (gi ) where, as above, f1 = ∑ri=1 gei

i is the factorization of thegenerator f1 of f ∩ k[X1].

radical computation: If f is in allgemeine position, one has√f = (g1, X2 − g2(X1), . . . , Xn − gn(Xn)),

where

• g1(X1) := SQFR( f1) = √f1,

• for each i > 1, gi (X1) is obtained from

Fi (X1, Xi ) ∈ (k[X1]/g1(X1)) [Xi ]


by dividing its coefficient of Xdi −1i by −di , where 8

Fi (X1, Xi ) := fi (X1, g2(X1), . . . , gi−1(X1), Xi )

= Xdii − di gi (X1)Xdi −1

i + · · ·or by simply performing division 9 of Fi by d−1

i F ′i in

(k[X1]/g1(X1)) [Xi ]

and using the formula

Rem(Fi (Xi ), d−1i F ′

i (X1)) = Xi − gi (Xi ).

Remark 35.5.2. Concerning the primality (respectively primariety, radicality)test, we must clarify that if the test fails because some f j , j > 1 does not havethe required shape the reason can be that f either is not prime (respectivelyprimary, radical) or that it is not in allgemeine position.10

The reasonable approach is to ‘locally’ change the frame of coordinates 11

before applying the general approach and testing whether f j is irreducible(resp. power of irreducible, squarefree) in L j−1[X j ] where L j−1 is the properring.

35.5.2 Higher-dimensional Allgemeine Ideals

Let us now move the discussion to the general case of a d-dimensional ideal f

in allgemeine position.The big advantage of the assumption that f, and so each of its components, is

in allgemeine position is that each δ-dimensional component has X1, . . . , Xδas a maximal set of independent variables.

Therefore, both in the GTZ- and in the ARGH-scheme we have

• dim(a(i + 1)) < dim(a(i)), for each i ,• a(i)ec is the intersection of all the primary components whose dimension is

dim(a(i)).

This gives an obvious advantage in the algorithms;

8 This formula also applies unless 0 = p := char(k) | di .It is easy to reformulate it in the case char(k) = p = 0; in this setting Fi is the polynomial

such that, for a suitable ei ≥ 0

Fi (X1, Xi )pei = fi (X1, g2(X1), . . . , gi−1(X1), Xi ),

Fi (X1, Xi ) = Xdii − di gi (X1)X

di −1i + · · ·

and gcd(p, di ) = 1.9 There is no need of Duval techniques in this trivial context.

10 This simply means that X1 is not an allgemeine coordinate of f.11 We will discuss ideas of this kind in the next section.


primality test: The algorithm is the same as before: testing the primality offe and the equality f = fec. The allgemeine position allows us to usea better algorithm for testing primality of fe.

primariety test: What we said for the primality test, applies here verb-atim.

radicality test: There is no advantage in being f in allgemeine position ex-cept that in testing each GTZ-(respectively ARGH-)component a(i)+(respectively a(i)√) we can apply the easiest Allgemeine Nulldimen-sionale Basissatz (Theorem 34.2.1).

equidimensionality test: The advantage provided by the allgemeine positionis now very effective: f is equidimensional iff f = fec.

primary decomposition: Once the GTZ-scheme is applied, primary decom-position is reduced to the factorization, for each i , of the polynomialgenerating

a(i)e ∩ k(X1, . . . , Xδ)[Xδ+1], δ := dim(a(i)).

prime decomposition: Once the GTZ-scheme is applied, prime decomposi-tion is reduced, for each i , to the reduction of the Grobner basis ofeach a(i)e by any irreducible component of the polynomial generat-ing

a(i)e ∩ k(X1, . . . , Xδ)[Xδ+1], δ := dim(a(i)).

In both the prime and the primary decomposition, the GTZ-schemeintroduces spurious components and the ARGH-scheme gets rid ofembedded components. If the ARGH-scheme is applied, such em-bedded components can be recovered by repeatedly applying thefollowing.

Lemma 35.5.3. Let f ⊂ k[X1, . . . , Xn] be an ideal and let us assumewlog that the irredundant primary representation f := ⋂r

i=1 qi isordered so that

qi is embedded ⇐⇒ i > ε

and let F := ⋂εi=1 qi . Then

f : F =r⋂

i=ε+1

Qi and√

Qi = √qi , for each i, ε < i ≤ r.


Proof. In fact

qi : F is

(1) ⇐⇒ F ⊆ qi ⇐⇒ 1 ≤ i ≤ ε,

pi − primary ⇐⇒ F ⊆ qi ⇐⇒ ε < i ≤ r

so that setting, for each i : ε < i ≤ r, Qi := qi : F ⊆ qi we have√Qi = √

qi and

f : F =r⋂

i=1

(qi : F) =r⋂

i=ε+1

Qi .

Note that primary-decomposing f1 := f : F is useless since Qi =qi ; one should therefore just prime-decompose f1 := f : F and thenrecover qi by means of Proposition 35.2.6.Note also that f1 can itself have some embedded components and sothis approach must be iterated as many times as the maximal lengthof a chain of associated primes.

equidimensional decomposition: Applying the GTZ- and ARGH-schemes,each component is equidimensional and the components have differ-ent dimensions. However, in the GTZ-scheme such components alsohave spurious components and in the ARGH-scheme the embeddedcomponents are lost. In the ARGH-scheme however they can be re-covered by applying Lemma 35.5.3.

top-dimensional component: The solution is trivial since Top(f) = fec.radical computation: Perform the ARGH-scheme and, for each component

a(i)0 ⊂ k(X1, . . . , Xδ)[Xδ+1, . . . , Xn],

which is necessarily returned by an allgemeine basis

G = f1, . . . , fn−δ, fi ∈ k(X1, . . . , Xδ)[Xδ+1, . . . , Xδ+i ],

return√

a(i)0 = (h1, q2 Xδ+2 − p2, . . . , qn−δ Xn − pn−δ), where (seeCorollary 34.3.5)

• g1(Xδ+1) := SQFR( f1) = √f1 ∈ k(X1, . . . , Xδ)[Xδ+1] and

• h1(X1, . . . , Xδ, Xδ+1) ∈ k[X1, . . . , Xδ][Xδ+1] is monic and asso-ciated to g1,

• for each i > 1,

qi ∈ k[X1, . . . , Xδ] and pi ∈ k[X1, . . . , Xδ][Xδ+1]


are obtained from 12

Fi (X1, . . . , Xδ, Xδ+1, Xδ+i ) ∈ R

by dividing its coefficient of Xdi −1δ+i by −di , where

R := (k(X1, . . . , Xδ)[Xδ+1]/h1(Xδ+1)

)[Xδ+i ]

and

Fi (X1, . . . , Xδ, Xδ+1, Xδ+i )

:= fi

(Xδ+1,

p2(Xδ+1)

q2, . . . ,

pi−1(Xδ+1)

qi−1, Xδ+i

)

= Xdiδ+i − di

pi (Xδ+1)

qiXdi −1

δ+i + · · · ,

or by simply performing division of Fi by d−1i F ′

i in R using theformula

Rem(Fi (Xδ+i ), d−1i F ′

i (Xδ+i )) = Xi − pi (Xδ+1)/qi .

minimal prime decomposition: The prime decomposition of the ARGH-components now reduces to simply univariate factorization.

equidimensional radical decomposition: Consists just of the collection ofthe ARGH-components.

35.6 Sparse Change of Coordinates

The strong improvement of the decomposition algorithms which is givenby the assumption of allgemeine position is tantalizing; however, it isclear that performing a generic change of coordinates is a nonsensical ap-proach: the improvement granted by the allgemeine position is completelylost since all the computations will need to be performed over fully densepolynomials.

This suggests investigating approaches which preserve sparsity as much aspossibile while giving the strong effect of genericity.

12 This formula also applies unless 0 = p := char(k) | di .When char(k) = p = 0, Fi is the polynomial such that, for a suitable ei ≥ 0

Fi (X1, . . . , Xδ, Xδ+1, Xδ+i )pei = fi

(Xδ+1, . . . ,

pl (Xδ+1)

ql, . . . , Xδ+i

),

Fi (X1, . . . , Xδ, Xδ+1, Xδ+i ) = Xdiδ+i − di

pi (Xδ+1)

qiX

di −1δ+i + · · ·

and gcd(p, di ) = 1.

35.6 Sparse Change of Coordinates 641

We will discuss here two such approaches, both variations of the PrimitiveElement Theorem:

• the first approach, suggested by Gianni, is mainly aimed at improving theprimality (primariety, radicality) tests on a zero-dimensional ideal f ⊂P = k[X1, . . . , Xn] and suggests the repeated performance on P of ‘local’changes of coordinates

L(X j ) =

X j if j = 1,X1 + cXi if j = 1,

which introduce density only on the polynomial subring k[X1, Xi ] and havethe generic effect that L(x1) is a primitive element in

k[x1, xi ] = k[X1, Xi ]/g, where g := k[X1, Xi ] ∩ f.

The net effect is that such repeated ‘local’ changes of coordinates reduceprimality (respectively radicality) tests to successive univariate polynomialirreducibility (respectively squarefree) tests, and radical computation to re-peated gcd computations;

• the other approach was proposed by Giusti and Heintz and is the core ideabehind the ARGH-scheme.Let f ⊂ P = k[X1, . . . , Xn] be a d-dimensional ideal for whichwlog X1, . . . , Xd is a maximal set of independent variables and letY := Xd+1 + ∑n

i=d+2 ci Xi . In the generic case Y is a primitive element 13

for each d-dimensional component g of f for which X1, . . . , Xd is a max-imal set of independent variables; moreover Giusti and Heintz proved that,if we set g(Y ) := f ∩ k[X1, . . . , Xd , Y ], g(Y ) /∈ p for each isolated prime p

of f, so that √f√ =

√f :

(fc0

)∞ =√

f : g(Y )∞,

thus making strongly effective the ARGH-scheme.

35.6.1 Gianni’s Local Change of Coordinates

Let us begin with Gianni’s approach and consider a zero-dimensional idealf ⊂ P = k[X1, . . . , Xn] and let us compute its Grobner basis

G := f1, . . . , fn ∪ hi j w.r.t. the lexicographical ordering induced by X1 < · · · < Xn where

13 And an allgemeine coordinate for ge .


• f j ∈ k[X1, . . . , X j−1][X j ] \ k[X1, . . . , X j−1],


• deg j ( f j ) = d j , for each j,• degl( f j ) < dl , for each l = j, for each j,• hi j ∈ k[X1, . . . , X j ] \ k[X1, . . . , X j−1],• degl(hi j ) < dl , for each l, i, j.

Then

• if G = f1, . . . , fn or f1 ∈ k[X1] is not irreducible, then f is not prime;• if G = f1, . . . , fn, f1 ∈ k[X1] is irreducible and X j ∈ T(G) for each

j > 1 then f is prime;• if G = f1, . . . , fn, f1 ∈ k[X1] is irreducible but there exists i > 1 : Xi ∈

T(G), that is di > 1, we cannot reach any conclusion.

Let us therefore assume that G = f1, . . . , fn, f1 ∈ k[X1] is irreduciblebut there exists i > 1 : di > 1 while X j ∈ T(G) for each j > i, and let uswrite

• K := k[X1]/ f1(X1),• k for the algebraic closure of k,• Z(f) = (a11, . . . , a1n), . . . , (ar1, . . . , arn) ⊂ kn .

The reasons why di > 1 are of course twofold: either

• fi is reducible and so f is not prime, for example f = (X2 − 2, Y 2 − 2) ⊂Q[X, Y ] or f = (X2 − 2, Y 2) ⊂ Q[X, Y ], or

• f is not in generic position, so that there are two roots whose first coordinatesare equal, for example f = (X2 − 2, Y 2 − X) ⊂ Q[X, Y ].

If we are sure that there is no pair of roots in Z(f) whose first coordinates areequal, which essentially means that f is in allgemeine position, from d j > 1we can deduce that f is not prime.

It is clear that it is sufficient to perform a ‘generic’ change of coordinates

L(X j ) =

X j if j = 1Xi + cX1 if j = 1

in order to establish that

aε j + caε1 = aδ j + caδ1, for each ε = δ.

For instance, for L(X) = Y − 1/2X, L(Y ) = Y and

• f = (X2 − 2, Y 2 − 2) we have

L(f) =((X2 − 1/2)(X2 − 9/2), 6Y + 2X3 − 13X

);


• f = (X2 − 2, Y 2) we have

L(f) =((X2 − 1/2)2, 2Y − 2X3 + X

);

• f = (X2 − 2, Y 2 − X) we have

L(f) =(

4X4 − 4X2 + 16X − 7, 6Y − 2X3 − 2X2 − 3X − 5)

.

Proposition 35.6.1 (Gianni). Let

f ⊂ k[X1, . . . , Xn] =: P

be a zero-dimensional ideal and let G be its Grobner basis w.r.t. the lexico-graphical ordering induced by X1 < · · · < Xn. Denote by

• f1(X1) the monic generator of f∩K [X1], which satisfies f1 = G∩K [X1],• i, 1 ≤ i ≤ n, the minimal value such that X j ∈ T(f) for each j > i,• for each c ∈ k, Lc : P → P the change of coordinates defined by

Lc(X j ) =

X j if j = 1,Xi + cX1 if j = 1,

and write• g1 := SQFR( f1),

• g := f + (g1).

Then:

(1) if f1 is not squarefree, then√

f = √g;

(2) if f1 is squarefree and i > 1, then exists f (Z) ∈ k[Z ] such that foreach c ∈ k we have

f (c) = 0 ⇒ X j ∈ T(Lc(f)) for each j ≥ i,

but the generator of Lc(f) ∩ K [X1] is not necessarily squarefree;(3) if f1 is squarefree and i = 1 then f is radical;(4) if f1 is irreducible and i = 1 then f is prime.

Proof.

(1) Clearly g1 ∈ f so that f g and√

f ⊆ √g; since f1 and g1 have the

same roots, g1 ∈ √f and

√g ⊆ √

f.

(2) For each j > i , if f j ∈ G is such that T( f j ) = X j then T(Lc( f j ) =T( f j ) = X j .Writing

Z (f ∩ K [X1, . . . , Xi ]) = (a11, . . . , a1i ), . . . , (ar1, . . . , ari ),


since f1 is squarefree we have aε1 = aδ1, for each ε = δ; there-fore the same argument as that proving the Primitive Element Theorem(Lemma 8.4.2) gives that if we take

f (Z) =∏

ε,δ:ε =δ

(Z − aδi − aεi

aε1 − aδ1

)

then for each c ∈ k : f (c) = 0

aεi + caε1 = aδi + caδ1, for each ε = δ.

Therefore the monic generator f ∗1 (X1) of Lc(f) ∩ k[X1] is such that

D := deg(√

f ∗1

)= # (Z(f ∩ K [X1, . . . , Xi ]) ;

the Chinese Remainder Theorem then grants the existence, for eachj ≤ i, of a polynomial h j ∈ k[X1], deg(h j ) < D ≤ deg( f ∗

1 ) such thatX j − h j (X1) ∈ Łc(f).

(3) f satisfies the Nulldimensionale Radikalbasissatz (Theorem 34.1.8).(4) f satisfies the Nulldimensionale Primbasissatz (Theorem 34.1.2).

Remark 35.6.2. This result is applied to decomposition algorithms to a zero-dimensional ideal f ⊂ k[X1, . . . , Xn] =: P, by successively producing a se-quence of ideals

f =: f1 ⊆ f2 ⊆ · · · ⊆ fl ⊆ · · ·and reducing the tests to the univariate case by testing the property on thepolynomials f (l)(X1) ∈ k[X1] which generate fl ∩ k[X1].

The main justification behind this approach is again complexity: it is moretime consuming testing irreducibility of a single polynomial over a fieldextension Q[X1]/ f1(X1) than testing irreducibility of several polynomialsover Q.

primality test: Setting f1 := f and l := 0, repeatedly:

• set l := l + 1;• compute the Grobner basis 14 Gl of fl w.r.t. the lexicographical or-

dering induced by X1 < · · · < Xn ; and• set f (l)(X1) ∈ k[X1] to be the monic generator of fl ∩ k[X1];

• if f (l) is not irreducible, then fl and f are not prime;• if f (l) is irreducible and X j ∈ T(fl) for each j > 1 then fi and f

are prime;• if f (l) is irreducible and there exists i > 1 such that Xi /∈ T(fl),

14 Once the Grobner basis of f is known, the computation of the Grobner bases of fl , l > 1 oftenmainly requires just a series of Buchberger reductions and so it is not too hard to perform.


then

let i be the maximal such value, choose randomly c ∈ k and apply the change of coordinates

Lc : P → P defined by

Lc(X j ) =

X j if j = 1Xi + cX1 if j = 1,

since fl and f are prime iff Lc(fl) is such, set fl+1 := Lc(fl);

• and repeat the same algorithm while f (l) is irreducible and thereexists i > 1 such that Xi /∈ T(fl);

radicality test: Apply the same algorithm proposed above substituting eachtest checking whether f (l)(X1) is irreducible with each squarefreetest gcd( f (l), ( f (l))′) = 1;

radical computation: This is another application of the same scheme. Settingf1 := f and l := 0, repeatedly

• set l := l + 1;• compute the Grobner basis Gl of fl w.r.t. the lexicographical order-

ing induced by X1 < · · · < Xn ; and• set f (l)(X1) ∈ k[X1] the monic generator of fi ∩ k[X1];• g(l)(X1) := SQFR( f (l)(X1));

• if f (l) is not squarefree, then set fl+1 := fl + (g(l)),• if f (l) is squarefree and there exists i > 1 such that Xi /∈ T(fl),

then

let i be the maximal such value, choose randomly c ∈ k and apply the change of coordinates

Lc : P → P defined by

Lc(X j ) =

X j if j = 1Xi + cX1 if j = 1,

set fl+1 := Lc(fl);

• and repeat the same algorithm until f (l) is squarefree and X j ∈T(fl) for each j > 1;

in which case√

f = fl .primariety test: Apply the radical computation algorithm returning fl = √

f;then f is primary iff f (l) is irreducible.

35.6.2 Giusti–Heintz Coordinates

Let us consider, as usual, a d-dimensional ideal f ⊂ P = k[X1, . . . , Xn] forwhich wlog X1, . . . , Xd is a maximal set of independent variables, and its


irredundant primary representation f := ⋂ri=1 qi , whose associated primes are

pi := √qi .

Let Y := Xd+1 + ∑ni=d+2 ci Xi , (cd+2, . . . , cn) ∈ C(n − d − 1, k) be a

generic linear form and let us consider the projection π : kn → kd+1 defined,for each (a1, . . . , an) ∈ kn, by

π(a1, . . . , an) :=(

a1, . . . , ad , ad+1 +n∑

i=d+2

ci ai

),

where k denotes the algebraic closure of k.Of course, for any ideal f ⊂ P = k[X1, . . . , Xn], we have

π(Z(f)) = Z (f ∩ k[X1, . . . , Xd , Y ]) .

Let us now consider the projections of the components Z(pi ): while

p j ∩k[X1, . . . , Xd , Y ] ⊂ pi ∩k[X1, . . . , Xd , Y ] ⇐⇒ π(Z(p j )) ⊃ π(Z(pi ))

and

p j ⊂ pi ⇒ p j ∩ k[X1, . . . , Xd , Y ] ⊂ pi ∩ k[X1, . . . , Xd , Y ],

the converse

p j ⊂ pi ⇒ p j ∩ k[X1, . . . , Xd , Y ] ⊂ pi ∩ k[X1, . . . , Xd , Y ]

does not necessarily hold, but it could be expected that it is true for the‘generic’ projection. This intuitive remark was formalized by Giusti and Heintzwhose argument just requires us to consider the zero-dimensional case.

For each γ ∈ k, write

Yγ := X1 +n∑

i=2

γ i−1 Xi =n∑

i=1

γ i−1 Xi .

Lemma 35.6.3 (Chistov–Grigoriev). Let Γ ⊂ k be a finite set of c := #(Γ )

elements and let M ⊂ kn \ 0 be a finite set of m := #(M) elements. Ifc > m(n − 1), then there is an element γ ∈ Γ such that

Yγ (a1, . . . , an) =n∑

i=1

γ i−1ai = 0 for each (a1, . . . , an) ∈ M.

Proof. Let us consider the polynomial

w(T ) :=∏

(a1,...,an)∈M

(n∑

i=1

ai Ti−1

)∈ k[T ]


whose degree is deg(w) = m(n − 1) < c. Therefore there is γ ∈ Γ such that

∏(a1,...,an)∈M

Yγ (a1, . . . , an) =∏

(a1,...,an)∈M

(n∑

i=1

aiγi−1

)= w(γ ) = 0.

Corollary 35.6.4. Let Γ ⊂ k, be a finite set of c := #(Γ ) elements and letM ⊂ kn\ be a finite set of m := #(M) elements. If c > m(m − 1)(n − 1), thenthere is an element γ ∈ Γ such that, for each (a1, . . . , an), (b1, . . . , bn) ∈ M,

Yγ (a1, . . . , an) = Yγ (b1, . . . , bn) ⇐⇒ (a1, . . . , an) = (b1, . . . , bn).

Proof. Apply the lemma above to the set a − b : a, b ∈ M, a = b.

Theorem 35.6.5 (Giusti–Heintz). Let f ⊂ P := k[X1, . . . , Xn] be a d-dimensional ideal for which wlog X1, . . . , Xd is a maximal set of indepen-dent variables, and let f := ⋂r

i=1 qi be its irredundant primary representation,whose associated primes are pi := √

qi .

Then, there are just a finite number of values γ ∈ k for which

p j ∩ k[X1, . . . , Xd , Yγ ] ⊂ pi ∩ k[X1, . . . , Xd , Yγ ] ⇒ p j ⊂ pi , for each i, j,

does not hold.

Proof. Let us denote by

φ : kn → kd and, for each γ ∈ k, πγ : kn → kd+1

the projections defined, for each (a1, . . . , an) ∈ kn, by

φ(a1, . . . , an) := (a1, . . . , ad), and

πγ (a1, . . . , an) :=(

a1, . . . , ad ,

n∑i=1

γ i−1ai

).

For each isolated primary component qi consider a point ai ∈ kn such that

ai ∈ Z(pi ), and ai ∈ Z(p j ), j = i.

Since dim(f) = d , for each i the set

Mi := b ∈ Z(pi ) : φ(b) = φ(ai )is finite.

Then by the lemma above, there are just a finite number of values γ ∈ k forwhich Yγ (a) = Yγ (b) for two distinct points a, b ∈ ⋃

i Mi .


For any other γ ∈ k, if we assume the existence of two isolated primaries qi

and q j such that

p j ∩ k[X1, . . . , Xd , Yγ ] ⊂ pi ∩ k[X1, . . . , Xd , Yγ ],

we deduce that πγ (M j ) ⊂ πγ (Mi ), obtaining the required contradiction.

Corollary 35.6.6. Let f ⊂ P := k[X1, . . . , Xn] be a d-dimensional idealfor which wlog X1, . . . , Xd is a maximal set of independent variables, andlet f := ⋂r

i=1 qi be its irredundant primary representation, whose associatedprimes are pi := √

qi .

Then there is a non-empty Zariski open set U ⊂ C(n − d, k) such that foreach c := (cd+1, . . . , cn) ∈ U, setting

Yc :=n∑

i=d+1

ci Xi ,

we have

p j ∩k[X1, . . . , Xd , Yc] ⊂ pi ∩k[X1, . . . , Xd , Yc] ⇒ p j ⊂ pi , for each i, j.

Definition 35.6.7. Let f ⊂ P := k[X1, . . . , Xn] be a d-dimensional idealfor which wlog X1, . . . , Xd is a maximal set of independent variables, andlet f := ⋂r

i=1 qi be its irredundant primary representation, whose associatedprimes are pi := √

qi .

Let Y be the linear form Y := ∑ni=d+1 ci Xi , (cd+1, . . . , cn) ∈ C(n − d, k).

Then Y is said to be a Giusti–Heintz coordinate for f if, for each i, j ,

p j ∩ k[X1, . . . , Xd , Y ] ⊂ pi ∩ k[X1, . . . , Xn, Y ] ⇐⇒ p j ⊂ pi .

Theorem 35.6.8 (Giusti–Heintz). Given a d-dimensional ideal f ⊂ P :=k[X1, . . . , Xn] and assuming wlog that the variables are ordered so thatX1, . . . , Xd are a maximal set of independent variables for f, let

Y := ∑ni=d+1 ci Xi , (cd+1, . . . , cn) ∈ C(n − d, k);

f ∈ k[X1, . . . , Xd ][Y ] the primitive generator of fe ∩ k(X1, . . . , Xd)[Y ];g := SQFR( f ) ∈ k[X1, . . . , Xd ][Y ] the primitive generator of√

fe ∩ k(X1, . . . , Xd)[Y ];

F := √fec ∩ √

f : g∞;L := f : F∞.

Then:

(1)√

f = F ∩ √L;

(2) Y is a Giusti–Heintz coordinate for f iff L = 1;


(3) if Y is a Giusti–Heintz coordinate for f then

(a)√

f : (fec)∞ = √f : g∞,

(b) the assignment of

f0 = fe and f√ := f : g∞

is an ARGH-decomposition.

Proof. Recall (Corollary 27.2.12) that for each g ∈ P and for each primaryq ⊂ P, p := √

q, one has

g ∈ p ⇐⇒ q : g∞ = q,

g ∈ p ⇐⇒ q : g∞ = (1).

Let f := ⋂ri=1 qi be an irredundant primary representation, where wlog, for

each i : pi := √qi and the primaries are ordered so that

i ≤ s ⇐⇒ X1, . . . , Xd is a maximal set of independent variablesfor qi ,

s < i ≤ u ⇐⇒ there exists j ≤ s :√

qi ⊃ √q j ,

u < i ≤ v ⇐⇒ g ∈ √qi and qi is an isolated component of f : g∞,

v < i ≤ t ⇐⇒ g ∈ √qi and qi is an embedded component of f : g∞,

t < i ≤ r ⇐⇒ g ∈ √q j .

Since g ∈ fe ∩ k[X1, . . . , Xn], then g ∈ fec and

g ∈ pi ⇐⇒ 1 ≤ i ≤ u or t < i ≤ r

so that we have

√f =

s⋂i=1

pi ∩v⋂

i=u+1pi ∩

r⋂i=t+1

pi , fe =s⋂

i=1qe

i ,

fec =s⋂

i=1qi ,

√fec =

s⋂i=1

pi ,

f : g∞ =v⋂

i=u+1pi ∩

t⋂i=v+1

pi ,√

f : g∞ =v⋂

i=u+1pi ,

F =s⋂

i=1pi ∩

v⋂i=u+1

pi , L =r⋂

i=t+1qi .

In order to complete the proof we need to prove that

qi : g∞ = qi for each i > u ⇐⇒ Y is a Giusti–Heintz coordinate for f,


since this gives the implications

Y is a Giusti–Heintz coordinate for f, ⇐⇒ qi : g∞ = qi for each i > u

⇐⇒ g /∈ √qi for each i > u

⇐⇒ t = r

⇐⇒ L = 1.

Let us therefore denote

ui := qi ∩ k[X1, . . . , Xd ][Y ] and vi := pi ∩ k[X1, . . . , Xd ][Y ];then ve

i = (1) if i > s while, for each i ≤ s, vi is principal and has anirreducible generator gi ∈ k[X1, . . . , Xd ][Y ]; therefore g = SQFR( f ) =∏s

i=1 gi .Now, for each i > u, since p j ⊆ pi , j ≤ s, we have

qi : g∞ = (1) ⇐⇒ g ∈ pi ∩ k[X1, . . . , Xd ][Y ] = vi

⇐⇒ there exists j ≤ s : g j ∈ vi (because vi is prime)

⇐⇒ there exists j ≤ s : v j ⊆ vi

⇐⇒ Y is not a Giusti–Heintz coordinate for f.

Remark 35.6.9. As suggested by Theorem 35.6.8, we can implement theARGH-scheme by setting

f√ := f : g∞,

and computing

a := f : g∞,b := √

a,F := √

fec ∩ b,L := f : F∞.

If L = (1) we know that Y is a Giusti–Heintz coordinate for f and the com-putation is complete; otherwise we also have to apply the algorithm to thecomponent L.

35.7 Linear Algebra and Change of Coordinates

Giusti and Heintz introduced their idea in order to show that minimal primedecomposition (and connected algorithms) has a good theoretical complexity.Such good theoretical complexity has a direct effect also on practical com-plexity: we have remarked that the ARGH-scheme is strongly effective if f is

35.7 Linear Algebra and Change of Coordinates 651

in allgemeine position but that putting it in such a position forces us to workwith dense polynomials.

The scheme suggested by Theorem 35.6.8 avoids completely any density,since

• once g(Y ) = g(∑n

i=d+1 ci Xi ) is obtained and a := f : g∞ is computed, allthe other computations, that is

b := √a, F :=

√fec ∩ b and L := f : F∞,

are performed within the original frame of coordinates and with polynomialideals which are the natural data, being proper intersections of the requireddata qi and pi ;

• as regards the computation of a := f : g∞, if

G is a basis of f,G ′ := G ∪ g(Y )T − 1, Y − ∑n

i=d+1 ci Xi , andd ⊂ k[X1, . . . , Xn, Y, T ] is the ideal generated by G ′,

we have (see Corollary 26.3.11) a = d ∩ k[X1, . . . , Xn]: the data in G ′ aretherefore as dense as those in G.

Algorithm 35.7.1 (Alonso–Raimondo). The ARGH-scheme requires as itscentral tool an algorithm for checking, given

a d-dimensional ideal f ⊂ P = k[X1, . . . , Xn] for which X1, . . . , Xd is amaximal set of independent variables and

a generic linear form Y := ∑ni=d+1 ci Xi , (cd+1, . . . , cn) ∈ C(n − d, k),

whether, for each associated prime p,

y :=n∑

i=d+1

ci xi ∈ P/p =: k[x1, . . . , xn] =: R

is a primitive element and, if so, for computing the polynomial g(Y ) ∈k(X1, . . . , Xd)[Y ] generating the principal ideal√

fe ∩ k(X1, . . . , Xd)[Y ].

An elementary modification of the FGLM algorithm gives an efficient toolfor doing that:

• we can assume that we have a Grobner basis of f ⊂ P and therefore also theGrobner basis G≺ of the zero-dimensional ideal

fe ⊂ k(X1, . . . , Xd)[Xd+1, . . . , Xn]

w.r.t. some term ordering ≺, so that we can merge the algorithms of


Figures 29.1 and 29.2 (or directly apply the algorithms of Figure 29.4) ob-taining the linear representation

(N≺(fe),M(N≺(fe))), M(N≺(fe)) =(

a(h)l j

),

of fe w.r.t. the lexicographical ordering ≺ induced by Xd+1 ≺ · · · ≺ Xn ;• this information can be easily transformed into a linear representation

(N≺(g),M(N≺(g))), of

g := fe +(

Y −n∑

h=d+1

ch Xh

)⊂ k(X1, . . . , Xd)[Y, Xd+1, . . . , Xn]

w.r.t. the lexicographical ordering ≺ induced by Xd+1 ≺ · · · ≺ Xn ≺ Y ,since N≺(g) = N≺(f) and we only have to compute the matrix storing themultipication by Y ; since Y = ∑n

h=d+1 ch Xh we only have to compute∑nh=d+1 cha(h)

l j , for each l, j ;• an application of the FGLM Algorithm (Figure 29.2) then allows the de-

duction of the linear representation of g w.r.t. the lexicographical orderinginduced by Y < Xd+1 < · · · < Xn and therefore the direct application ofthe result of Proposition 35.6.1;

• we obtain the monic generator f (Y ) of g ∩ k(X1, . . . , Xd)[Y ] and the min-imal value 15 i, d ≤ i < n, such that X j ∈ T(g) for each j > i, and

• if f is squarefree and i = d we have found the required solution;• if f is not squarefree, we can apply the algorithm of Figure 29.3 in order

to deduce the Grobner basis G and the linear representation of

g + SQFR( f )w.r.t. the lexicographical ordering induced by Y < Xd+1 < · · · < Xn ;

• if f is squarefree and i > d , then, setting

h := g ∩ k(X1, . . . , Xd)[Y, Xd+1, . . . , Xi ]

and

H := G ∩ k(X1, . . . , Xd)[Y, Xd+1, . . . , Xi ],

we know (Theorem 34.2.1) that there are polynomials

gi ∈ k(X1, . . . , Xd)[Y ] such that

G = H ∪ Xi+1 + gi (Y ), . . . , Xn + gn(Y );since N<(h) = N<(g) we also have the Grobner representation and thelinear representation of h w.r.t. the lexicographical ordering induced by

15 Note that i < n since Xn = T(h) for h := Y − ∑nh=d+1 ch Xh ∈ T(g).

35.7 Linear Algebra and Change of Coordinates 653

Y < Xd+1 < · · · < Xi and we can iteratively apply the same algorithmuntil we obtain a generic linear form

Z :=i∑

h=d+1

dh Xh + cY =i∑

h=d+1

(cch + dh)Xh +n∑

h=i+1

cch Xh

and the required monic generator h(Z) of√h ∩ k(X1, . . . , Xd)[Z ] =

√fe ∩ k(X1, . . . , Xd)[Z ].

Note that in the ARGH-scheme the zero-dimensional Grobner basis com-puted in each step belongs to the polynomial ring

k(X1, . . . , Xd)[Xd+1, . . . , Xn]

and the linear algebra of Figure 29.2 is to be performed within the fieldk(X1, . . . , Xd); therefore the complexity evaluation O(ns3) given in Sec-tion 29.4 is not applicable here; therefore the ARGH-scheme is not of poly-nomial complexity; its only advantage is to avoid density.

Algorithm 35.7.2 (Krick–Logar). The algorithm above is essentially a refine-ment of the proposal by Krick and Logar of using the Seidenberg Algorithm(Corollary 35.2.3) in order to compute the radical of a zero-dimensional ideal.

Given a zero-dimensional ideal f ⊂ k[X1, . . . , Xn] by a Grobner basis, lin-ear algebra allows us to find, for each i ≤ n, the minimal polynomial fi (Xi ) ∈f ∩ k[Xi ] and its squarefree associate gi (Xi ) so that

√f = f + (g1, . . . , gs).

Algorithm 35.7.3 (Singular). An extension of the algorithm above allows us toperform a partial decomposition

f =(⋂

iqi

)∩

(⋂jI j

)⊂ k[X1, . . . , Xn]

of f into components such that

each qi is a primary whose associated prime is pi ,each I j , while not yet completely decomposed, is ‘smaller’;

we therefore need to perform Gianni’s local change of coordinates only onsuch components I j .

The algorithm factorizes each fi into powers of irreducible polynomials and,for all n-tuples (h1(X1), . . . , hn(Xn)), where each hi is a factor of fi , com-putes the Grobner basis of f + (h1, . . . , hn) w.r.t. the lexicographical ordering


induced by X1 < · · · < Xn , thus checking whether such a component is pri-mary – in which case it is labelled qi and returned – or not – in which case itis labelled I j , submitted to a local change of coordinates and, iteratively, to thesame algorithm.

Example 35.7.4 (Partini). This algorithm is, however, unable, without localchange of coordinates, to produce a complete factorization as is shown by thefollowing example: let us consider the ideal

I := X2Y 2 Z2 − 1, X2 + Y 2 + Z2 ⊂ k[X, Y, Z ]

which is reducible since

X2Y 2 Z2 − 1 = (XY Z − 1)(XY Z + 1).

The minimal polynomial in I ∩ k(X)[Y ], which is

X2Y 4 + (X4 − X2)Y 2 + 1,

is irreducible and, by symmetry, the same happens for any other choice ofvariables.

35.8 Direct Methods for Radical Computation

In the early 1990s, Eisenbud, Huneke and Vasconcelos proposed a differentapproach to decomposition algorithm, the use of

‘direct methods’, in the sense that they do not require this reduction [to the one-polynomial case].Why should one want to avoid this reduction? To answer questions . . . by the methodsusing projections one needs ‘sufficiently generic’ projections. In practice, this currentlymeans that one takes [. . . ] random linear forms [as a new frame of coordinates], check-ing afterwards that the choice was ‘random enough’. Unfortunately this randomnessdestroys whatever sparseness and symmetry the original problem may have had, andleads to computations which are often extremely slow.D. Eisenbud, C. Huneke, and W. Vasconcelos, Direct Methods for Primary Decompo-sition, Inventiones Math. 110 (1992), p. 209.

Since most of their algorithms require advanced theoretical tools which areoutside the scope of this book, I limit myself to presenting an improved andsimplified version (due to Fortuna, Gianni and Trager) of their radical compu-tation algorithm, which takes advantage of the Nulldimensionalen Basissatzein order to produce the complete intersection required by the original state-ment.

Let I ⊂ P := k[X1, . . . , Xn] be a zero-dimensional ideal and let I =⋂rj=1 q j be an irredundant primary representation of f and, for each j , p j be

the associated prime of q j .

35.8 Direct Methods for Radical Computation 655

Proposition 35.8.1. Assume that the reduced Grobner basis G of I w.r.t. thelexicographical ordering induced by X1 < · · · < Xn consists of exactly nelements, say

G = γ1, . . . , γn ⊂ k[X1, . . . , Xn],

and let

F :=n∏

i=1

∂γi

∂ Xi.

Then

(1) (I : F) = ⋂F /∈q j

p j ,

(2) if F /∈ q j , then the field P/p j is separable over k,(3) if char(k) = 0 or char(k) > maxdegi (γi ), then

√I = (I : F).

Proof.

(1) Since (I : F) = ⋂F /∈q j

(q j : F) we need only to prove that, for any j ,

F ∈ q j ⇒ (q j : F) = p j .

Let us fix a value j and let us consider

the reduced Grobner basis G ′ of q j w.r.t the lexicographical orderinginduced by X1 < . . . < Xn ,

the polynomials f1, . . . , fn ⊂ G ′ such that T( fi ) = Xdii ,

the primbasis g1, . . . , gn of p j ,

so that (Theorem 34.1.5(2)(e)) for each i

fi = gsii +

i−1∑h=1

pih gh

for suitable si ∈ N, pih ∈ k[X1, . . . , Xi ] and gh ∈ k[X1, . . . , Xh] foreach h; in particular f1 = gs1

1 .In order to prove that (q j : F) = p j it is sufficient to prove that gi ∈(q j : F) for each i .If we write

Fi :=i∏

h=1

∂γh

∂ Xh

so that F = Fn , we will prove the claim by inductively proving thatgi Fi ∈ q j for each i .


Note that, for each i , γi ∈ q j ∩ k[X1, . . . , Xi ] and is monic in Xi sothat it must be reduced to 0 modulo G ′ ∩ k[X1, . . . , Xi ]; therefore, forsuitable ui , cih ∈ k[X1, . . . , Xi ], we have

γi = ui gsii +

i−1∑h=1

cih gh .

We immediately see that, since γ1 = u1gs11 , we have

g1 F1 = g1∂γ1

∂ X1= g1

∂u1

∂ X1gs1

1 + g1s1∂g1

∂ X1u1gs1−1

1

= gs11

(g1

∂u1

∂ X1+ u1s1

∂g1

∂ X1

)

= f1

(g1

∂u1

∂ X1+ u1s1

∂g1

∂ X1

)∈ q j .

This allows us to perform an inductive proof; so let us assume thatgl Fl ∈ q j (and hence also gl Fm ∈ q j for m ≥ l) for each l < i and letus prove that gi Fi ∈ q j .Observe that

gsii Fi−1 = fi Fi−1 −

i−1∑h=1

pih gh Fi−1 ∈ q j .

We also have

∂γi

∂ Xi= gsi

i∂ui

∂ Xi+ si ui g

si −1i

∂gi

∂ Xi+

i−1∑h=1

gh∂cih

∂ Xi

so that

gi Fi = gi∂γi

∂ XiFi−1 =

(gi

∂ui

∂ Xi+ si ui

∂gi

∂ Xi

)gsi

i Fi−1

+ gi

i−1∑h=1

∂cih

∂ Xigh Fi−1

and gi Fi ∈ q j because both gsii Fi−1 and each gh Fi−1 are in q j .

(2) We need to show that, if F /∈ q j , then P/p j is separable over k and wewill do it by showing that ∂gi/∂ Xi /∈ p j for each i .Noting that g1, . . . , gn is the reduced Grobner basis of p j , and that∂gi/∂ Xi cannot be reduced by any of the gls we can deduce that∂gi/∂ Xi ∈ p j ⇒ ∂gi/∂ Xi = 0.

35.8 Direct Methods for Radical Computation 657

Then let us assume that ∂gi/∂ Xi = 0 so that we have

Fi = Fi−1∂γi

∂ Xi= Fi−1

(gsi

i∂ui

∂ Xi+ si ui g

si −1i

∂gi

∂ Xi+

i−1∑h=1

gh∂cih

∂ Xi

)

= ∂ui

∂ Xigsi

i Fi−1 +i−1∑h=1

∂cih

∂ Xigh Fi−1

and Fi ∈ q j because both gsii Fi−1 and each gh Fi−1 are in q j . This

implies that also F ∈ q j , giving the required contradiction.(3) The argument above shows that each P/p j is separable over k when

char(k) = 0 or char(k) > maxdegi ( fi ). Thus we just need to showthat the separability of P/p j implies F /∈ q j . The argument is by in-duction on the number of variables. If n = 1, then I = (γ1), q j =( f1), p j = (g1) and f1 = gs1

1 , γ1 = u1gs11 , gcd(u1, g1) = 1;

also, by separability, ∂γ1/∂ X1 /∈ p j . We therefore have

F1 = ∂γ1

∂ X1= ∂u1

∂ X1gs1

1 + ∂g1

∂ X1s1u1gs1−1

1 = gs1−11

(∂u1

∂ X1g1 + ∂g1

∂ X1s1u1

).

If we assume F1 ∈ q j , since gs1−11 /∈ q j , we deduce (∂u1/∂ X1)g1 +

(∂g1/∂ X1)s1u1 ∈ p j and (∂g1/∂ X1)s1u1 ∈ p j .

Now the fact that u1 /∈ p j and the assumption on the characteristicforce ∂g1/∂ X1 ∈ p j and contradict the separability of P/p j .We can therefore perform induction: setting

q′j := q j ∩ k[X1, . . . , Xn−1],

p′j := q j ∩ k[X1, . . . , Xn−1] = (g1, . . . , gn−1),

the separability of P/p j implies ∂gi/∂ Xi /∈ p j and so the separabilityof k[X1, . . . , Xn−1]/p′

j . By induction Fn−1 = ∏n−1i=1 (∂γi/∂ Xi ) /∈ q′

jand Fn−1 /∈ q j .Also we have

F = Fn−1∂γn

∂ Xn= gsn−1

n Fn−1

(∂un

∂ Xngn + ∂gn

∂ Xnsnun

)+

n−1∑h=1

∂cnh

∂ Xngh Fn−1.

As we already proved above,

n−1∑h=1

∂cnh

∂ Xngh Fn−1 ∈ q j ;


moreover gsn−1n Fn−1 /∈ q j , because neither gsn−1

n nor Fn−1 is in q j :since Fn−1 ∈ k[X1, . . . , Xn−1],

Fn−1 ∈ q j , ⇒ Fn−1 ∈ q j ∩ k[X1, . . . , Xn−1] = q′j ,

giving a contradiction.Therefore the assumption F ∈ q j implies (∂un/∂ Xn)gn +(∂gn/∂ Xn)snun ∈ p j and (∂gn/∂ Xn)snun ∈ p j .

This gives the required contradiction ∂gn/∂ Xn ∈ p j since sn = 0 andun /∈ p j .

Let J ⊂ P := k[X1, . . . , Xn] be a zero-dimensional ideal; then from thereduced Grobner basis of J w.r.t. the lexicographical ordering induced byX1 < · · · < Xn it is possible to extract the unique polynomials γ1, . . . , γn

such that, for each i , T<(γi ) = Xdii . Let I denote the ideal generated by

G := γ1, . . . , γn which is, by construction, the reduced Grobner basis ofI w.r.t. the lexicographical ordering induced by X1 < · · · < Xn . Then, settingF := ∏n

i=1 ∂γi/∂ Xi , we are under the assumptions of Proposition 35.8.1 sothat we can conclude, with the notation above:

Proposition 35.8.2 (Eisenbud–Huneke–Vasconcelos). We have√

J =√

I :(√

I : J)

.

Proof. Let√

I = ⋂si=1 pi be the primary decomposition of

√I. Noting that

I ⊂ J so that each associated prime of J is associated also to I, we can wlogenumerate the pi s so that

√J = ⋂r

i=1 pi with r ≤ s.Then

√I : J = ⋂s

i=r+1 pi and

√I :

(√I : J

)=

r⋂i=1

pi =√

J.

35.9 Caboara–Conti–Traverso Decomposition Algorithm

The decomposition algorithm proposed by Caboara, Conti and Traverso aimsto investigate whether it is possible to adapt the ARGH-scheme in order toavoid any change of coordinates;16 in other words: what results can be obtainedin Theorem 35.6.8 and in the ARGH-decomposition if we simply put Y :=Xd+1?

16 While Giusti and Heintz’ position is density-free it has the disadvantage of destroying the struc-ture of binomial ideals.

35.9 CCT Decomposition Algorithm 659

Their CCT-scheme consists of iteratively computing a CCT-decomposition

g :=⋂

j

a j ⊃ f

where each component a j is unmixed and√

f = √g; this is sufficient at least

to obtain the prime decomposition of each a j and (by Proposition 35.2.6) theprimary decomposition.

Such a CCT-decomposition is obtained inductively, finding for any ideal a

either

a proof that a is unmixed ora splitting

√a = √

(a : h) ∩ √a + (h) for a suitable polynomial h, on whose

components the algorithm is iteratively applied.

Via a preprocessing Grobner computation and a renumbering of the vari-ables we can wlog assume we know that dim(f) = d and that X1, . . . , Xd isa maximal set of independent variables for f.

Then:

(1) we compute a Grobner basis G of f for any term ordering < un-der which X j > t for any j > d + 1 and any term t ∈k[X1, . . . , Xd , Xd+1], thus getting the Grobner basis G0 := G ∩k[X1, . . . , Xd , Xd+1] of the ideal

F := f∩k[X1, . . . , Xd , Xd+1] ⊂ k(X1, . . . , Xd)[Xd+1], dim(F) = 1.

(2) We now consider f := gcd(h ∈ G0) ∈ k(X1, . . . , Xd)[Xd+1] and thepolynomial g := Prim(SQFR( f )) ∈ k[X1, . . . , Xd , Xd+1]. If g /∈ f

we obtain the splitting√f =

√f : g∞ ∩

√f + (g).

(3) If, instead, g ∈ f, then G ∩ k[X1, . . . , Xd , Xd+1] = g and g ∈k[X1, . . . , Xd ][Xd+1] is squarefree and primitive.Each element h ∈ G \ g can be considered to be a polynomial in

k[X1, . . . , Xd , Xd+1][Xd+2, . . . , Xn]

and uniquely expressed as

h =∑

t∈T [d+2,n]

gt (X1, . . . , Xd , Xd+1)t;

we write

T(h) := max<

t ∈ T [d + 2, n] : gt = 0, Lp(h) := gT(h).

We can now extract from G a subset H ⊂ G \ g such that, for each


h ∈ G \g, there is an element h′ ∈ H such that T(h′) | T(h). Clearly,for any h ∈ H , g Lp(h) since G is reduced.Now if some h ∈ H is such that gcd(Lp(h), g) = 1 then we obtain thesplitting

√f =

√f + (gcd(Lp(h), g)) ∩

√f +

(g

gcd(Lp(h), g)

).

(4) If we reach this step, we know that g is squarefree and has no commonfactor with g′ := ∏

h∈H Lp(h) so that g′ /∈ √f and we can compute

f′ := f : g′∞. If f′ = f then we obtain the splitting√f =

√f′ ∩

√f + (g′).

(5) If instead we have f′ = f we are through since this implies that noprime, associated to f, contains g′, whence f is unmixed.

35.10 Squarefree Decomposition of a Zero-dimensional Ideal

Let I ⊂ P := k[X1, . . . , Xn] be a zero-dimensional ideal and let I = ⋂ri=1 qi

be its irredundant primary representation; for each i let mi := √qi be the

associated (maximal) prime and ρi the characteristic number of qi . Denoteρ := maxi (ρi ), and, for each h, 1 ≤ h ≤ ρ,

Ih :=⋂ρi =h

qi and Rh :=⋂ρi =h

mi .

Definition 35.10.1 (Heiß–Oberst–Pauer). The squarefree decomposition ofthe zero-dimensional ideal I is the unique sequence R1, . . . , Rρ.

Proposition 35.10.2 (Heiß–Oberst–Pauer).

(1) For each h, 1 ≤ h ≤ ρ, Ih = I + Rhh;

(2) I = ⋂ρh=1 Ih = ⋂ρ

h=1

(I + Rh

h

).

Proof. (1) For each j for which ρ j = h we have

I + Rhh = I +

( ⋂ρi =h

mi

)h

⊆ I + mhj = q j

whence I + Rhh ⊆ ⋂

ρi =h qi = Ih .

35.10 Squarefree Decomposition 661

Conversely

Ih =⋂ρi =h

qi

=∏ρi =h

qi

=∏ρi =h

(I + mh

i

)

⊆ I +∏ρi =h

mhi

= I +( ⋂

ρi =h

mi

)h

= I + Rhh .

(2) Obvious.

Knowing a basis of I allows us to compute

• a basis G := g1, . . . , gs ⊂ P of√

I – by any algorithm discussed in thischapter – and

• a linearly independent set L = 1, . . . , r ⊂ P∗ such that Spank(L) =L(I) – for instance L = γ ( · , t, <), t ∈ N<(I), where < is any termordering

and this information is sufficient for computation of, with good complexity, thesquarefree decomposition of I.

Let us denote the P-module structure of P∗ by : P × P∗ → P∗ where,for each ∈ P∗ and each f ∈ P , f denotes the functional defined by

( f )(g) := ( f g) for each g ∈ P;we also write, for each ideal P ⊂ P and each P-module L ⊂ P∗,

P L := f : f ∈ P, ∈ L.

Lemma 35.10.3. Let J1 and J2 be two ideals in P; let G be a finite basis ofJ2 and L ⊂ P∗ be a finite set such that Spank(L) = L(J1). Then

J2 L(J1) = ( f : f ∈ G, ∈ L) .

Proof. A trivial consequence of the fact that Spank(L)= L(J1) is aP-module.


Lemma 35.10.4 (See Corollary 30.2.8). Let J1 and J2 be two ideals in P;then

L(J1 : J2) = J2 L(J1).

Proof. Since LP(J2 L(J1)) = J2 L(J1) it is sufficient to prove that(J1 : J2) = P(J2 L(J1)) which is true because

(J1 : J2) = f ∈ P : f g ∈ J1 for each g ∈ J2= f ∈ P : ( f g) = 0 for each g ∈ J2, ∈ L(J1)= f ∈ P : (g )( f ) = 0 for each g ∈ J2, ∈ L(J1)= P(J2 L(J1)).

With the notation above:

Proposition 35.10.5 (Heiß–Oberst–Pauer). We have:

(1) L(I) = ⊕i L(qi );

(2) mi L(q j ) = L(q j ) if i = j ;(3)

√I L(q j ) = m j L(q j ) for each j;

(4)√

I L(I) = ⊕j m j L(q j );

(5)√

Ih L(I) = ⊕

j mhj L(q j ) for each h;

(6) mρ j −1j L(q j ) = L(m j ) for each j;

(7)√

Ih−1 L(I) = L(Rh) ⊕

(⊕ρ j >h mh−1

j L(q j ))

for each h,

2 ≤ h ≤ ρ;

(8) L(Rρ) = √Iρ−1 L(I),

(9) for 1 ≤ h ≤ ρ − 1, L(Rh) =(∏ρ

i=h+1 Ri−h+1i

)

(√Ih−1 L(I)

).

Proof.

(1) Trivial.(2) If i = j then mi and q j are comaximal so that mi + q j = P; also

q j L(q j ) = 0 so that

mi L(q j ) = (mi + q j

) L(q j ) = P L(q j ) = L(q j ).

(3)√

I L(q j ) = (∏i mi

) L(q j ) = m j

(∏i = j mi

) L(q j ) = m jL(q j ).

(4)√

I L(I) = ⊕j

√I L(q j ) = ⊕

j m j L(q j ).

(5) Trivial.

35.10 Squarefree Decomposition 663

(6) We have mρ jj ⊆ q j and m

ρ j −1j ⊆ q j so that

m j ⊆(q j : m

ρ j −1j

)= P;

since m j is maximal we have m j =(q j : m

ρ j −1j

)and the claim fol-

lows by Lemma 35.10.4.(7) If ρ j < h then mh−1

j ⊆ q j so that mh−1j L(q j ) = 0. Therefore

√Ih−1 L(I) =

⊕jmh−1

j L(q j ) =⊕ρ j ≥h

mh−1j L(q j ).

Also ⊕ρ j =h

mh−1j L(q j ) =

⊕ρ j =h

L(m j ) = L(Rh).

(8) Follows by the result above.(9) We have (

ρ∏i=h+1

Ri−h+1i

)

(√Ih−1 L(I)

)

=(

ρ∏i=h+1

Ri−h+1i

)

⎛⎝⊕

ρ j ≥h

mh−1j L(q j )

⎞⎠

=⊕ρ j ≥h

((ρ∏

i=h+1

Ri−h+1i

)mh−1

j

) L(q j ).

For each j, ρ j > h,((ρ∏

i=h+1

Ri−h+1i

)mh−1

j

)⊂ mi

j ⊂ mρ jj ⊆ q j

so that ((ρ∏

i=h+1

Ri−h+1i

)mh−1

j

) L(q j ) = 0

and (ρ∏

i=h+1

Ri−h+1i

)

(√Ih−1 L(I)

)

=⊕ρi =h

((ρ∏

i=h+1

Ri−h+1i

)mh−1

j

) L(q j ).


On the other hand, for each j, ρ j = h, the ideals(∏ρ

i=h+1 Ri−h+1i

)and m j are coprime so that

∏ρi=h+1 Ri−h+1

i ⊆ m j , whence(ρ∏

i=h+1

Ri−h+1i

) L(q j ) = L(q j )

and ((ρ∏

i=h+1

Ri−h+1i

)mh−1

j

) L(q j )

= mh−1j

((ρ∏

i=h+1

Ri−h+1i

) L(q j )

)

= mρ j −1j L(q j )

= L(m j ).

In conclusion(ρ∏

i=h+1

Ri−h+1i

)

(√Ih−1 L(I)

)=

⊕ρ j =h

L(m j ) = L(Rh).

Algorithm 35.10.6 (Heiß–Oberst–Pauer). The results above allow us to com-pute the squarefree decomposition of I as follows:

• compute iteratively, starting with r0 := L(I), ri := √I ri−1 until ri = 0;

• set ρ := i and L(Rρ) := rρ−1;• compute iteratively, for h = ρ − 1, . . . , 1,

Rh+1 := PL(Rh+1) and L(Rh) :=ρ∏

i=h+1

Ri−h+1i · rh−1.

36

Macaulay III

This chapter continues my report on Macaulay’s analysis of the structure ofthe Hilbert function.

The starting point is the same as it will be later in Grobner’s introduction ofthe notion of Prombasis: an admissible sequence (g1, . . . , gr ) defines an idealof rank r and it is to be expected that, in general, an ideal generated by r poly-nomials has rank r . This led Kronecker to generalize the notion of principalideal to that of ideal of principal class (nowadays complete intersections):

This term was used by Kronecker, though it seems to have gone out of use and noother term has replaced it. It is not what is called a principal ideal (or ideal of rank1 with a basis (F) consisting of a single member) but an ideal of rank r with a basis(F1, F2, . . . , Fr ) consisting of r members only.[1]

Macaulay evaluated (Section 36.1) the Hilbert function of a complete in-tersection and used the same technique in order (Section 36.2) to characterizethe coefficients of the Hilbert function of a homogeneous ideal I(0) := I ⊂k[Y0, . . . , Yn] – where the coordinates are generic – in terms of those of theideals I(i) := I + (Y0, . . . , Yi−1).

If, equivalently, for each i < dim(I) := d + 1,

I(i) = I(i)sat,I(i)irr = (Y0, . . . , Yn),

I(i) : Yi = I(i),

the Hilbert function of the zero-dimensional ideal I(d+1) – which is obtained bysimply counting, for any term ordering <, the terms belonging to NN<(I(d+1)) –allows us to deduce the Hilbert polynomial and the Hilbert function of each I(i)

by simply comparing iteratively the difference between the Hilbert function

1 F. S. Macaulay, Some Properties of Enumeration in the Theory of Modular Systems, Proc. Lon-don Math. Soc. 26 (1927), p. 548.

665

666 Macaulay III

and the Hilbert polynomial of I(i+1). Macaulay (Section 36.3) introduced thenotion of perfectness to characterize the ideals I having this property.

36.1 Hilbert Function and Complete Intersections

Let

hP := k[X0, . . . , Xn],I ⊂ hP be a homogeneous ideal,f1, . . . , fr ∈ hP \ 0 be a sequence of homogeneous polynomials,Y0, Y1, . . . , Yn be a system of coordinates for hP .

Definition 36.1.1. Then:

• f1, . . . , fr is called a regular sequence for I if, for each i ≥ 0, fi+1 is anon-zero divisor of hP/(I + ( f1, . . . , fi )), that is I : f1 = I and, for i ≥ 1,(I + ( f1, . . . , fi )) : fi+1 = I + ( f1, . . . , fi );

• f1, . . . , fr is called a regular sequence if, for i ≥ 1, ( f1, . . . , fi ) : fi+1 =( f1, . . . , fi );

• any homogeneous ideal ( f1, . . . , fr ) generated by a regular sequence iscalled a complete intersection;

• the depth of I, depth(I), is the maximal λ for which there is a regular se-quence of linear forms Y0, . . . , Yλ−1 for I;

• the index of regularity, γ (I), of I is the minimal value δ for which

hHI(l) = hH(l; I) for each l ≥ δ.

Recall that for any homogeneous ideal I ⊂ hP we denote by

hHI(T ) = k0

(T + d

d

)+ k1

(T + d − 1

d − 1

)+ · · · + kd−1T + kd =

= k0(I)(

T + d

d

)+ k1(I)

(T + d − 1

d − 1

)+ · · · + kd−1(I)T + kd(I)

its Hilbert polynomial where

• d := deg(hHI) = dim(I) − 1,2

• k0(I) is the degree of I.

After the trivial remark that:

2 In order to avoid ambiguities let me stress that for an affine ideal I ⊂ k[X1, . . . , Xn ] and ahomogeneous ideal J ⊂ k[X0, X1, . . . , Xn ] related by J = h I, I = aJ, I consider valid therelation

dim(I) = dim(J) − 1, r(I) = r(J).

36.1 Hilbert Function and Complete Intersections 667

Lemma 36.1.2. Let a, b ⊂ hP be homogeneous ideals, then

hH(T ; a) + hH(T ; b) = hH(T ; a + b) + hH(T ; a ∩ b),

clearly a reformulation of Lemma 23.5.1 and Corollary 23.5.3, allows us tocompute the Hilbert function of a complete intersection and to connect theHilbert function of a homogeneous ideal I ⊂ hP with that of the ideals I +(Y0, . . . , Yi ) where Y0, Y1, . . . , Yn is a system of coordinates for hP .

Proposition 36.1.3. Let ∈ hP be a homogeneous element such thatdeg() = δ, and let f ⊂ P be a homogeneous ideal. If f : = f we have

hH(T ; f + ()) = hH(T ; f) − hH(T − δ; f),

dim(f + ()) = dim(f) − 1.

Proof. We have (see Lemma 26.3.6)

f ∩ () = (f : ()) = f,

so that

hH(T ; f) + hH(T ; ()) = hH(T ; f + ()) + hH(T ; f).

Clearly we have, for l ≥ δ,(l + n

n

)− hH(l; f) =

(l − δ + n

n

)− hH(l − δ; f),

hH(l; ()) =(

l + n

n

)−

(l − δ + n

n

)= hH(l; f) − hH(l − δ; f),

hH(l; f + ()) = hH(l; f) + hH(l; ()) − hH(l; f)

= hH(l; f) − hH(l − δ; f).

As regards the second statement, it is sufficient to prove it when f is primeand this follows directly from the proof of Lemma 27.10.3.

Lemma 36.1.4.(T + n

n

)−

(T + n − δ

n

)= δ

(T + n

n − 1

)+

δ∑i=2

(i − 1)

(T + n − δ + i

n − 2

).

668 Macaulay III

Proof. We have

(T + n

n

)−

(T + n − δ

n

)

=(

T + n − 1

n

)+

(T + n

n − 1

)−

(T + n − δ

n

)

=(

T + n − j

n

)+

j−1∑i=0

(T + n − i

n − 1

)−

(T + n − δ

n

)

=δ−1∑i=0

(T + n − i

n − 1

)

=δ∑

i=1

(T + n − δ + i

n − 1

)

=(

T + n − δ + 1

n − 1

)+

(T + n − δ + 2

n − 1

)+

δ∑i=3

(T + n − δ + i

n − 1

)

=(

T + n − δ + 2

n − 2

)+ 2

(T + n − δ + 2

n − 1

)+

δ∑i=3

(T + n − δ + i

n − 1

)= · · ·

=j∑

i=2

(i − 1)

(T + n − δ + i

n − 2

)+ j

(T + n − δ + j

n − 1

)

+δ∑

i= j+1

(T + n − δ + i

n − 1

)

=δ∑

i=2

(i − 1)

(T + n − δ + i

n − 2

)+ δ

(T + n

n − 1

).

Proposition 36.1.5 (Macaulay). Let I = ( f1, . . . , fr ) be a complete intersec-tion; writing δi := deg( fi ) for each i we have

k0(I) = ∏ri=1 δi ,

γ (I) = 1 + ∑ri=1(δi − 1),

r(I) = r, dim(I) = n + 1 − r.

Proof. Writing hl := ( f1, . . . , fl), for l, 1 ≤ l ≤ r , since hl : fl+1 = hl , foreach l we can inductively apply the result of Lemma 36.1.4 to the formula of

36.1 Hilbert Function and Complete Intersections 669

Proposition 36.1.3. We then begin with

hH(T ; h1) =(

T + n

n

)−

(T + n − δ1

n

)= δ1

(T + n

n − 1

)+ · · ·

and inductively obtain

hH(T ; hl+1) = hH(T ; hl) − hH(T − δl+1; hl)

=(

l∏i=1

δi

) ((T + n

n − l

)−

(T + n − δl+1

n − l

))+ · · ·

=(

l+1∏i=1

δi

) (T + n

n − l − 1

)+ · · · .

Also (see Lemma 23.5.3)

hH(h1, T ) =∞∑

t=0

(t + n

n

)T t −

∞∑t=δ1

(t + n − δ1

n

)T t

= (1 − T )−n−1 − T δ1(1 − T )−n−1

= (1 − T δ1)(1 − T )−n−1

and, inductively,

hH(hl+1, T ) = hH(hl , T ) −h H(hl , T − δl+1)

= (1 − T δl+1)hH(hl , T )

= (1 − T )−n−1l+1∏i=1

(1 − T δi )

so that

hH(I, T ) = (1 − T )−n−1r∏

i=1

(1 − T δi )

= (1 − T )−n−1+rr∏

i=1

δi −1∑j=0

T j .

Corollary 36.1.6. For any homogeneous ideal I ⊂ k[X0, . . . , Xn], r :=r(I) = n + 1, dim(I) = 0, generated by a basis ( f1, . . . , fs), deg( fi ) ≤ D(I)for each i , we have

• γ (I) ≤ 1 + r(I)(D(I) − 1),

• k0(I) ≤ D(I)r(I).

670 Macaulay III

Proof. Let us consider r = n + 1 generic 3 linear combinations

gi :=s∑

j=1

λi j f j , 1 ≤ i ≤ r,

and let us write, for each l ≤ r, hl := (g1, . . . , gl).

Then, for almost all choices 4 of g1, . . . , gr we have

r(hl) = l,hl : gl+1 = hl for each l,g1, . . . , gr is a complete intersection, andhr ⊂ I.

Then

k0(I) ≤ k0(hr ) ≤ D(I)r

γ (I) ≤ γ (hr ) ≤ 1 + r(D(I) − 1).

36.2 The Coefficients of the Hilbert Function

Lemma 36.2.1. Let I ⊂ hP := k[X0, . . . , Xn] be a homogeneous ideal andIsat = ⋂

i qi be an irredundant primary representation of the saturation Isat

of I.Then there is at least a linear form

Y :=∑

j

c j X j ∈⋃

i

√qi , (c0, . . . , cn+1) ∈ kn+1 \ 0

and for any such linear form we have:

(1) (I : Y ∞) = Isat;(2) (I : Y ) = I ⇐⇒ Isat = (I : Y ∞) = (I : Y ) = I;(3) (I : Y ) = I ⇐⇒ Iirr = (X0, . . . , Xn);(4) (I : Y ) = I ⇐⇒ Y ∈ Iirr;(5) (Y ) (I : Y ) = I ∩ (Y ).

Proof. Denoting, for each c := (c0, . . . , cn) ∈ kn+1 \ 0, Yc the linear formYc := ∑

i ci Xi , each condition Yc ∈ √qi imposes constraints on kn+1 \ 0;

therefore there is a Zariski open set M ⊂ kn+1 such that

Yc ∈⋃

i

√qi for each c ∈ M.

3 That is let us consider any matrix L ∈ L where L denotes the set of all r × s matrices L := (λi j )with coefficients in the infinite field k.

4 That is there is a non-empty Zariski open set U ⊂ L such that for each L = (λi j ) ∈ U thestatement holds for gi := ∑s

j=1 λi j f j , 1 ≤ i ≤ r.

36.2 The Coefficients of the Hilbert Function 671

Then we have:

(1) Recall that for any linear form Y and any primary q we have(q : Y ∞) = (q : Y ) = q ⇐⇒ Y ∈ √

q,

(1) = (q : Y ∞) ⊇ (q : Y ) ⊃ q ⇐⇒ Y ∈ √

q, Y /∈ q,

(1) = (q : Y ∞) = (q : Y ) ⇐⇒ Y ∈ q.

Therefore, for any linear form Y ∈ ⋃i√

qi , we have

(I : Y ∞) = (

Iirr : Y ∞) ∩(⋂

i

(qi : Y ∞)) = (1) ∩

(⋂i

qi

)= Isat,

proving the first claim.(2) This then follows from the trivial equality

I = (I : Y ) ⇐⇒ I = (I : Y ∞)

.

(3) This follows from the maximality of Iirr and the homogeneity of I andIsat, giving

Iirr = (X0, . . . , Xn) ⇐⇒ I = Isat ∩ (X0, . . . , Xn) ⇐⇒ I = Isat.

(4) As a consequence we have

Y ∈ Iirr ⇒ Iirr = (X0, . . . , Xn) ⇐⇒ I = Isat ⇐⇒ (I : Y ) = I,

while, from

(I : Y ) = (Iirr : Y ) ∩ (∩i (qi : Y )) = (Iirr : Y ) ∩ Isat,

we obtain

Y ∈ Iirr ⇐⇒ (Iirr : Y ) = (1) ⇒ Isat = (I : Y ) ⇒ I = (I : Y ) .

(5) This follows directly by Lemma 26.3.6.

Corollary 36.2.2. Let I ⊂ hP := k[X0, . . . , Xn] be a homogeneous ideal.Then the following conditions are equivalent:

• Iirr = (X0, . . . , Xn);• there is a linear form Y such that (I : Y ) = I;• for almost all linear forms 5 Y := ∑

i ci Xi , we have (I : Y ) = I.

Let

hP := k[X0, . . . , Xn],

5 That is if, for each c := (c0, . . . , cn) ∈ kn+1 \ 0, Yc denotes the linear form Yc := ∑i ci Xi ,

there is a Zariski open set M ⊂ kn+1 such that the statement (I : Y ) = I holds for each linearform Yc for which c ∈ M.

672 Macaulay III

I ⊂ hP be a homogeneous ideal,Y0, Y1, . . . , Yn be a system of coordinates for hP

and let us define

I(0) := I,J(δ) := I(δ)sat , 0 ≤ δ ≤ dim(I),

L(δ) := I(δ)irr , 0 ≤ δ ≤ dim(I),I(δ+1) := I(δ) + (Yδ), 0 ≤ δ ≤ dim(I).

Lemma 36.2.3. With the notation above, we have:

(1) I(δ) = J(δ) ∩ L(δ), for each δ ≤ dim(I);(2) L(δ) is maximal among the irrelevant ideals satisfying (1), for each

δ ≤ dim(I);(3) I(δ+1) = I + (Y0, . . . , Yδ) for each δ ≤ dim(I);(4) in generic position 6

(a) dim(I(δ)) = dim(J(δ)) = dim(I) − δ, for each δ ≤ dim(I),(b) r(I(δ)) = r(J(δ)) = r(I), for each δ ≤ dim(I),(c)

(J(δ) : Yδ

) = J(δ) for each δ ≤ dim(I),(d)

(I(δ) : Yδ

) = I(δ) for each δ, 0 ≤ δ < depth(I),(e) Y0, . . . , Yλ−1, λ = depth(I), is a regular sequence for I,(f) I(δ) = J(δ), for each δ, 0 ≤ δ < depth(I),(g) L(δ) = (X0, . . . , Xn) for each δ, 0 ≤ δ < depth(I);

(5) I(d+1) = I + (Y0, . . . , Yd), d := dim(I) − 1, is irrelevant.

We are now able to present the characterization given by Macaulay ofthe coefficients of the Hilbert polynomial hHI(T ) of a homogeneous idealI ⊂ k[X0, . . . , Xn] =: hP; this discussion will also give a direct proof ofthe properties of the Hilbert function and polynomial already discussed, inparticular the relation

deg(hHI) + 1 = deg(HI) = dim(I).

Recall that, for an affine ideal I ⊂ k[X1, . . . , Xn] = P , we have

H(T ; I) = hH(T ; h I) and dim(h I) = dim(I) + 1

so these results can be directly extended to affine ideals giving

deg(HI) = deg(hHh I) = dim(h I) − 1 = dim(I).

6 That is there is a non-empty Zariski open set U ⊂ GL(n + 1, k) such that the statements holdfor each M := (ci j ) ∈ U and each Yi = ∑

j ci j X j .


We express again the Hilbert polynomials hHI(T ) in terms of the linear basis(T + i

i

): i ∈ N

,

obtaining the representation

hHI(T ) = k0

(T + d

d

)+ k1

(T + d − 1

d − 1

)+ · · · + kd

= k0(I)(

T + d

d

)+ k1(I)

(T + d − 1

d − 1

)+ · · · + kd(I)

and we will write

σ(I, T )) := hH(T ; I) − hHI(T ).

Let us begin by disposing of the extreme case of an irrelevant ideal (see alsoProposition 27.12.5) q by fixing any degree-compatible term ordering < andconsidering the set N<(q):

Lemma 36.2.4. With the notation above and denoting by ρ the characteristicnumber of q we have

hH(t; q) = #τ ∈ N<(q) : deg(τ ) = t,maxdeg(τ ) : τ ∈ N<(q) = ρ − 1,hHq = 0, andhHq(t) = hH(t; q) ⇐⇒ t ≥ ρ =: γ (q).

Corollary 36.2.5. If dim(I) = 1, let Y be a linear form such that I : Y = I andq := I + (Y ). Then for any degree-compatible term ordering < we have:

hH(t; I) = #τ ∈ N<(q), deg(τ ) ≤ t, for each t;γ (I) = maxdeg(τ ) : τ ∈ N<(q);k0(I) = #N<(q);hHI(T ) = k0(I)

(T0

);

for each t ∈ N, hHI(t) − hH(t; I) = #τ ∈ N<(q), deg(τ ) > t;σ(I, t) =

−#τ ∈ N<(q), deg(τ ) > t if t < γ (I),0 if t ≥ γ (I).

Let us now note the relation between the Hilbert functions of an ideal I andthat of its saturation:

Lemma 36.2.6. For the homogeneous ideal

I = Isat ∩ Iirr ⊂ hP

we have

674 Macaulay III

• hH(t; I) ≥ hH(t; Isat),• hH(t; I) = hH(t; Isat) if t ≥ γ (Iirr).

Proof. The result being trivial if Iirr = (X0, . . . , Xn), for which γ (Iirr) = 1 letus assume this is not the case; then 7 Iirr + Isat Iirr is also irrelevant and wehave, writing ρ := γ (Iirr),

Iirr + Isat ⊃ Iirr ⊃ (X0, . . . , Xn)ρ

so that

hH(t; Isat), ≥ hH(t; Iirr + Isat),hH(t; Isat) = hH(t; Iirr + Isat) = 0 if t ≥ ρ.

The claim then follows substituting these results into

hH(t; I) = hH(t; Iirr ∩ Isat) = hH(t; Isat) + hH(t; Iirr) − hH(t; Iirr + Isat).

We can now reformulate Proposition 36.1.3 as

Lemma 36.2.7. Let ∈ hP be a homogeneous linear form, that is deg() = 1,and let f ⊂ P be a homogeneous ideal. If f : = f, and we set g := f + ()

and d := dim(f) − 1, we have:

• ki (f) = ki (g), for each i < d,• kd(f) = ∑γ (g)−1

l=0 σ(g, l)),• γ (f) = γ (g) − 1.

Proof. Setting γ := γ (g), for each t ∈ N we have

hH(t; f) − hH(t − 1; f) = hH(t; g)

= hHg(t) − σ(g, t)

= k0(g)

(t + d − 1

d − 1

)+ · · · + k j (g)

(t + d − 1 − j

d − 1 − j

)+ · · · + kd−2(g)(t + 1) + kd−1(g) + σ(g, t),

7 Iirr + Isat = Iirr ⇒ Isat ⊂ Iirr ⇒ Iirr = (X0, . . . , Xn).


from which one gets8

hH(t; f) =(

t∑l=1

hH(l; f) − hH(l − 1; f)

)+ hH(0; f)

=t∑

l=0

H(l; g)

= k0(g)

t∑l=0

(l + d − 1

d − 1

)+ · · · + k j (g)

t∑l=0

(l + d − 1 − j

d − 1 − j

)+ · · ·

+t∑

l=0

kd−1(g) +t∑

l=0

σ(g, l))

= k0(g)

(t + d

d

)+ · · · + k j (g)

(t + d − j

d − j

)+ · · · + kd−1(g)(t + 1)

+ kd(f) −γ−1∑

l=t+1

σ(g, l)

where kd(f) = ∑∞l=0 σ(g, l) = ∑γ−1

l=0 σ(g, l), and we have

σ(f, t) := ∑γ−1

l=t+1 σ(g, l) for t < γ − 1,

0 for t ≥ γ − 1.

Applying this to the ideals J( j) and I( j), 0 ≤ j ≤ dim(I), we obtain

Theorem 36.2.8. With the notation above and assuming we are in generic po-sition, we have

(1) dim(I) = d + 1 = deg(hHI) + 1;(2) for each i ≤ d and each j < i

kd−i (I( j)) = kd−i (J( j)) = kd−i (J(i));

(3) k0(J(d)) = #(N<(J(d+1))) where < is any term ordering;(4) for each i < d

kd−i (I(i)) ≥ kd−i (J(i)) =γ (J(i−1))∑

l=0

HJ(i−1) (l) − H(l; J(i−1));

8 Using the combinatorial formula

t∑l=0

(l + i

i

)=

(t + i + 1

i + 1

).

676 Macaulay III

(5) kd−i (I(i)) = kd−i (J(i)) ⇐⇒ i < depth(I);(6) for γ := maxγ (L( j)), depth(I) ≤ j ≤ dim(I), we have hH(t; I) =

hHI(t), for each t ≥ γ ;(7) γ (I) ≤ maxγ (L( j)), depth(I) ≤ j ≤ dim(I).

Proof. For each j we have

I( j) = J( j) ∩ L( j),dim(I( j)) = dim(I) − j ,I( j) = J( j) ⇐⇒ j < depth(I),L( j) = (X0, . . . , Xn) ⇐⇒ j < depth(I).

Therefore

kd−i (I( j)) = kd−i (J( j)), for each i > j ,kd− j (I( j)) ≥ kd− j (J( j)), by Lemma 36.2.6, andkd− j (I( j)) = kd− j (J( j)) ⇐⇒ L( j) = (X0, . . . , Xn) ⇐⇒ j < depth(I).

Moreover if in Lemma 36.2.7 we set: g := J( j), := Y j , f := J( j−1), weobtain

kd−i (J( j−1)) = kd−i (J( j)), for each i > j

and we reduce the evaluation of each kd−i (J( j)) to the evaluation of the termskd−i (J(i)); if instead we set g := J(i+1), := Yi+1, f := J(i), we obtainkd−i (J(i)) in terms of kd−l(J(l)), l > i , and we reduce each evaluation to thatof k0(J(d)).

This is done by applying Corollary 36.2.5 which gives

hHJ(d) (T ) = k0(J(d)) = #N<(J(d+1))

and completes the evaluation of each kd−i (I( j)).In these iterative computations the differences between the Hilbert polyno-

mials and the corresponding Hilbert functions are due to the contribution ofL( j) (see Lemma 36.2.6); we therefore obtain, for any t ≥ γ (L( j)),

hH(t; I( j)) = hH(t; J( j)) = hHI( j) (t)

so that

γ (I( j)) ≤ max(γ (J( j)), γ (L( j))

)≤ max

γ (L( j)), depth(I) ≤ j ≤ dim(I)

.


Corollary 36.2.9. For a homogeneous ideal I ⊂ k[X0, . . . , Xn] its Hilbertpolynomial

hHI(T ) =d∑

i=0

ki (I)(

T + d − i

d − i

)

satisfies

deg(hHI) = d = dim(I) − 1,ki (I) ∈ Z, for each i ,k0(I) > 0.

For an affine ideal I ⊂ k[X1, . . . , Xn] its Hilbert polynomial

HI(T ) =d∑

i=0

ki (I)(

T + d − i

d − i

)

satisfies

deg(HI) = d = dim(I),ki (I) ∈ Z, for each i ,k0(I) > 0.

Example 36.2.10. Let us consider hP = k[Y0, Y1, Y2, Y3] and I = (Y 43 ) so that

dim(I) = depth(I) = 3, r(I) = 1, d = 2.

We have

I(3) = (Y0, Y1, Y2, Y 43 ), N(I(3)) = 1, Y3, Y 2

3 , Y 33 ,

I(2) = (Y1, Y2, Y 43 ), H(t; I(2)) =

t + 1 iff 0 ≤ t ≤ 2,

4 iff t > 2,HI(2) = 4 = 4

(T0

),

γ (I(2)) = 3,∑

t σ(I(2), t) = −6,

I(1) = (Y2, Y 43 ), H(t; I(1)) =

⎧⎨⎩

1 iff t = 0,

3 iff t = 1,

4t − 2 iff t > 1,

HI(1) = 4T − 2 = 4(T +1

1

) − 6(T

0

),

γ (I(1)) = 2,∑

t σ(I(2), t) = 4,

I(0) = (Y 43 ), H(t; I(0)) =

1 iff t = 0,

2t2 + 2 iff t > 0,

HI(0) = 2T 2 + 2 = 4(T +2

2

) − 6(T +1

1

) + 4(T

0

),

γ (I(0)) = 1.

678 Macaulay III

36.3 Perfectness

Let us use the same notation as before: in particular, let us consider

hP := k[X0, . . . , Xn],I ⊂ hP a homogeneous ideal,Y0, Y1, . . . , Yn a system of coordinates for hP ,d := dim(I) − 1, r := n−d = r(I), λ := depth(I),I(δ) := I + (Y0, . . . , Yδ−1), 0 ≤ δ ≤ d + 1 = dim(I).

In connection with Theorem 36.2.8, it is easy to deduce that

Corollary 36.3.1. The following conditions are equivalent

(1) dim(I) = depth(I),(2) I(δ) = I(δ)sat for each δ ≤ d = dim(I) − 1:(3) I(δ)irr = (X0, . . . , Xn) for each δ ≤ d = dim(I) − 1.

Moreover these equivalent conditions imply that knowledge of the set 9

#(N(I(d+1))), where < is any term ordering, is sufficient to compute hHI:

Proposition 36.3.2. With respect to the degrevlex ordering < induced by Yn <

. . . < Y0 the following conditions are equivalent:

(1) dim(I) = depth(I);

(2) k[Y0, . . . , Yn] = I ⊕∑

τ∈N(I(d+1)) bτ τ, bτ ∈ k[Y0, . . . , Yd ]

;

(3) for each term ω ∈ k[Y0, . . . , Yd ] and each term τ ∈ k[Yd+1, . . . , Yn]

ωτ ∈ T(I) ⇒ τ ∈ T(I).

Definition 36.3.3 (Macaulay). The ideal I is called perfect if it satisfies theconditions of Proposition 36.3.2.

Historical Remark 36.3.4. The notion of perfectness, which we have alreadydiscussed in Section 30.5, on the basis of his book – where the notion is directlyrelated to condition (3) of Proposition 36.3.1 – was introduced by Macaulay,on the basis of condition (2), in connection with his study of the structure ofthe Hilbert function described in the section above in his 1913 paper, where hewrote:

The H-module (M, xr+1, . . . , xn) is to all intents and purposes the same as the modulein r variables obtained from M by putting xr+1 = · · · = xn = 0. In particular theHilbert numbers of the two modules for any degree are equal. If (M, xr+1, . . . , xn) isa given simple H-N-module [i.e. an irrelevant ideal] and we regard M as being built

9 Remember (see Historical Remark 30.4.17) that Macaulay has explicitly the concept of linearrepresentation so applying the notation of Grobner theory is not a strain.

36.3 Perfectness 679

up from (M, xr+1, . . . , xn), then the Hilbert numbers and Hilbert function of M havecertain higher limits which can be reached but not exceeded. The module M will becalled a perfect module if its Hilbert function reaches its higher limit.A K-module [i.e. an affine ideal] is called perfect if its equivalent H-module is perfect;but, for the sake of clearness, we shall only consider H-modules. That a perfect H-module can be built up from any given simple H-N-module [i.e. an irrelevant ideal]follows from the fact that a simple H-N-module in r variables x1, x2, . . . , xr becomesa perfect H-module in n variables on changing xr to

xr + ar+1xr+1 + · · · + an xn .

To prove the property mentioned above, let H(l), Hl denote the Hilbert numbers ofM and (M, xn) for degree l, and χ(l), χl the Hilbert functions.10 Then H(l) is thenumber of independent modular equations of (M, xn) of degree l, added to the numberof independent modular equations of M/(xn) of degree l − 1 [ . . . ]. The former numberis Hl , and the latter ≤ H(l − 1). Hence

H(l) = Hl + H(l − 1) − αl ,

where αl is a positive integer, which is not zero for all values of l except in the case thatM/(xn) = M , that is, the case when M does not contain a relevant simple N-module[i.e. a zero-dimensional ideal].[11] Thus

H(l) = (H0 + H1 + · · · Hl ) − (α1 + α2 + · · · + αl ).

Hence the highest limit of H(l) regarded as depending on (M, xn) is H0 + H1 +· · · Hl .

Also the highest limit of χ(l) is H0 + H1 +· · · Hl , when l is taken large enough;[12] butthe actual value of χ(l) is less than this by a constant, equal to the sum of all the α’s; forαl is zero when l is large enough. From this it follows that χ(l), regarded as dependingon (M, xr+1, . . . , xn) reaches its highest limit when, and only when, no-one of themodules M, (M, xn), . . . (M, xr+2, . . . , xn) contains a relevant simple N-module, andin this case all the Hilbert numbers of M also reach their highest limits.F. S. Macaulay, On the Resolution of a given Modular System into Primary Systemsincluding some Properties of Hilbert Numbers, Math. Ann. 74 (1913), Section 66,pp. 114–5.

In order to read correctly Macaulay’s quotation we need to relate it to thenotation we are using; in these quotations, Macaulay relates the homogeneousideal M ⊂ k[x1, . . . , xn] to two other ideals

(M, xr+1, . . . , xn), andthe ideal 13 Mxr+1= ··· =xn=0 in k[x1, . . . , xr ] obtained by setting xr+1 = · · · =

xn = 0.

10 Macaulay calls ‘Hilbert numbers’ what we call ‘Hilbert function’ and ‘Hilbert function’ whatwe call ‘Hilbert polynomial’.

11 Here Macaulay formulates Lemma 36.2.7 where M = f and xn = ; if M/(xn) = M , that isf : = f, αl is the contribution of girr.

12 I have the impression that this is the first introduction of the notion of ‘index of regularity’ andof the implicit formula γ (I) ≤ maxγ (L( j)), depth(I) ≤ j ≤ dim(I).

13 This is Macaulay’s notation.

680 Macaulay III

If we begin with M = I ⊂ k[X0, . . . , Xn−1],14 dim(I) = d = n − r , ifY0, Y1, . . . , Yn−1 is generic, we know that I∩ k[Y0, . . . , Yd−1] = 0 and, foreach i, 1 ≤ i ≤ r , there is a monic polynomial

gi ∈ k(Y0, . . . , Yd−1)[Yn−i ] such that Prim(gi ) ∈ I

and we can renumber the variables 15 as x1, . . . , xn where xi := Yn−i for eachi so that, if dim(I) = depth(I) = n − r, xn, . . . , xr+1 is a regular sequence; inconnection with this notation Macaulay also introduced the ideal

M (r) := Mk(xr+1, . . . , xn)[x1, . . . , xr ] ∩ k[x1, . . . , xn].

Therefore, for M = I, what Macaulay denoted

• (M, xr+1, . . . , xn) is what I denote I(d) = I + (Y0, . . . , Yd−1);• the second ideal is the image π(I) of I under the projection

π : k[Y0, . . . , Yn−1] → k[Yd , . . . , Yn−1]

defined by

π( f ) = f (0, . . . , 0, Yd , . . . , Yn−1] for each f ∈ k[Y0, . . . , Yn−1]

• and M (r) is

Iec = Ik(Y0, . . . , Yd−1)[Yd , . . . , Yn−1] ∩ k[Y0, . . . , Yn−1].

In connection with these objects Macaulay remarked that:

Lemma 36.3.5. With the notation above

π(I) = I(d) ∩ k[Yd , . . . , Yn−1].

Proof. For any element f ∈ k[Y0, . . . , Yn−1] there is a unique element g ∈k[Yd , . . . , Yn−1] and there are elements h0, . . . , hd−1 ∈ k[Y0, . . . , Yn−1] suchthat

f = g +d−1∑i=0

hi Yi .

14 Unlike the current usual notation, Macaulay considered homogeneous ideals in polynomialrings with no homogenizing variable. The curious enumeration is justified by the note below.

15 It is perhaps fascinating and probably not misleading re-interpreting these operations in termsof Grobner technology: in order to detect dim(I) we need to compute a Grobner basis of I w.r.t.the lexicographical ordering induced by Y0 < Y1 < · · · < Yn−1. Under the renumberingxi := Yn−i the same ordering becomes the degrevlex ordering induced by x1 < · · · < xn .

This justifies that Proposition 36.3.2 is stated for degrevlex ordering < induced by Yn <

· · · < Y0; for homogeneous ideals this coincides with the lexicographical ordering induced byY0 < Y1 < · · · < Yn .


Then, for each f ∈ I,

π( f ) = g = f −d−1∑i=0

hi Yi ∈ I(d) ∩ k[Yd , . . . , Yn−1].

Conversely if f ′ ∈ I(d) ∩ k[Yd , . . . , Yn−1] then there are f ∈ I andh′

0, . . . , h′d−1 ∈ k[Y0, . . . , Yn−1] such that

f ′ = f +d−1∑i=0

h′i Yi .

Also, there is a unique g ∈ k[Yd , . . . , Yn−1] and there are elements

h0, . . . , hd−1 ∈ k[Y0, . . . , Yn−1] : f = g +d−1∑i=0

hi Yi .

In conclusion f ′ = g + ∑d−1i=0 (hi + h′

i )Yi , whence

f ′ ∈ I(d) ∩ k[Yd , . . . , Yn−1] ⇒ f ′ = g = π( f ) ∈ π(I).

Let us now recall that for each i, 1 ≤ i ≤ r , there is a monic polynomialgi ∈ M (r) ∩ k(xr+1, . . . , xn)[xi ]; therefore we know that 16

M (r) = Mxr+1= ··· =xn=0

has a linear representation. More precisely it consists of a subset of

xa11 · · · xar

r : ai < deg(gi ).If we now impose on k[x1, . . . , xn] the degrevlex ordering 17 induced by

x1 < · · · < xn we have an extra bonus (Lemma 26.3.12):

xi | T<(g) ⇒ xi | g for each g ∈ k[xi , . . . , xn].

Proof of Proposition 36.3.2.

(1) ⇒ (2) We only need to prove that there is no

0 = g :=∑

τ∈N(I(d+1))

bτ τ ∈ I, bτ ∈ k[Y0, . . . , Yd ].

For any such g there is some τ ∈ N(I(d+1)) for which T<(g) =T<(bτ )τ. Moreover, there are h ∈ k[Yd+1, . . . , Yn] and hi ∈k[Yi , . . . , Yn] such that g = h + ∑d

i=0 hi Yi .

16 The equality is just a reformulation of Lemma 36.3.5.17 Which, by the way, on the basis of the previous footnote is the more natural choice.

682 Macaulay III

Now the property of < stated in Lemma 26.3.12 implies that

Y a00 · · · Y ad

d = ω := T<(bτ ) | g.

Also dim(I) = depth(I), implying, for δ ≤ d,

I(δ) : Yδ = I(δ),

gives that

Y −a00 g = Y −a0

0 h +d∑

i=0

Y −a00 hi Yi ∈ I

so that

Y −a00 h +

d∑i=1

Y −a00 hi Yi ∈ I(1)

and, recursively, that, for each j ,

Y −a00 . . . Y

−a jj h +

d∑i= j+1

Y −a00 . . . Y

−a jj hi Yi ∈ I( j+1),

thus allowing us to conclude that

h′ := ω−1h ∈ I(d+1) and T<(h′) = ω−1T<(h) = τ ∈ N(I(d+1)),

giving the required contradiction w.r.t. the assumption ωτ = T<(g) ∈T(I).

(2) ⇒ (3) Let

g :=∑

υ∈T [d+1,n]

bυυ ∈ I, bυ ∈ k[Y0, . . . , Yd ]

be such that g ∈ I and T<(g) = ωτ , so that τ = max<(υ : bυ = 0)

and ω = T<(bτ ). Since g ∈ I, then

g /∈⎧⎨⎩

∑τ∈N(I(d+1))

bτ τ, bτ ∈ k[Y0, . . . , Yd ]

⎫⎬⎭ ,

whence τ /∈ N(I(d+1)), τ ∈ T(I(d+1)) ∩ T [d + 1, n] and τ ∈ T(I).(3) ⇒ (1) Assume that (I : Y0) I; then, necessarily, there is some ele-

ment f ∈ (I : Y0) such that τ := T( f ) /∈ T(I); this gives a contradic-tion since Y0 f ∈ I, Y0τ ∈ T(I) and, by (3), τ ∈ T(I).This is sufficient to perform induction: we can assume that I(i) : Yi

= I(i) for each i < δ ≤ d and let us prove that I(δ) : Yδ = I(δ); if this


is not the case we can choose some element f ∈ I(δ) : Yδ such thatτ := T( f ) /∈ T(I(δ)); f can be expressed as

f = g +δ−1∑i=0

hi Yi

for suitable g ∈ k[Yδ, . . . , Yn] and h0, . . . , hδ−1 ∈ k[Y0, . . . ,

Yn].Since, under the degrevlex ordering <, we have, for each i < δ, Yi |T( f ) implies Yi | f , Y −1

i f ∈ I(i) : Yi = I(i), we can wlog assumethat τ = T( f ) = T(g) ∈ T [δ, n]. We then obtain again the samecontradiction as above: Yδ f ∈ I(δ), Yδτ ∈ T(I(δ)) and, by (3), τ ∈T(I(δ)).

The construction performed in the proof above requires some comments: let

Y := Y a00 . . . Y an

n : (a0, . . . , an) ∈ Nn+1;

let < be the degrevlex ordering induced by Y0 < Y1 < · · · < Yn and let usdenote, for each i , by <i its restriction to k[Yi , . . . , Yn]; if we have a homoge-neous polynomial f ∈ k[Y0, . . . , Yn] (see Lemma 23.1.4), there are homoge-neous polynomials M(g) ∈ k[Y1, . . . , Yn] and R(g) ∈ k[Y0, . . . , Yn] and aninteger a0 such that

g = Y a00 (M(g) + Y0R(g)),

M(g) = H(ag),deg(g) = a0 + deg(M(g)) = a0 + 1 + deg(R(g)),

T<(g) = Y a00 T<(M(g));

therefore, to any homogeneous ideal I ⊂ k[Y0, . . . , Yn] we can associate thehomogeneous ideal M(I) = H(a I) ⊂ k[Y1, . . . , Yn].

Since, for each t1, t2 ∈ Y , we have

t1 < t2 ⇐⇒ deg(t1) < deg(t2) or deg(t1) = deg(t2),at1 <1

at2,

by Corollary 23.2.8 the Grobner basis of M(I) w.r.t. <1 computationally liftsto the Grobner basis of I w.r.t. <.

The operation can, of course, be iterated, considering Yδ as a homogenizingvariable of k[Yδ, . . . , Yn], at least while δ < d + 1 = depth(I).

Therefore writing

M0(g) := M(g), R0(g)) := R(g)),

684 Macaulay III

Mδ(g) := M(Mδ−1(g)), Rδ(g) := R(Mδ−1(g)) for each δ < d + 1 =depth(I),

we obtain:

Lemma 36.3.6. Any homogeneous polynomial g ∈ K [Y0, . . . , Yn] can beuniquely expressed as

g = Y a00 . . . Y ad

d

(Md(g) +

d∑i=0

YiRi (g)

),

Md(g) ∈ k[Yd+1, . . . , Yn] and Ri (g) ∈ k[Yi , . . . , Yn], for each i. Moreover

T<(g) = Y a00 . . . Y ad

d T<(Md(g)).

Moreover, if we associate to I ⊂ k[Y0, . . . , Yn], depth(I) = d + 1, the ideal

Md(I) := Md(g) : g ∈ I ⊂ k[Yd+1, . . . , Yn],

since, for each i and each t1, t2 ∈ Y ∩ k[Yi , . . . , Yn], we have

t1 <i t2 ⇐⇒ deg(t1) < deg(t2) or deg(t1) = deg(t2),at1 <i+1

at2,

we can iteratively apply Corollary 23.2.8; therefore the Grobner basis of Md(I)w.r.t. < computationally lifts iteratively to the Grobner basis of I w.r.t. <.

Also, in this context, denoting by

π : k[Y0, . . . , Yn] → k[Yd+1, . . . , Yn]

the projection defined by

π( f ) = f (0, . . . , 0, Yd+1, . . . , Yn] for each f ∈ k[Y0, . . . , Yn],

Lemma 36.3.5 can be reformulated as

Corollary 36.3.7. With the notation above,

Md(I) = π(I) = π(I(d+1)) = I(d+1) ∩ k[Yd+1, . . . , Yn].

Remark 36.3.8. Macaulay’s construction therefore also answers the query Iposed on Remark 23.10.4.

If I ⊂ k[X1, . . . , Xn] = k[Y1, . . . , Yn] – where Y1, . . . , Yn is a genericsystem of coordinates – is an affine ideal, depth(h I) = λ, given by a basisG then in order to compute the Grobner basis of I w.r.t. the degree reverse


lexicographical ordering induced by Y1 < · · · < Yn , it is sufficient to computethe Grobner basis of

Mλ−1(I) = (Mλ−1(g), g ∈ G) ⊂ k[Yλ, . . . , Yn]

and iteratively lift it.Alternatively, in the frame of Section 34.3 and Section 35.5, one can com-

pute the Grobner basis of Ie in K (X1, . . . , Xd)[Yd+1, . . . , Yn]; naturally, ifX1, . . . , Xd are generic 18 and depth(h I) = dim(h I) so that

λ = depth(h I) = dim(h I) = dim(I) + 1 = d + 1,

the two computations are essentially equivalent.

18 They are not, in order to avoid denseness.

37

Galligo

Throughout this chapter I assume char(k) = 0.Within the framework of Hironaka’s theory, Galligo gave a strong and fruit-

ful description of the structure of T<(I): he considered all changes of coordi-nates M ∈ GL(k, n) over the polynomial ring P := k[X1, . . . , Xn] and provedthat the generic initial ideal

ε(I) = T<(M(I))

is stable within a non-empty Zariski open set of GL(k, n) and described thestructure of the corresponding generic escalier T \ ε(I) = N<(M(I)).

This chapter is devoted to Galligo’s Theorem: I begin by stating (Sec-tion 37.1) the result, introducing notation and informally discussing the ar-gument before giving a formal proof.

The crucial property of the generic escalier is that it is a Borel ideal, that isa monomial ideal stable under Borel transformations:

ε(I) = M(ε(I)), for each M ∈ B(n, k);Section 37.2 introduces notation, informally discusses the property on ele-mentary examples and introduces Gjunter–Marinari combinatorial notation todeal with Borel ideals; Section 37.3 gives a proof of Galligo’s result and Sec-tion 37.4 describes the structure of the generic escalier deducible from it.

Finally Section 37.5 is devoted to the resolution deduced by Eliahou andKervaire for stable monomial ideals, a class including Borel ideals.

37.1 Galligo Theorem (1): Existence of Generic Escalier

Let us consider, using the same notation as in Remark 24.5.5 (respectivelyRemark 24.6.14), the graded (respectively valuation) ring

P := k[X1, . . . , Xn] respectively P := k[[X1, . . . , Xn]]

686

37.1 Galligo Theorem (1) 687

having the graduation (respectively valuation) vw induced by the weight vector

w := (w1, . . . , wn) ∈ Rn, wi ≥ 0, respectively wi ≤ 0.

Let us now consider on P any term ordering < and let us denote by ≺ therefinement of vw with < defined by

t1 ≺ t2 ⇐⇒ vw(t1) < vw(t2) or vw(t1) = vw(t2), t1 < t2.

Let GL(n, k) be the general linear group, that is the set of all invertiblen × n square matrices M := (

ci j)

with entries in k.For any matrix M := (

ci j) ∈ GL(n, k) we will still denote M the linear

transformation M : P → P defined by

M(Xi ) =∑

jci j X j for each i.

In this setting, Galligo’s Theorem describes the behaviour of T(I) when I istransformed by the application of a generic M ∈ GL(n, k).

Theorem 37.1.1 (Galligo). For any ideal I ⊂ P , there are a non-emptyZariski open set U ⊂ GL(n, k) and a monomial ideal ε(I) such that

ε(I) = T(M(I)), for each M ∈ U.

Our aim in this section is not only to give a proof of Galligo’s Theorembut also to present the structural properties of ε(I) which his seminal paper 1

highlighted.Let us first remark that since we have T≺(I) = T≺(Lw(I)), it is sufficient to

restrict the problem to the case of homogeneous ideals 2

I ⊂ k[X1, . . . , Xn] =: P.

Following Galligo 3 we will assume < satisfies X1 < X2 < · · · < Xn ; note

1 A. Galligo, A propos du theoreme de preparation de Weierstrass, L. N. Math. 409 (1974),Springer, 543–579.

2 More generally, when P := k[[X1, . . . , Xn ]], the same theorem holds considering, instead ofGL(n, k), the group of all automorphisms of P . Clearly also in this case it is sufficient to restrictoneself to the case of homogeneous ideals and linear changes of coordinates.

3 This is not a support of my left-brained choice but it is due to Buchberger’s parity switch; infact in his result, which is completely independent of Buchberger’s and depends on Hironaka’s,Galligo used as ordering < the lex ordering induced by Xn < · · · < X1, but, within the frame ofHironaka’s standard basis, he, a la Macaulay, considered as leading term the minimal monomial.

Therefore, if his result is read within Grobner theory, it applies to the (deg)revlex orderinginduced by X1 < · · · < Xn .

Most of the statements will therefore be stated for an ordering such that X1 < X2 < · · · <

Xn , but the relevant ones will also be stated for the orderings such that X1 > X2 > · · · > Xn .

688 Galligo

that, since, for each f ∈ P homogeneous, T≺( f ) = T<( f ), we have alsoX1 ≺ · · · ≺ Xn .

In this setting let us recall the usual notation: T will denote the set of mon-omials in k[X1, . . . , Xn],

T := Xa11 . . . Xan

n , (a1, . . . , an) ∈ Nn,

and, for each d ∈ N, and any set W ⊂ P , Wd will denote the set of allhomogeneous polynomials f ∈ W such that vw( f ) = d. In particular

Td := µ ∈ T : vw(µ) = d,Pd := Spank(Td), Id = I ∩ Pd .

Needing to use the set of the terms generated by some subsets of variables,we denote for each i, j, 1 ≤ i < j ≤ n, T [i, j] the monomials generated byXi , . . . , X j ,

T [i, j] =

Xaii . . . X

a jj , (ai , . . . , a j ) ∈ N

j−i+1

,

and T [i, j]d denotes those terms of degree d .We will also use the Hilbert function hH(d; I) := #Td −#Id which of course

satisfieshH(d; I) = hH(d;M(I)), for each M ∈ GL(n, k).

We will finally use the shorthand k[Xi j ] and k(Xi j ) to denote, respectively,the polynomial ring generated over k by the variables


For each χ, 1 ≤ χ ≤ n, we will denote by

φχ : k[X1, . . . , Xn] → k[Xχ+1, . . . , Xn]

the projection defined 4 by

φχ( f ) = f (1, . . . , 1, Xχ+1, . . . , Xn)

and we will set φn( f ) = 1, for each f.When it is possible, we will illustrate the structure of ε(I) by figures analo-

gous to the ones used in Examples 21.2.4 and 22.3.1 5 when

P = k[T, X, Y ] = k[X1, X2, X3], T < X < Y.

4 Note that φ0( f ) = f, for each f.5 Not casually: the presentation of Buchberger’s theory used in this book is strongly indebted to

Galligo.


Most of the figures will just describe the structure of the monomials inT [2, 3], that is the subset T a1 Xa2 Y a3 ∈ T , a1 = 0; ‘geometrically’ T is theaxis perpendicular to the illustrated plane, which in this context is the planeT = 0; similar figures can however describe:

• the subset T a1 Xa2 Y a3 ∈ T , a1 = d or the plane T = d for some d > 0;• the ‘generic’ subset T a1 Xa2 Y a3 ∈ T , a1 = d and plane T = d for all

d 0;• the ‘projection along the T -axis’ of a subset W ⊂ T ,

(a2, a3) : there exists a1 : T a1 Xa2 Y a3 ∈W = φ1(W) ⊂ T [2, 3],

where, according to the definition above, φ1 : k[T, X, Y ] → k[X, Y ] is theprojection defined by φ1( f ) = f (1, X, Y ).

Let δ(1) ≥ 1 be the minimal value such that Iδ(1) = 0. This implies thatvw( f ) ≥ δ(1) for each f ∈ I and the existence of some f1 ∈ Iδ(1).

Let us consider a generic change of coordinates M = (ci j ) ∈ GL(n, k);clearly there are polynomials Ct (Xi j ) ∈ k[Xi j ] indexed by the terms t ∈ Tδ(1),such that

M( f1) =∑

t∈Tδ(1)

Ct (ci j )t, for each M = (ci j ) ∈ GL(n, k).

Write

µ1 := max<t ∈ Tδ(1) = X δ(1)n ,

P1(Xi j ) := Cµ1(Xi j ) ∈ k[Xi j ]\0 so that, for each M = (ci j ) ∈ GL(n, k),

P1(ci j ) = 0 ⇐⇒ T(M( f1)) = µ1,

U1 := M ∈ GL(n, k) : P1(ci j ) = 0 = M ∈ GL(n, k) : T(M( f1)) = µ1,J1 := ( f1),

M1 := (µ1),L1 := (µ1) andN1 := T \ M1

so that we have

• U1 is a non-empty Zariski open set,• M1 = T(M(J1)), for each M ∈ U1,

• L1 = µ1t : t ∈ T ,• T = L1 N1

690 Galligo

and we have...

......

......

......

...

• • • • • • • • · · ·• • • • • • • • · · ·•Y δ(1) • • • • • • • · · · · · · · · · · · · · · · · · ·

where

represents the terms t ∈ N1,• represents the terms t ∈ L1.

Now there are two possibilities: either

#Iδ(1) > 1, or#Iδ(1) = 1.

In the first case, in which we set δ(2) := δ(1), there is at least one polyno-mial f ∈ Iδ(2) which is linearly independent with f1.

Again, for any such polynomial f , there are polynomials Dt f (Xi j ) ∈ k[Xi j ]indexed by the terms t ∈ Tδ(2), such that

M( f ) =∑

t∈Tδ(2)

Dt f (ci j )t, for each M = (ci j ) ∈ GL(n, k).

In particular, unless Dµ1 f (ci j ) = 0, we will have T(M( f )) = µ1; in anycase, noting that P1(ci j ) = 0 for each M ∈ U1, for any such f and any suchM, we can consider the polynomial

R( f, M) := M( f )− Dµ1 f (ci j )P1(ci j )−1M( f1)

=∑

t∈Tδ(2)

P1(ci j )−1

(P1(ci j )Dt f (ci j )− Ct (ci j )Dµ1 f (ci j )

)t.

Remark 37.1.2. Clearly, T(R( f, M)) < µ1 so that T(R( f, M)) ∈ (N1)δ(2);may we state

T(R( f, M)) = max<t ∈ (N1)δ(2) =: τ?

Of course, this would happen iff

P1(ci j )Dτ f (ci j )− Cτ (ci j )Dµ1 f (ci j ) = 0;


clearly, there could be some f and M for which this is false, but it could betrue for a proper choice of f and for most choices of M; so we can try toreformulate our question defining

µ2 := max<T(R( f, M)) : f ∈ Iδ(2), M ∈ U1.

The question now becomes whether

µ2 = max<t ∈ (N1)δ(2)?

We will see later that, while the answer is still negative, µ2 will be a minimalelement in (N1)δ(2) under a suitable partial ordering→ satisfying ν → µ ⇒ν > µ.

We postpone the discussion of that to the next section; our temporary aim isjust to reduce the proof of Theorem 37.1.1 to that of a (weak) lemma.

In the next section we will then state a stronger version of that lemma, proveit and deduce from it the structural properties of ε(I).

Therefore we limit ourselves to writing

µ2 := max<T(R( f, M)) : f ∈ Iδ(2), M ∈ U1,f2 ∈ Iδ(2) for a ‘suitable’6 element such that µ2 = T(R( f2, M)) for some

M ∈ U1,P2(Xi j ) := P1(Xi j )Dµ2 f2(Xi j )−Cµ2(ci j )Dµ1 f2(Xi j ) ∈ k[Xi j ]\0, so that

for each M = (ci j ) ∈ U1 since P1(ci j ) = 0, we have

P2(ci j ) = 0 ⇐⇒ T(M( f2)) = µ2,

U2 := M ∈ U1 : Pl(ci j ) = 0, 1 ≤ l ≤ 2= M ∈ U1 : T(M( fl)) = µl , 1 ≤ l ≤ 2,

J2 := ( f1, f2),

M2 := (µ1, µ2),L2 := M2 \ M1, andN2 := T \ M2,

so that we have

U2 ⊆ U1 is a non-empty Zariski open set,M2 = T(M(J2)), for each M ∈ U2,

6 In order to be able to prove the required formula

t → µ2, t ∈ (N1)δ(2) ⇒ t = µ2

and deduce from it the structural properties of ε(I) we will need to impose some further satisfi-able conditions on f2 and even f1; but this will be the argument of the next section.

In order to prove only Theorem 37.1.1, we only need

there exists M ∈ U1 : µ2 = T(R( f, M)).

692 Galligo

L2 = µ2t : t ∈ N1,T = N2 L1 L2,M2 ⊃ M1.

It is best to stress immediately the role of the property µ2 = max<t ∈(N1)δ(2); in the same figure as above we have

max<t ∈ (N1)δ(2) = XY δ(1)−1

and we have...

......

......

......

...

• • • • • • • • · · ·• • • • • • • • · · ·•Y δ(1) • • • • • • • · · · XY δ(1)−1 · · · · · · · · · · · · · · ·

where

represents the terms t ∈ N2,• represents the terms t ∈ L1, represents the terms t ∈ L2.

If it happens that µ2 = max<t ∈ (N1)δ(2), and for example µ2 :=X2Y δ(1)−2, the figure would be

......

......

......

......

• • • • • • • • · · ·• • • • • • • • · · ·•Y δ(1) • • • • • • • · · · · · · X2Y δ(1)−2 · · · · · · · · · · · ·

Let us now consider the second case, in which #Iδ(1) = 1 and in which wemust compare hH(d; I) with hH(d;M1), recalling that

T(M(J1)) = M1 ⊆ T(M(I))

for each M ∈ U1:


if, for each d ≥ δ(1), hH(d; I) = hH(d;M1) then we are through since, foreach M ∈ U1, T(M(J1)) = M1 = T(M(I)) and M( f ) generates M(I);

otherwise, there is a minimal value δ(2) > δ(1) such that hH(δ(2); I) <hH(δ(2);M1).

In this case, for each d, δ(1) ≤ d < δ(2), since M1 = T(M(J1)) we deducefrom hH(d; I) = hH(d;M1) that, for each M ∈ U1,

M(J1)d = SpanktM( f1) : t ∈ Td−δ(1) = M(Id);this implies, in particular, that the canonical form of any element M( f ), f ∈ Idw.r.t. M( f1) is 0:

Can(M( f ), M(J1), <) : f ∈ Id = 0.For δ(2), since hH(δ(2); I) < hH(δ(2);M1) this is no longer true; how-

ever, for each element f ∈ Iδ(2), and for each M ∈ U1 one can computeCan(M( f ), M(J1), <) which will be a combination of terms t ∈ (N1)δ(2);moreover since hH(δ(2); I))− hH(δ(2);M1) < 0, we can deduce that the vec-torspace

Can(M( f ), M(J1), <) : f ∈ Iδ(2) = 0.Therefore if we consider any polynomial f ∈ Iδ(2), we can deduce that thereare polynomials Dt f (Xi j ) ∈ k[Xi j ] indexed by the terms t ∈ Tδ(2) and a valuer( f ) ∈ N such that, for each M = (ci j ) ∈ U1,

R( f, M) := Can(M( f ), M(J1), <) =∑

t∈(N1)δ(2)

P1(ci j )−r( f ) Dt f (ci j )t.

As in the previous case, we cannot claim that there is a proper choice of fand M such that T(R( f, M)) = max<t ∈ (N1)δ(2) but just

t → T(R( f, M)), t ∈ (N1)δ(2) ⇒ t = T(R( f, M))

and we limit ourselves to setting

µ2 := max<T(R( f, M)) : f ∈ Iδ(2), M ∈ U1,f2 ∈ Iδ(2) to be a ‘suitable’ element such that µ2 = T(R( f2, M)) for some

M ∈ U1,P2(Xi j ) := Dµ2 f2(Xi j ) ∈ k[Xi j ]\ 0, so that for each M = (ci j ) ∈ U1 since

P1(ci j ) = 0, we have

P2(ci j ) = 0 ⇐⇒ T(M( f2)) = µ2,

U2 := M ∈ U1 : Pl(ci j ) = 0, 1 ≤ l ≤ 2= M ∈ U1 : T(M( fl)) = µl , 1 ≤ l ≤ 2,

694 Galligo

J2 := ( f1, f2),

M2 := (µ1, µ2),L2 := M2 \ M1 andN2 := T \ M2,

so that we have

U2 ⊆ U1 is a non-empty Zariski open set,M2 = T(M(J2)), for each M ∈ U2,

L2 = µ2t : t ∈ N1,T = N2 L1 L2,M2 ⊃ M1.

In this situation, the corresponding figures in which

µ2 = max<t ∈ (N1)δ(2) = X δ(2)−δ(1)+1Y δ(1)−1, andµ2 = max<t ∈ (N1)δ(2) and we assume µ2 = X δ(2)−δ(1)+2Y δ(1)−2

are, respectively

......

......

......

......

• • • • • • • • · · ·• • • • • • • • · · ·•Y δ(1) • • • • • • • · · · X δ(2)−δ(1)+1Y δ(1)−1 · · · · · · · · · · · · · · ·

and

......

......

......

......

• • • • • • • • · · ·• • • • • • • • · · ·•Y δ(1) • • • • • • • · · · · · · X δ(2)−δ(1)+2Y δ(1)−2 · · · · · · · · · · · ·


where

represents the terms t ∈ N2,• represents the terms t ∈ L1, represents the terms t ∈ L2.

Remark 37.1.3. This discussion suggests proving the theorem by iterativelyproducing for h = 1, 2, . . .

a degree δ(h) ≥ δ(h − 1),

the monomial

µh := max<T(Can(M( f ), M(Jh−1)δ(h))), f ∈ Iδ(h), M ∈ Uh−1,

fh ∈ Iδ(h), a ‘suitable’ element such that

µh = T(Can(M( fh), M(Jh−1)δ(h)))

for some M ∈ Uh−1,a polynomial Ph(Xi j ) ∈ k[Xi j ] \ 0,the non-empty Zariski open set

Uh := M ∈ Uh−1 : Pl(ci j ) = 0, 1 ≤ l ≤ h= M ∈ Uh−1 : T(M( fl)) = µl , 1 ≤ l ≤ h,

Jh := ( fl : 1 ≤ l ≤ h),

Mh := (µl : 1 ≤ l ≤ h),Lh := Mh \ Mh−1,Nh := T \ Mh ,

so that we have

Uh ⊆ Uh−1 is a non-empty Zariski open set,Mh = T(M(Jh)), for each M ∈ Uh,

Lh = µht : t ∈ Nh−1,T = Nh L1 · · · Lh ,Mh ⊃ Mh−1.

Before continuing the discussion, we must explain the notation

Can(M( fh), M(Jh−1)δ(h))

used in the Remark above; clearly M( fl), 1 ≤ l < h is not a Grobner basisuntil the iteration terminates, giving, as the lemma below will prove,

T(M(Jκ)) = Mκ = T(M(I)) and Jκ = I

for each M ∈ Uκ .

696 Galligo

The indexing of M(Jh−1)δ(h) by δ(h) indicates that we are thinking not ofthe ideal Jh−1 but just of the vectorspace

(Jh−1)δ(h) = f ∈ Jh−1 homogeneous , vw( f ) = δ(h)which has an echelon basis

Bh−1 := t fl : vw(t fl) = δ(h), t ∈ Ll , 1 ≤ l < hsuch that (Jh−1)δ(h) = Spank(Bh−1).

In conclusion the notation Can refers here not to the definition ofLemma 22.2.12 but to that of Corollary 21.2.16.

Lemma 37.1.4. Let us assume we have, for each h, 1 ≤ h < λ, elementsδ(h), fh , µh, Ph(Xi j ), Uh, Jh, Mh, Lh, Nh, satisfying the conditions of Re-mark 37.1.3. Then, either

hH(d; I) = hH(d;Mλ−1), for each d ∈ N

or there are elements δ(λ), fλ, µλ, Pλ(Xi j ), Uλ, Jλ, Mλ, Lλ, Nλ, satisfying theconditions of Remark 37.1.3.

Proof (of Theorem 37.1.1). Since, for each M ∈ Uh,

Mh−1 ⊂ Mh = T(M(Jh)) ⊂ T(M(I)),

Gordan’s Lemma implies that such iterative production is necessarily finite,and that there is a value κ such that

Mκ = T(M(I)), for each M ∈ Uκ .

Therefore, the theorem is proved by setting

U := Uκ and ε(I) := Mκ .

Proof (of Lemma 37.1.4). Let δ(λ) be the minimal value such that

hH(δ(λ); I) < hH(δ(λ);Mλ−1)).

Clearly we have

• δ(λ) ≥ δ(λ− 1); and• if δ(λ) > δ(λ−1) then M(Jλ−1)d = M(I)d for each d, δ(λ−1) ≤ d < δ(λ),

and each M ∈ Uλ−1; so that• Can(M( f ), M(Jλ−1)d) = 0, for each f ∈ Id , δ(λ − 1) ≤ d < δ(λ), and

each M ∈ Uλ−1;• Can(M( f ), M(Jλ−1)δ(λ)) : f ∈ Iδ(λ), M ∈ Uλ−1 = 0.

37.2 Borel Relation 697

Therefore if we consider any polynomial f ∈ Iδ(λ), we can deduce thatthere are polynomials Dt f (Xi j ) ∈ k[Xi j ] indexed by the terms t ∈ Tδ(λ) anda polynomial Q f (Xi j ) := ∏λ−1

h=1 Ph(Xi j )ah such that for each M = (ci j ) ∈

Uλ−1

Can(M( f ), M(Jλ−1)δ(λ)) =∑

t∈(Nλ−1)δ(λ)

Q f (ci j )−1 Dt f (ci j )t.

As a consequence we can write

• µλ := max<T(Can(M( f ), M(Jλ−1)δ(λ))) : f ∈ Iδ(λ), M ∈ Uλ−1,• fλ ∈ Iδ(λ) for a ‘suitable’ element such that

µλ = T(Can(M( fλ), M(Jλ−1)δ(λ)))

for some M ∈ Uλ−1,• Pλ(Xi j ) := Dµλ fλ(Xi j ) ∈ k[Xi j ] \ 0, so that for each M = (ci j ) ∈ Uλ−1

since Q(ci j ) = 0, we have

Pλ(ci j ) = 0 ⇐⇒ T(M( fλ)) = µλ,

• the non-empty Zariski open set

Uλ := M ∈ Uλ−1 : Pl(ci j ) = 0, 1 ≤ l ≤ λ= M ∈ Uλ−1 : T(M( fl)) = µl , 1 ≤ l ≤ λ,

• Jλ := ( fl : 1 ≤ l ≤ λ),

• Mλ := (µl : 1 ≤ l ≤ λ),• Lλ := Mλ \ Mλ−1,• Nλ := T \ Mλ.

so that the conditions of Remark 37.1.3 are trivially satisfied.

Definition 37.1.5. For any ideal I ⊂ P , the monomial ideal ε(I) whose exis-tence is proved by Theorem 37.1.1 was called by Galligo the Grauert invariant;it is usually called the generic initial ideal and denoted gin(I).

Its complement T \ gin(I) which is often more relevant, is called the genericescalier.7

37.2 Borel Relation

In order to understand what kind of maximality is verified by µh among theelements of (Nh−1)δ(h) and to be able to introduce the Borel relation → and

7 The term, introduced by Galligo, in French reads generique escalier and in English generic stairsbut in the technical lingo Galligo’s term remained untranslated and for any ideal I its escalier isN(I), whence the term generic escalier.

698 Galligo

to prove (as suggested in Remark 37.1.2) that µh is a →-minimal element in(Nh−1)δ(h), that is

µh ∈µ ∈ (Nh−1)δ(h) : t → µ, t ∈ (Nh−1)δ(h) ⇒ t = µ

,

it is better to begin by considering an example.

Example 37.2.1. Let I = (X23, X2 X3, X1 X3) which satisfies

hH(d; I) =⎧⎨⎩

d + 1 if d ≥ 2,3 if d = 1,1 if d = 0,

and let us compute M(I), for each M = (ci j ) ∈ GL(n, k); we have

M(X23) =

∑i j

c3i c3 j Xi X j

= c233 X2

3 + 2c32c33 X2 X3 + c232 X2

2 + 2c31c33 X1 X3 + · · ·

and we can set

f1 := X23, µ1 := X2

3 = max<

(T2), U1 =M = (ci j ) : c33 = 0

.

Then we have

M(X2 X3) =∑i, j

c2i c3 j Xi X j

= c23c33 X23 + (c22c33 + c32c23) X2 X3

+ c22c32 X22 + (c21c33 + c31c23) X1 X3 + · · ·

M(c33 X2 X3 − c23 f1) =∑i, j

(c33c2i c3 j − c23c3i c3 j

)Xi X j

= c33 (c22c33 − c32c23) X2 X3

+ c32 (c22c33 − c32c23) X22

+ c33 (c21c33 − c31c23) X1 X3 + · · ·

and we can set

f2 := c33 X2 X3 − c23 f1,

µ2 := X2 X3 = max<

((N2)2),

P2 := c22c33 − c32c23,

U2 := M = (ci j ) : c33 = 0, P2 = 0.


The next computation is

M(X1 X3) =∑

i j

c1i c3 j Xi X j

= c13c33 X23 + (c12c33 + c32c13) X2 X3

+ c12c32 X22 + (c11c33 + c31c13) X1 X3 + · · · ,

M(c33 X1 X3 − c13 f1) =∑

i j

(c33c1i c3 j − c13c3i c3 j

)Xi X j

= c33 (c12c33 − c32c13) X2 X3

+ c32 (c12c33 − c32c13) X22

+ c33 (c11c33 − c31c13) X1 X3 + · · · ,M(g) = c3

33 (−c11c22c33 + c11c32c23 + c21c12c33) X1 X3

− c333(c21c32c13−c31c12c23+c31c22c13) X1 X3+· · ·

where

g := P2c33 X1 X3 − P2c13 f1 − (c12c33 − c32c13) f2

and we must set f3 := g and µ3 := X1 X3.

Until now we have had no need to make reference to <; our first choiceX3 > X2 > X1 gave us that the maximal element in T2 is X2

3 and the secondone is X2 X3; now, however the choice of the third maximal element in T2

depends on <; we have in fact two candidates, X22 and X1 X3:

• the choice X22 > X1 X3, together with the ordering of the variables, imposes

on T2 the ordering

X23 > X2 X3 > X2

2 > X1 X3 > X1 X2 > X21

which is satisfied, for example by rev-lex,• while the choice X2

2 < X1 X3, together with the ordering of the variables,imposes on T2 the ordering

X23 > X2 X3 > X1 X3 > X2

2 > X1 X2 > X21

which is satisfied for example by lex.8

Since, for each M ∈ U2, the coefficient of X22 is 0 in M( f3) we have just one

choice for µ3, µ3 := X1 X3.

8 Note that there is no other ordering on T2 satisfying the fixed ordering on the variables.But none of them is able to force a unique ordering on T3; compare the discussion in Remark

37.2.13 below.

700 Galligo

Apparently, the reason is ‘geometrical’: if we choose µ3 := X22 then M3 :=

(X23, X2 X3, X2

2) has the Hilbert function

hH(d;M3) = 3 if d ≥ 1,

1 if d = 0.

The ideal M3 is 1-dimensional, while I is 2-dimensional. But the geometricalexplanation is not correct; in fact the same computation would apply 9 to thezero-dimensional ideal J = (X2

3, X2 X3, X1 X3, X32).

Example 37.2.2. It is worth continuing with this example by considering theideal I = (X2

3, X2 X3, X22) and computing M(I), for each M = (ci j ) ∈

GL(n, k); the previous computation holds and gives

f1 := X23, µ1 := X2

3,

f2 := c33 X2 X3 − c23 f1, µ2 := X2 X3,

P2 := c22c33 − c32c23 andU2 = (ci j ) : c33 = 0, P2 = 0.

The next computation is

M(X22) =

∑i, j

c2i c2 j Xi X j

= c223 X2

3 + 2c22c23 X2 X3 + c222 X2

2 + 2c21c23 X1 X3 + · · ·M(c2

33 X22 − c2

23 f1) = 2c23c33 (−c32c23 + c22c33) X2 X3

+(−c2

32c223 + c2

22c233

)X2

2

+ 2c23c33 (−c31c23 + c21c33) X1 X3 + · · ·M(g) = DX2

2 g X22 + · · ·

where

g := P2c233 X2

2 − P2c223 f1 −

(−2c32c2

23 + 2c22c23c33

)f2

DX22 g := −c3

32c323 + 3c22c2

32c223c33 − 3c2

22c32c23c233 + c3

22c333

and we must set f3 := g and µ3 := X22.

The example therefore is symmetric to the previous one; the solution is(X2

3, X2 X3, X22), whatever is the ordering.

9 Since it is performed by increasing degree.


Example 37.2.3. This suggests that we try a third example.The computation of M(I), M = (ci j ) ∈ GL(n, k), for the ideal I = (X2

3,

X2 X3, X1 X2) which satisfies

hH(d; I) = d + 1 if d ≥ 2,

3 if d = 1,1 if d = 0

gives

M(X1 X2) = c13c23 X23 + (c22c13 + c12c23) X2 X3

+ c12c22 X22 + (c21c13 + c11c23) X1 X3 + · · · ,

M(g′) = c33 (−2c32c13c23 + c22c13c33 + c12c23c33) X2 X3

+(−c2

32c13c23 + c12c22c233

)X2

2

+ c33 (−2c31c13c23 + c21c13c33 + c11c23c33) X1 X3 + · · · ,M(g) = DX2

2 g X22 + DX1 X3g X1 X3 + · · · ,

where

g′ := c233 X1 X2 − c13c23 f1,

g := P2g′ − (−2c32c13c23 + c22c13c33 + c12c23c33) f2,

DX22 g := −c3

32c13c223 + 2c22c2

32c13c23c33 + c12c232c2

23c33

− c222c32c13c2

33 − 2c12c22c32c23c233 + c12c2

22c333,

DX1 X3g := −c31c22c13c23c233 + c21c32c13c23c2

33 + c31c12c223c2

33

− c11c32c223c2

33 − c21c12c23c333 + c11c22c23c3

33.

Since the coefficients in g of both X22 and X1 X3 are not zero, this time we

have two alternatives:

• if X22 > X1 X3, we must set µ3 := X2

2 and we get

ε(I) := (X23, X2 X3, X2

2);• if X2

2 < X1 X3, we must set µ3 := X1 X3 and, after a computation in degree3, we get

ε(I) := (X23, X2 X3, X1 X3, X3

2).

Note that the computation for the ideal I = (X23, X2 X3, X2

1) would give asimilar result.

These examples show that we cannot hope to prove a relation µh =max<((Nh−1)δ(h)) and we must look for a more subtle relation between µh and(Nh−1)δ(h), knowing that there are two possible alternatives: X2

2 and X1 X3.

702 Galligo

Such a more subtle relation was found by Galligo in the case in which thevaluation is the classical degree v( f ) = deg( f ), for each f ∈ P .

From now on, therefore, we will assume that P is the classical polynomial(respectively: series) ring. In this context, all the previous results (and nota-

tions) still hold and we will write, for each λ ≤ κ, µλ := Xaλ

11 . . . X

aλh

h . . . Xaλ

nn ;

moreover we will denote χ(λ) := minh : aλh = 0.

Galligo proved

Theorem 37.2.4 (Galligo). For each λ ≤ κ, , ′, 1 ≤ < ′ ≤ n, p ≤ aλ ,

we have

Xaλ

11 . . . X

aλ−1

−1 Xaλ−p

Xaλ+1

+1 . . . Xaλ′−1

′−1 Xaλ′+p

′ Xaλ′+1

′+1 . . . Xaλ

nn ∈ L1 · · · Lλ−1.

Corollary 37.2.5. For each λ ≤ κ Lλ = µλt, t ∈ T [1, χ(λ)].Proof. Clearly, for each j > χ(λ), µλ X j ∈ L1 · · · Lλ−1 so that

Lλ = µλt ∈ Nλ−1 ⊆ µλt, t ∈ T [1, χ(λ)],and we need to prove only the converse inclusion; let us therefore assume thatthere is some t ∈ T [1, χ(λ)] such that µλt ∈ L1· · · Lλ−1; this implies thatthere are j < λ and τ ∈ T [1, χ( j)] such that µλt = µ jτ .

Then, either

χ( j) > χ(λ) and µ j | µλ, a contradiction, orχ( j) ≤ χ(λ) and τ | t , so there is ω ∈ T [χ(λ) + 1, χ( j)] such that t =

τω, ωµλ = µ j so that µ j ∈ Lλ, another contradiction.

Proposition 37.2.6 (Galligo). Let I be a monomial ideal. The following con-ditions are equivalent:

(1) For each , ′, 1 ≤ < ′ ≤ n,

Xa11 . . . Xan

n ∈ I ⇒ Xa11 . . . Xa−1

. . . Xa′+1′ . . . Xan

n ∈ I.

(2) For each , ′, 1 ≤ < ′ ≤ n, and each p ≤ a

Xa11 . . . Xan

n ∈ I ⇒ Xa11 . . . Xa−p

. . . Xa′+p′ . . . Xan

n ∈ I.

(3) For each , ′, 1 ≤ < ′ ≤ n, β ∈ k denote N = B(, ′;β) thechange of coordinates such that

N(Xh) =

X + β X′ if h = ,Xh if h = ;

then I = N(I).


(4) I = M(I), for each M := (ci j

) ∈ GL(n, k), which is upper triangular,that is

i > j ⇒ ci j = 0.

(5) For each , ′, 1 ≤ < ′ ≤ n,

Xa11 . . . Xan

n /∈ I ⇒ Xa11 . . . Xa+1

. . . Xa′−1′ . . . Xan

n /∈ I.

(6) For each , ′, 1 ≤ < ′ ≤ n, and each p ≤ a′

Xa11 . . . Xan

n /∈ I ⇒ Xa11 . . . Xa+p

. . . Xa′−p′ . . . Xan

n /∈ I.

Proof.

(1) ⇐⇒ (2) ⇐⇒ (5) ⇐⇒ (6) are trivial.(2) ⇐⇒ (3) For t := Xa1

1 . . . Xahh . . . Xan

n ∈ I, we have

N(t) = t ′a∑

p=0

(a

p

)β p Xa−p

Xa′+p′ ∈ N(I)

where

t ′ := Xa11 . . . Xa−1

−1 Xa+1+1 . . . X

a′−1′−1 Xa+1

′+1 . . . Xann .

Then (2) ⇒ (3) is trivial, while its converse is a consequence ofthe fact that N(I) = I is a monomial ideal.

(3) ⇐⇒ (4) Each upper triangular matrix M := (ci j

) ∈ GL(n, k) is theproduct of the matrices B(i, j; ci j ) : M = ∏

i< j B(i, j; ci j ).

Definition 37.2.7. A monomial ideal I which satisfies the equivalent condi-tions above is called a Borel ideal.

Corollary 37.2.8. A generic initial ideal is a Borel ideal and conversely.

Proof. That a generic initial ideal is a Borel ideal is stated in Theorem 37.2.4.That a Borel ideal is the generic initial ideal of itself is the content of Proposi-tion 37.2.6

Example 37.2.9. In general, if an ideal I is such that T(I) is a Borel ideal, onecannot deduce that T(I) = ε(I).

The easiest example is I = (X23, X2 X3, X2

2 + X1 X2), for which, under anordering such that X2

2 < X1 X3, one has T(I) = (X23, X2 X3, X2

2) and ε(I) =(X2

3, X2 X3, X1 X3).

704 Galligo

Lemma 37.2.10. For all , ′, 1 ≤ < ′ ≤ n and each µ :=Xa1

1 . . . Xann such that a = 0 we have 10

Xa11 . . . Xa−1

−1 Xa−1 Xa+1

+1 . . . Xa′−1′−1 X

a′+1′ X

a′+1′+1 . . . Xan

n =: ν µ.

Proof. Let

τ := gcd(µ, ν) = Xa11 . . . Xa−1

−1 Xa−1 Xa+1

+1 . . . Xa′−1′−1 X

a′′ X

a′+1′+1 . . . Xan

n ;then, since X ≺ X′ we have µ = τ X ≺ τ X′ = ν.

Definition 37.2.11 (Gjunter–Marinari). The Borel relation is the relation→generated on each Td by the formulas 11

Xa11 . . . Xah−1

h−1 Xahh . . . Xan

n → Xa11 . . . Xah−1+1

h−1 Xah−1h . . . Xan

n ,

for each h, 1 ≤ h ≤ n, with ah > 0.

Since, in this notation, one has

Xn ← X2 · · · ← X1,

and X1 < X2 < · · · < Xn the result above can be read as 12

ν ← µ ⇒ µ ≺ ν.

Example 37.2.12. For instance, for the polynomial ring

P = k[X, Y, Z ] = k[X1, X2, X3]

the monomials in Td , 1 ≤ d ≤ 3, can be represented by the diagrams

X ← Y↑Z

X2 ← XY ← Y 2

↑ ↑X Z ← Y Z

↑Z2

X3 ← X2Y ← XY 2 ← Y 3

↑ ↑ ↑X2 Z ← XY Z ← Y 2 Z

↑ ↑X Z2 Y Z2

↑Z3

10 Remember that we are assuming X1 < Xn ; for an ordering for which Xn < X1 the statementis µ ν.

11 The definition is to be considered to be independent of the ordering on the variables.12 And µ ← ν ⇒ µ ν in the case Xn < · · · < X1.


and the generic Td by

Xd ← Xd−1Y ← Xd−2Y 2 ← · · · ← X2Y d−2 ← XY d−1 ← Y d

↑ ↑ ↑ ↑ ↑Xd−1 Z ← Xd−2Y Z ← · · · ← X2Y d−3 Z ← XY d−2 Z ← Y d−1 Z

↑ ↑ ↑ ↑Xd−2 Z2 ← · · · ← X2Y d−4 Z2 ← XY d−3 Z2 ← Y d−2 Z2

↑ ↑ ↑...

.

.

.

.

.

.

↑ ↑ ↑X2 Zd−2 ← XY Zd−2 ← Y 2 Zd−2

↑ ↑X Zd−1 ← Y Zd−1

↑Zd

Remark 37.2.13. The Borel relation and the corresponding diagram are a goodtool for describing the structure of Borel and generic initial ideals. For instance:

• Galligo’s result (Theorem 37.2.4) can be stated as:For each i ≤ κ , µi is a minimal element in (Ni−1)δ(i) under→, that is

t → µi , t ∈ (Ni−1)δ(i) ⇒ t = µi .

In the examples we have discussed, we had

(N2)2 = X22, X1 X3, X1 X2, X2

1and the→-minimal elements are X2

2 and X1 X3.• Borel ideals I are those monomial ideals such that, for each d, Id is stable

under→.Following again our examples, for a Borel ideal I such that

hH(0; I) = 1, hH(1; I) = 3, and hH(2; I) = 3,

there are only two subsets of T2 with cardinality 3 which are stable under→, namely

X23, X2 X3, X2

2 and X23, X2 X3, X1 X3.

• As remarked by Marinari,13 the diagrams allow us also to read the relevantterm ordering:Beginning from the top-left corner and moving against the arrows within the rows(respectively: columns) one reads, increasingly, deg-lex (respectively: deg-rev-lex)induced by X < Y < Z .Conversely, beginning from the bottom-right corner and moving along the arrowswithin the rows (respectively: columns) one reads, increasingly, deg-rev-lex (respec-tively: deg-lex) induced by X > Y > Z .

13 M. G. Marinari, Sugli ideali di Borel, Boll. UMI (2000).

706 Galligo

• These diagrams can easily help to describe the orderings on Td ; for instance,in order to obtain any ordering on T2 induced by X < Y < Z (and socompatible with→) one just needs to impose a diagonal on the square

XY ← Y 2

↑ ↑X Z ← Y Z

• if we set

XY Y 2

↑ ↑X Z Y Z

, that is Y 2 < X Z , we obtain deg-lex, while

• settingXY ← Y 2

X Z ← Y Z

, that is Y 2 > X Z we get deg-rev-lex.

Note that, when we move to consider the orderings on T3, there are still tiesto be solved:

• if we have fixed Y 2 < X Z most of the terms are uniquely ordered except

for

Y 3

↑XY Z Y 2 Z↑

X Z2

and we must solve a tie between Y 3 and X Z2;

• and, if we fixed Y 2 > X Z , most of the terms are uniquely ordered except

forXY 2 ← Y 3

X2 Z ← XY Z

and we must solve a tie between Y 3 and X2 Z .

37.3 *Galligo Theorem (2): The Generic Initial Ideal is Borel Invariant

For each series f := ∑t∈T c( f, t)t ∈ k[[X1, . . . , Xn]] we will write

‖ f ‖ :=∑t∈T

|c( f, t)|

and for each ρ := (ρ1, . . . , ρn) ∈ Qn, ρ j > 0, for each j ,

‖ f ‖ρ :=∑t∈T

|c( f, t)|t (ρ1, . . . , ρn).

Lemma 37.3.1. Let δ ∈ Q, δ > 0.In the construction of Lemma 37.1.4 we can assume that, for each h, 1 ≤

h < λ, fh , among the other properties of Remark 37.1.3, also satisfies

Can(M( fh), M(Jh−1)δ(h)) = µh + rh, ‖rh‖ < δ.

37.3 *Galligo Theorem (2) 707

Proof. By Bayer’s result (Proposition 24.9.7) there is a weight w :=(w1, . . . , wn) such that

t1 < t2 ⇒ vw(t1) < vw(t2) for each t1, t2 ∈ T (δ()).

Then, writing, for each h, 1 ≤ h ≤ ,

fh = µh + rh, rh =∑

t∈Tδ(h)

c( fh, t)t =∑

t∈Nh−1

c( fh, t)t,

and

σh := max vw(t) : t ∈ Nh−1, c( fh, t) = 0 < vw(µh), 14

if we choose ρ ∈ Q such that ‖rh‖ < δρvw(µh)−σh , for each h, and writeρ j := ρw j for each j , we obtain, for each h,

‖rh‖ρ =∑

t∈Nh−1

|c( fh, t)|ρvw(t) ≤ ρσh∑

t∈Nh−1

|c( fh, t)| < δρvw(µh) = δµh(ρ).

Since U is Zariski open, we can also choose ρ in such a way that

M ∈ U ⇒ DρM ∈ U,

where Dρ denotes the change of coordinates defined by Dρ(X j ) = ρw j X j , foreach j; in this way we have

Can

(DρM

(fh

µh(ρ)

), DρM(Jh−1)δ(h)

)= µh + rh(ρw1 X1, . . . , ρ

wn Xn)

µh(ρ)

=: µh + r ′hand

‖r ′h‖ =‖Dρ(rh)‖µh(ρ)

=∑

t∈Nh−1|c( fh, t)|ρvw(t)

µh(ρ)= ‖rh‖ρ

µh(ρ)<

δµh(ρ)

µh(ρ)= δ.

For each λ ≤ κ, and each , ′, 1 ≤ < ′ ≤ n, let us denote by N thechange of coordinates defined by

N(Xh) =

X + β X′ if h = ,Xh if h = ,

where δ and β, 0 < δ β 1, are chosen so that N ∈ Uλ.

To simplify the notation, let us assume wlog 15 that the identity belongs toUλ, so that

Can( fh, (Jh−1)δ(h)) = µh + rh, rh =∑

t∈Nh−1

c( fh, t)t, ‖rh‖ < δ,

14 Since µh > t and vw(µh) > vw(t), for each t ∈ T , c( fh , t) = 0.15 Because we might effectively perform a change of coordinate M ∈ Uλ.

708 Galligo

for each h ≤ λ.

Then:

Lemma 37.3.2. If g is such that ‖g‖ < δ, then

• ‖N(g)‖ < δ,• ‖Can(N(g), N(Jh))‖ < δ, for each h ≤ λ.

Proof. For any term t = Xa11 . . . Xan

n write

t ′ := Xa11 . . . Xa−1

−1 Xa+1+1 . . . X

a′−1′−1 Xa+1

′+1 . . . Xann .

One has

N(t) = t ′a∑

p=0

(a

p

)β p Xa−p

Xa′+p′

so that

‖N(t)‖ ≤a∑

p=0

(a

p

)β p ≤ 1

and for g = ∑t c(g, t)t we have

‖N(g)‖ =∥∥∥∑

t

c(g, t)N(t)∥∥∥ ≤ ∑

t

|c(g, t)| = ‖g‖ < δ.

Assume

N(g) = aτµh +∑

t

at t, a ∈ k, at ∈ k, τ ∈ T ,

so that |a| +∑t |at | = ‖N(g)‖ < δ and let g′ = N(g)− aτ fh ; then

N(g′) = aN(τ )N(µh)+∑

t

at N(t)− aN(τ )N(µh)− aN(τ )N(rh)

=∑

t

at N(t)− aN(τ )N(rh)

and

‖N(g′)‖ =∑

t

|at | + |a|‖N(rh)‖ ≤∑

t

|at | + |a| < δ.

This shows that the claim holds after one step of reduction and thereforeholds for the canonical form.

Corollary 37.3.3. Assume the statement of Theorem 37.2.4 holds for each h <

λ, then for each h < λ

Can(N( fh), N(Jh−1)δ(h)) =: µh + r ′h, ‖r ′h‖ < δ and T(r ′h) < µh .


Proof. Since the statement follows directly from Lemma 37.3.1 for h = 1, letus prove it by induction.

We have

Can(N( fh), N(Jh−1)δ(h)) =: Can(N(µh), N(Jh−1)δ(h))

+Can(N(rh), N(Jh−1)δ(h)).

The norm of the second addend is less than δ by the lemma above; as regardsthe first addend, writing

t ′ := Xah

11 . . . X

ah−1

−1 Xah+1

+1 . . . Xah′−1

′−1 Xah+1

′+1 . . . Xah

nn ,

we can rewrite Can(N(µh), N(Jh−1)δ(h)) as

a∑p=0

(a

p

)β p Can

(t ′Xah

−p X

ah′+p

′ , N(Jh−1)δ(h)

).

By the assumption each term t ′Xah−p

Xah′+p

′ except µh is a member in L1· · · Lh−1 and its coefficient satisfies

(a

p

)β p 1 so that

Can

(t ′Xah

−p X

ah′+p

′ , N(Jh−1)δ(h)

)= bpµh + gp,

with ‖gp‖ < δ and bp 1, so that∑

p |bp| 1.

Therefore the coefficient of µh in Can(N( fh), N(Jh−1)δ(h)) is 1 +∑p bp,

0 = 1+∑p bp ≈ 1.

Since T(Can(N( fh), N(Jh−1)δ(h))) ≤ µh by definition, this proves theclaim.

Proof (of Theorem 37.2.4). In the same way as in the corollary above, write

t ′ := Xaλ

11 . . . X

aλ−1

−1 Xaλ+1

+1 . . . Xaλ′−1

′−1 Xaλ+1

′+1 . . . Xaλ

nn ,

Can(N( fλ), N(Jλ−1)δ(λ)) is a combination of:

• Can(N(rλ), N(Jλ−1)δ(λ)) whose norm is less than δ;

• the elements, if any, Can

(t ′Xaλ

−p X

aλ′+p

′ , N(Jλ−1)δ(λ)

), 0 ≤ p ≤ a,

such that t ′Xaλ−p

Xaλ′+p

′ ∈ L1 · · · Lλ−1 and whose norm is less than δ;

• the elements(a

p

)β pt ′Xaλ

−p X

aλ′+p

′ , 0 ≤ p ≤ a, such that t ′Xaλ−p

Xaλ′+p

′∈ L1 · · · Lλ−1.

710 Galligo

Since the following holds

• µλ < t ′Xaλ−p

Xaλ′+p

′ , for each p,

• in Can(N( fλ), N(Jλ−1)δ(λ)) the coefficient of µλ is 1+∑p bp = 0,

• by construction

µλ = maxT(Can(M( fλ), M(Jλ−1)δ(λ))) : M ∈ Uλ≥ T(Can(N( fλ), N(Jλ−1)δ(λ))),

the existence of some p > 0 for which

t ′Xai−p

Xai′+p

′ ∈ L1 · · · Li−1

would give a contradiction.

37.4 *Galligo Theorem (3): The Structure of the Generic Escalier

Let us introduce some further notation:

• Fj := i : χ(i) = j,• L j := φ j (t) : t ∈ Lh, h ∈ Fj ,• B j := Xa

j β : a ∈ N, β ∈ φ j (gin(I)) \ φ j−1(gin(I)),

where each φ j is the projection φ j : k[X1, . . . , Xn] → k[X j+1, . . . , Xn] and,for each i , χ(i) := minh : ai

h = 0.Lemma 37.4.1 (Galligo). The following holds

(1) i ∈ Fj ⇒ φ j (µi ) /∈ φ j (Lh), for each h such that χ(h) > j ;(2) l < j, i ∈ Fl ⇒ φ j (µi ) ∈ L j ;(3) for each j, B j is finite;(4) for each j, #(B j ) =

∑i∈Fj

aij .

Proof.

(1) Trivial since µi = Xai

jj φ j (µi ) and µh = φ j (µh), for each h such that

χ(h) > j.

(2) For µi := Xai

ll . . . X

ain

n , writing

d :=j∑

h=l

aih and ν := Xd

j Xai

j+1j+1 . . . X

ain

n ,

we have µi ← ν ∈ L1 · · · , Li−1. For the result above φ j (µi ) /∈φ j (Lh), for each h such that χ(h) > j. Therefore, there is h ∈ Fj suchthat ν ∈ Lh and φ j (µi ) = φ j (ν) ∈ L j .


(3) The proof is the description of

Xaj β : a ∈ N, β ∈ φ j (gin(I)) =

κ⋃i=1

Xaj β : a ∈ N, β ∈ φ j (Li )

where we set

Li := Xaj β : a ∈ N, β ∈ φ j (Li )

= Xaj φ j (µi )φ j (t) : a ∈ N, t ∈ T [1, χ(i)].

We have:

• if χ(i) > j + 1,

Li = Xaj φ j (µi )t : a ∈ N, t ∈ T [ j + 1, χ(i)]

= φ j−1(µi )t : t ∈ T [ j, χ(i)]⊂ φ j−1(gin(I));

• if χ(i) = j + 1, then φ j (T [1, χ(i)]) = Xbj+1 b ∈ N and µi =

φ j (µi ) = φ j−1(µi ) so that

Li = Xaj Xb

j+1µi , a, b ∈ N= φ j−1(Xa

j Xbj+1µi ), a, b ∈ N

⊂ φ j−1(gin(I));• if χ(i) = j , then φ j (T [1, χ(i)]) = 1 and µi = φ j−1(µi ) =

Xai

jj φ j (µi ) so that

Li := Xaj φ j (µi ), a ∈ N

= Xaj φ j (µi ), a ∈ N, a < ai

j ∪ Xaj φ j−1(µi ), a ∈ N

so that

Li \ φ j−1(gin(I)) = Xaj φ j (µi ), a ∈ N, a < ai

j ;• if χ(i) < j , then φ j (µi ) ∈

⋃χ(h)< j

φ j (Lh) gives no contribution.

In conclusion

B j = Xaj φ j (µi ), µi ∈ Fj , a ∈ N, a < ai

j .(4) This is a direct consequence of the formula above.

Theorem 37.4.2 (Galligo). The generic escalier E(I) := T \ gin(I) of I satis-fies

E(I) = T \ gin(I) = τγ : γ ∈ B j , τ ∈ T [1, j − 1], 1 ≤ j ≤ n.

712 Galligo

Proof. Setting, with a slight abuse of notation, T [1, 0] = 1 and noting thatT [1, n] = T , φ0(gin(I)) = gin(I) and φn(gin(I)) = 1, one has

E(I) = T∖

gin(I)

=τβ : β ∈ 1, τ ∈ T

∖τβ : β ∈ gin(I), τ ∈ 1

=

τβ : β ∈ φn(gin(I)), τ ∈ T [1, n]

∖

τβ : β ∈ φ0(gin(I)), τ ∈ T [1, 0]

=n⋃

j=1

τβ : β ∈ φ j (gin(I)), τ ∈ T [1, j]

∖ n⋃

j=1

τβ : β ∈ φ j−1(gin(I)), τ ∈ T [1, j − 1]

=n⋃

j=1

τ Xa

j β : a ∈ N, β ∈ φ j (gin(I)), τ ∈ T [1, j − 1]

∖ n⋃j=1

τγ : γ ∈ φ j−1(gin(I)), τ ∈ T [1, j − 1]

=n⋃

j=1

τγ : β ∈ φ j (gin(I)), γ := Xa

j β ∈ φ j−1(gin(I)), τ ∈T [1, j − 1]

=n⋃

j=1

τγ : τ ∈ T [1, j − 1], γ ∈ B j

.

Definition 37.4.3. The decomposition

E(I) = T \ gin(I) = τγ : γ ∈ B j , τ ∈ T [1, j − 1], 1 ≤ j ≤ nis called the escalier decomposition of I w.r.t. <.

Example 37.4.4. To illustrate Galligo’s result, let us build a Borel ideal 16 I ⊂K [T, X, Y ] =: P whose Hilbert function satisfies

hH(d; I) =

⎧⎪⎪⎨⎪⎪⎩

hH(d;P) if d ≤ 4,

20 if d = 5,

21 if d = 6,

2d + 8 if d ≥ 7;

16 The reader is advised to follow the argument in the figures of Example 37.2.12.


• since hH(5;P) = 21 we know that #I5 = 1 and we set µ1 := Y 5, M1 :=(µ1);

• therefore #(M1)6 = 3 < 7 = 28− 21 = #I6 and we have to add to M1 fourterms of degree 6; the first choice is forced and we set µ2 := X2Y 4;

• then we can arbitrarily choose either X3Y 3 or T XY 4 and we choose µ3 :=X3Y 3;

• this leaves to us as third choice T XY 4 or X4Y 2 and we take µ4 := T XY 4;• then among T 2Y 4, T X2Y 3 and X4Y 2 we choose µ5 := T 2Y 4;• therefore M5 := (Y 5, X2Y 4, X3Y 3, T XY 4, T 2Y 4) and hH(6, 1) = 21 but

#T7 − #(M5)7 = 36− 12 = 24 > 22 = hH(7; I);we are therefore required to add two more terms of degree 7; the candidatesare T 2 X2Y 3 and X5Y 2 and we choose µ6 := X5Y 2;

• for the last choice, among T 2 X2Y 3, T X4Y 2 and X6Y we take µ7 :=T X4Y 2.Therefore

M7 := (Y 5, X2Y 4, X3Y 3, T XY 4, T 2Y 4, X5Y 2, T X4Y 2)

and since hH(d; I) = hH(d;M7), for each d ≥ 7, we are through.

The situation can be pictured as

......

......

......

......

• • • • • • • • · · ·• • • • • • • • · · ·•Y 5 • • • • • • • · · ·∗T 2Y 4 ∗T XY 4 X2Y 4 · · · X3Y 3 · · · ∗T X4Y 2 X5Y 2 · · · · · · · · ·

representing (at the same time) the projection φ1(I) and the generic plane T =d for all d ≥ 2 where

represents the terms in the generic escalier;• represents the terms t ∈ Li , i ∈ 1 = F3, represents the terms t ∈ Li , i ∈ 2, 3, 6 = F2,∗ represents the terms t ∈ Li , i ∈ 4, 5, 7 = F1.

714 Galligo

With this figure it should be clear that we have

B3 = 1, Y ,B2 = Y 2, XY 2, X2Y 2, X3Y 2, Y 3, XY 3, X2Y 3,B1 = X4Y 2, Y 4, T Y 4, XY 4.

We report here also the picture of the plane T = 0:

......

......

......

......

• • • • • • • • · · ·• • • • • • • • · · ·•Y 5 • • • • • • • · · · X2Y 4 · · · X3Y 3 · · · X5Y 2 · · · · · · · · ·

The structure of the generic escalier, which is a direct consequence ofTheorem 37.2.4 and is made clear from these figures, was described by Galligoas follows:17

One can deduce that

Fn−1 = (0, . . . , 0, α j , s − j) : j = 1 . . . #(Fn−1)with α j strictly increasing. The complement of ε(I) ∩ T [n − 1, n] is therefore an ‘es-calier avec des marches du hauteur 1’.In higher dimension the configuration of ε(I) ∩ T [ j, n] is more difficult to visualize;but it can be figuratively said that the natural generalization of the escalier avec desmarches du hauteur 1 in N

j is a domain in Nj such that, if one arbitrarily fixes all

coordinate values except two, one always obtains un escalier in N2 avec des marches

du hauteur 1.

For instance the set of the elements T a XbY c ∈ I such that

c = 4 is the Borel ideal (T 2, T X, X2),

b = 4 is the Borel ideal (T Y 2, Y 3).

37.5 Eliahou–Kervaire Resolution

Let P := k[X1, . . . , Xn] and T := Xa11 . . . Xan

n : (a1, . . . , an) ∈ Nn.

17 Where s is defined by µ1 = (0, . . . , 0, s).

37.5 Eliahou–Kervaire Resolution 715

For a monomial τ := Xa11 · · · Xan

n ∈ T we write

max(τ ) := maxi : ai = 0,min(τ ) := mini : ai = 0,φ(τ ) := ∑n

i=1(n − i)ai .

Let I ⊂ P be a monomial ideal and G := t1, . . . , ts its minimal basis.

Definition 37.5.1 (Eliahou–Kervaire). The ideal I is called stable if for eachτ ∈ I ∩ T and each j > µ = min(τ ), τ X j/Xµ ∈ I.

For each term τ ∈ I ∩ T a representation τ = υti with υ ∈ T and ti ∈ G iscalled a canonical decomposition if max(υ) ≤ min(ti ).

Note that Borel ideals are stable.

Lemma 37.5.2. Canonical decompositions, if they exist, are unique.

Proof. If τ = υti = ωt j with υ, ω ∈ T , ti , t j ∈ G, max(υ) ≤ min(ti ) andmax(ω) ≤ min(t j ) then both ti and t j are final segments of τ , which impliesthat one of them must divide the other, but, both being elements in G, thisforces ti = t j .

Proposition 37.5.3. The following conditions are equivalent:

(1) I is stable,(2) each term τ ∈ I ∩ T has a (unique) canonical decomposition,(3) there is a function m : I ∩ T → G which satisfies, for each τ ∈ I ∩ T

and each ω ∈ T(a) m(τ ) | τ ,(b) m(ωτ) = m(τ ) ⇐⇒ max(ω) ≤ min(m(τ )).

Proof.

(1) ⇒ (2) Let τ ∈ I ∩ T and let

τ = υti , υ ∈ T , ti ∈ G

be a representation for which max(υ) > min(ti ) and let j > µ =min(ti ) be an index such that X j | υ. Then ti X j/Xµ ∈ I has a de-composition ti X j/Xµ = υ ′tι which gives the decomposition

τ = Xµυ

X j· ti X j

Xµ

=(

Xµυ

X jυ ′

)· tι

with φ(tι) ≤ φ(ti X j/Xµ) = φ(ti )−( j−µ) < φ(t j ). Therefore, afterfinitely many such rewritings we obtain a canonical decomposition.

716 Galligo

(2) ⇒ (1) Let τ ∈ I ∩ T and j > µ = min(τ ) and let X jτ = υti be thecanonical decomposition. Then υ = 1 since ti ∈ G; therefore Xµ

divides υ because max(υ) ≤ min(ti ) and µ = min(τ ) = min(X jτ).Setting υ = Xµω we have τ X j/Xµ = ωti ∈ I.

(2) ⇒ (3) For any τ ∈ I ∩ T , let us write m(τ ) := ti where τ = υti is theunique canonical decomposition of τ . Clearly we have m(τ ) | τ .Assume m(ωτ) = m(τ ) so that the canonical decomposition of ωτ is

ωτ = υm(ωτ) = υm(τ ) with max(υ) ≤ min(m(τ )),

for some υ ∈ T ; since m(τ ) | τ , then ω | υ and max(ω) ≤ max(υ) ≤min(m(τ )).

Conversely, if max(ω) ≤ min(m(τ )), the canonical decomposi-tion τ = υm(τ ), max(υ) ≤ min(m(τ )), gives the decompositionωτ = ωυm(τ ); since both max(ω) ≤ min(m(τ )) and max(υ) ≤min(m(τ )), we have max(ωυ) ≤ min(m(τ )), that is ωτ = ωυm(τ )

is the unique canonical decomposition of ωτ , that is m(ωτ) = m(τ ).(3) ⇒ (2) Let us begin by remarking that, for each ti ∈ G, (a) implies

m(ti ) = ti .For any τ ∈ I ∩ T let ω := τ

m(τ )so that τ = ωm(τ ) and m(τ ) =

m(ωm(τ )). Setting υ := m(τ ) we have

m(ωυ) = m(ωm(τ )) = m(τ ) = υ = m(υ)

and, by (b), max(ω) ≤ min(m(υ)) = min(m(τ )). Hence τ = ωm(τ )

is a canonical decomposition.

For any term τ ∈ I ∩ T , if τ = υti is its unique canonical decomposition,we write

m(τ ) := ti ∈ G and g(τ ) := i ∈ 1, . . . , s.Lemma 37.5.4. Let I be a stable monomial ideal; then for any term τ ∈ I ∩ Tthe following hold:

(1) for any i , m(Xi m(τ )) = m(Xiτ),(2) for any i , min(m(Xiτ)) ≥ min(m(τ )),(3) for any term υ ∈ T , m(υm(τ )) = m(υτ),(4) for any term υ ∈ T , min(m(υτ)) ≥ min(m(τ )).

Proof.

(1) If i ≤ min(m(τ )) then both m(Xiτ) = m(τ ) and m(Xi m(τ )) =m(m(τ )) = m(τ ) hold by condition (b) of Proposition 37.5.3, whencethe claim.


If i > min(m(τ )), let us consider the canonical decomposition

τ = υm(τ ), max(υ) ≤ min(m(τ )) :

multiplying by Xi and applying m we obtain m(Xiτ) = m(Xiυm(τ )).Since m(Xi m(τ )) | Xi m(τ ) we get

min(m(Xi m(τ ))) ≥ min(Xi m(τ )) = min(m(τ )) ≥ max(υ),

whence by condition (b) of Proposition 37.5.3 m(υ · Xi m(τ )) =m(Xi m(τ )) and m(Xiτ) = m(Xiυm(τ )) = m(Xi m(τ )).

(2) If i ≤ min(m(τ )) then, by condition (b) of Proposition 37.5.3,m(Xiτ) = m(τ ) and min(m(Xiτ)) = min(m(τ )).

If i ≥ min(m(τ )) then, since m(Xiτ) = m(Xi m(τ )) we have

min(m(Xiτ)) = min(m(Xi m(τ ))) ≥ min(Xi m(τ )) = min(m(τ )).

(3) By induction on deg(υ) we have

m(Xiυm(τ )) = m(Xi m(υm(τ ))) = m(Xi m(υτ)) = m(Xiυτ).

(4) By induction on deg(υ) we have

min(m(Xiυτ)) ≥ min(m(υτ)) ≥ min(m(τ )).

Lemma 37.5.5. For each τ ∈ I ∩ T and each υ ∈ T , the following hold:

(1) deg(m(υτ) ≤ deg(m(τ ));(2) if < is the degrevlex ordering induced by X1 < · · · < Xn, then

deg(m(υτ)) = deg(m(τ )) ⇒ m(υτ) ≥ m(τ ).

Proof.

(1) If max(υ) ≤ min(m(τ )), m(υτ) = m(τ ) follows by condition (b) ofProposition 37.5.3.If max(υ) > min(m(τ )), let us consider the canonical decomposi-tion 18

υm(τ ) = ωm(υm(τ )) = ωm(υτ), where max(ω) ≤ min(m(υτ)).

Since m(υτ) ∈ G is not a multiple of m(τ ), necessarily deg(ω) ≥deg(υ) and deg(m(υτ)) ≤ deg(m(τ )).

(2) Continuing the argument with the same notation, we can restrict our-selves to the following assumptions:

18 The equality m(υm(τ )) = m(υτ) follows from Lemma 37.5.4.

718 Galligo

max(υ) > min(m(τ ));deg(ω) = deg(υ), since deg(m(υτ)) = deg(m(τ ));min(m(υτ)) = min(m(τ )); in fact min(m(υτ)) ≥ min(m(τ )) and

min(m(υτ)) > min(m(τ )) ⇒ m(υτ) > m(τ ).

Moreover, if υ = Xiς and deg(m(υτ)) = deg(m(τ )) we have

deg(m(υτ)) = deg(m(Xiςτ)) ≤ deg(m(Xiτ)) ≤ deg(m(τ ))

= deg(m(υτ)),

that is deg(m(Xiτ)) = deg(m(τ )).Therefore the general case follows by induction on deg(υ), if we as-sume deg(υ) = 1, υ = Xi for some i, 1 ≤ i ≤ n, and prove

deg(m(Xiτ)) = deg(m(τ )) ⇒ m(Xiτ) ≥ m(τ ).

Since deg(ω) = deg(υ) = 1 we have ω = X j for some j, 1 ≤ j ≤ n.From i = max(υ) > min(m(τ )) and j = max(ω) ≤ min(m(Xiτ)) wehave

j = max(ω) ≤ min(m(Xiτ)) = min(m(τ )).

Since the exponent of X j in m(Xiτ) is strictly smaller than the one inX j m(Xiτ) = Xi m(τ ) we have m(Xiτ) > m(τ ).

We now write, for 0 < q ,

• Iq := (i1, . . . , iq) : n ≥ i1 > i2 > · · · > iq ≥ 1,• Cq := (i, i) : 1 ≤ i ≤ s, i ∈ Iq,• Lq := (i, i) ∈ Cq : iq > min(ti ),• Nq := (i, i) ∈ Cq : iq ≤ min(ti ),and we set C0 := L0 := (i) : 1 ≤ i ≤ s and N0 := ∅.

Let us then write, for 0 ≤ q ,

• sq := #Cq , rq := #Lq ;• e(i, i) : (i, i) ∈ Cq for the canonical basis of the P-module Psq ;• Prq for the P-module whose canonical basis is e(i, i) : (i, i) ∈ Lq;• Ψq : Psq → Prq for the morphism such that, for each (i, i) ∈ Cq ,

Ψq(e(i, i)) :=

e(i, i) if (i, i) ∈ Lq ,0 if e(i, i) ∈ Nq ;

• for each (i, i) ∈ Cq , i := (i1, . . . , iq),

• T(i, i) := Xi1 · · · Xiq ti ,• for each j, 1 ≤ j ≤ q,


i j := (i1, . . . , i j−1, i j+1, . . . , iq) ∈ Iq−1, g( j) := g(Xi j ti ), m( j) := m(Xi j ti ) = tg( j),

υ j := Xi j ti m( j)−1,

µ j := min(m( j)),

• for each j, l, 1 ≤ l < j ≤ q,

i (l, j) := (i1, . . . , il−1, il+1, . . . , i j−1, i j+1, . . . , ik), e(i, i; l, j) := e(i, i (l, j)), g(l, j) := g(Xi j Xil ti )), m(l, j) := m(Xi j Xil ti )) = tg(l, j),

υ(l, j) := Xi j Xil ti m(l, j)−1.

Lemma 37.5.6. For each q, 0 < q and (i, i) ∈ Lq , i := (i1, . . . , iq), writing

A(i, i) := j : 1 ≤ j ≤ q, µ j < minil , l = j,

we have

(1) j ∈ A(i, i) ⇐⇒ (i1, . . . , i j−1, i j+1, . . . , iq , µ j ) ∈ Lq ,(2) q ∈ A(i, i),(3) for j > q, j ∈ A(i, i) ⇐⇒ µ j < iq .

Proof. The only statement which is not trivial is (2): Xiq ti = υqm(q) andXiq υq , otherwise ti = m(q), and

iq > min(ti ) = min(m(q)) ≥ max(υq) ≥ iq ,

a contradiction.So Xiq | m(q), iq ≥ µq min(m(q)) and (i1, . . . , iq−1, µq) ∈ Lq .

We also set

• δ0 to be the map δ0 : Pr0 → P defined by δ0(e(i)) = ti ;• δq , 0 < q, to be the map δq : Prq → Prq−1 defined by

δq(e(i, i)) =q∑

j=1

(−1) j Xi j e(i, i j)−∑

j∈A(i,i)

(−1) jυ j e(g( j), i j);

• γq , 0 < q, to be the map γq : Psq → Psq−1 defined by

γq(e(i, i)) =q∑

j=1

(−1) j Xi j e(i, i j);

720 Galligo

• χq , 0 < q, to be the map χq : Psq → Psq−1 defined by

χq(e(i, i)) =q∑

j=1

(−1) jυ j e(g( j), i j);

• ∆q , 0 < q, to be the map ∆q : Psq → Psq−1 defined by

∆q(e(i, i)) := γq(e(i, i))− χq(e(i, i)).

Lemma 37.5.7. For each q, 0 < q and (i, i) ∈ Nq , i := (i1, . . . , iq),

∆q(e(i, i)) ∈ ker(Ψq−1).

Proof. Since, for each j < q , we have, by Lemma 37.5.4(2),

min(m( j)) = min(m(Xi j ti )) ≥ min(ti ) ≥ iq

and iq is the last index in each i j then e(i, i j) ∈ Nq−1 and e(g( j), i j) ∈Nq−1 for each j < q . Moreover iq ≤ min(ti ) implies also g(Xiq ti )) = i andυq = Xiq . Therefore

Ψq−1∆q(e(i, i)) =q∑

j=1

(−1) j Xi j Ψq−1(e(i, i j))

−q∑

j=1

(−1) jυ jΨq−1(e(g( j), i j))

= (−1)q(Xiq − υq)Ψq−1(e(i, i q))

= 0.

Easy and straightforward verification, in the same way as for Lemma 23.4.1,allows us to prove that:

Lemma 37.5.8. With the notation above, for each q > 0, we have

(1) γq−1γq = 0,

(2) γq−1χq = −χq−1γq .

Still in the same mood, we also have

Lemma 37.5.9. With the notation above, for each q > 0, we have χq−1χq =0.

Proof. Since

g(l, j) := g(Xi j Xil ti ) = g(Xi j , g(Xil ti ))


and g(l, j) = g( j, l), we obtain

χq−1χq(e(i, i)) =q∑

j=1

(−1) jυ jχq−1(e(g( j), i j))

=q∑

j=1

(−1) jυ j

j−1∑l=1

(−1)lυ(l, j)e(i, i; l, j)

+q∑

j=1

(−1) jυ j

q∑l= j+1

(−1)l−1υ( j,l)e(i, i; j, l)

=q∑

j=1

j−1∑l=1

((−1) j+l + (−1) j+l+1

)υ jυ(l, j)e(i, i; l, j)

= 0.

Proposition 37.5.10. For each q > 0, we have δq−1δq = 0.

Proof. Since

∆q−1∆q = γq−1γq − χq−1γq − γq−1χq + χq−1χq = 0

the claim follows from Lemma 37.5.7.

If we impose a T -degree on each Prq by defining

T -deg(e(i, i) := T(i, i)

then each module Im(δq) is T -homogeneous and each morphism is T -homogeneous of T -degree 1.

We can now impose a T -degree-compatible ordering ≺ on each k-basis

Bq := ωe(i, i) : ω ∈ T , e(i, i) ∈ Lqof Prq by setting

ωe(i, i) ≺ υe( j, j) ⇐⇒

⎧⎪⎪⎨⎪⎪⎩

deg(ti ) < deg(t j ),

ti > t j if deg(ti ) = deg(t j ),

Xiq . . . Xi1 > X jq . . . X j1 if ti = t j ,

ω < υ if e(i, i) = e( j, j),

where i = (i1, . . . , iq) and j = ( j1, . . . , jq) and < is the degrevlex orderinginduced by X1 < · · · < Xn .

Definition 37.5.11. A term ωe(i, i) ∈ Bq , i = (i1, . . . , iq), is called normal if

ω = 1 or max(ω) ≤

i1 if q ≥ 1,min(ti ) if q = 0.

722 Galligo

Proposition 37.5.12. For each q, if Nq and Tq denote the sets of all normal(respectively non-normal) terms in Bq , the following hold:

(1) for e(i, i) ∈ Bq+1, q ≥ 0, i = (i0, i1, . . . , iq), then

T≺(δq+1(e(i, i)) = Xi0e(i, j), j = (i1, . . . , iq);

(2) Tq ⊂ T≺(Im(δq+1));(3) let β ∈ Nq , β ′ ∈ Bq and let

δq(β ′) =∑

b∈Bq−1

c(δq(β ′), b)b ∈ Prq−1;

then c(δq(β ′), T≺(δq(β))) = 0 ⇒ β β ′;(4) Spank(Nq) ∩ ker(δq) = (0);(5) Im(δq+1) = ker(δq).

Proof.

(1) We have

δq+1(e(i, i)) =q∑

j=0

(−1) j+1Xi j e(i, i j)−∑

j∈A(i,i)

(−1) j+1υ j e(g( j), i j);

since i j > min(ti ) we have, by condition (b) of Proposition 37.5.3,

m( j) = m(Xi j ti ) = m(ti ) = ti ,

whence either deg(m( j)) < deg(ti ) or m( j) > ti ; therefore all terms inthe second sum are smaller than Xi0 e(i, j). The same is also true forthe first sum since

Xiq · · · Xi j+1 Xi j · · · Xi1 < Xiq · · · Xi j+1 Xi j−1 · · · Xi1 Xi0

for each j .(2) For any ωe(i, i) ∈ Tq , i = (i1, . . . , iq), we can express ω as ω = τ Xi0

with i0 = max(ω) and

i0 >

i1 if q > 0min(ti ) if q = 0

since ωe(i, i) is non-normal. Setting j = (i0, i1, . . . , iq), by the resultabove we have

ωe(i, i) = τ Xi0e(i, i)

= τT≺(δq+1(e(i, j)))

= T≺(τδq+1(e(i, j))) ∈ T≺(Im(δq+1)).


(3) We need different proofs according to whether q = 0 or q > 0.If q = 0 we can assume β = ωe(i) and β ′ = τe( j), so that δ0(β) =ωti , δ0(β

′) = τ t j and the assumption amounts to ωti = τ t j ; moreovermax(ω) ≤ min(ti ) since β is normal. Thus either deg(ti ) < deg(t j ) orti = m(τ t j ) ≥ m(t j ) = t j and β β ′.

If q > 0 we can assume β = ωe(i, i) and β ′ = τe( j, j), i =(i1, . . . , iq) and j = ( j1, . . . , jq). By assumption, for

γ := T≺(δq(β)) = Xi1ωe(i, i 1), where i 1 = (i2, . . . , iq),

we have c(δq(β ′), γ ) = 0. Either

γ = Xi1ωe(i, i 1) = τυle(g(X jl t j ), j l) where υlm(X jl t j ) = X jl t j

and ti = m(X jl t j ), so that either deg(ti ) < deg(t j ) or ti =m(X jl t j ) > m(t j ) = t j and, in both cases, β ≺ β ′; or

γ = Xi1ωe(i, i 1) = τ X jl e(t j , j l), ti = t j and we need to compareXi q · · · Xi1 with X j q · · · X j1 :

if l > 1, we have

Xiq · · · Xi2 = X jq · · · X jl+1 X jl−1 · · · X j1

ia = ja for q ≥ a > l, and il = jl−1 > jl so thatXi q · · · Xi 1 > X j q · · · X j 1 and β ≺ β ′;

if l = 1, then i 1 = j 1 so that Xi1ω = X j1τ ; since β isnormal i1 = max(Xi1ω) = max(X j1τ) and i1 ≥ j1so that

β = β ′ if i1 = j1 andβ ≺ β ′ if i1 > j1.

(4) Let f = ∑β ′∈Spank (Nq ) c( f, β ′)β ′ ∈ Spank(Nq) \ 0, β := T≺( f ) and

γ := T≺(δq(β)); by the last result we know that

c( f, β ′) = 0 ⇒ c(δq(β ′), γ ) = 0 for each β ′ ≺ β.

Therefore

c(δq( f ), γ ) = c( f, β)+∑

β′∈Spank (Nq )

β′ =β

c( f, β ′)c(δq(β ′), γ ) = c( f, β) = 0,

δq( f ) = 0 and f /∈ ker(δq).

(5) Since, by (2), we have

Prq = Im(δq+1)+ Spank(Nq) = ker(δq)+ Spank(Nq),

724 Galligo

(4) allows us to conclude that

Prq = Im(δq+1)⊕ Spank(Nq) = ker(δq)⊕ Spank(Nq)

and Im(δq+1) = ker(δq).

Theorem 37.5.13 (Eliahou–Kervaire). For a stable monomial ideal M =(t1, . . . , ts) ⊂ P, using the notation above, the sequence

0 → Prnδn−→ Prn−1 · · ·Prq+1

δq+1−→ Prqδq−→ Prq−1 · · ·Pr1

δ1−→ Pr0δ0−→ M

is a free resolution (the Eliahou–Kervaire resolution) of M.

Corollary 37.5.14. For a stable monomial ideal M = (t1, . . . , ts) ⊂ P,

writing, for each i, 1 ≤ i ≤ s, ν(i) := n −min(ti ), then:

(1) for each q, rq := ∑si=1

(ν(i)

q

)(2) H(I, T ) = ∑s

i=1 T deg(ti )(1− T )−n+ν(i).

Proof.

(1)(ν(i)

q

)is the cardinality of the set

(i1, . . . , iq) ∈ Iq : n ≥ i1 > i2 > · · · > iq > min(ti ).(2) Each element e := e(i, i), (i, i) ∈ Lq , i := (i1, . . . , iq) contributes

(−1)q T deg(e)(1− T )−n to the Hilbert series H(I, T ).

Since deg(e) = q + deg(ti ), we have

H(I, T ) =∑

q

(−1)qs∑

i=1

(ν(i)

q

)T q+deg(ti )(1− T )−n

=s∑

i=1

( ∞∑q=0

(−1)q(

ν(i)

q

)T q

)T deg(ti )(1− T )−n

=s∑

i=1

T deg(ti )(1− T )−n+ν(i).

38

Giusti

Throughout this chapter I assume char(k) = 0.The results of Macaulay on complete intersections (mainly Corollary 36.1.6)

and those of Galligo on the structure of the generic escalier are the two centraltools in the deep analysis performed by Giusti on the complexity of Buch-berger’s algorithm: the problem (as was stated at the end of Chapter 22) is toevaluate G<(I), the maximal degree of the elements of the Grobner basis w.r.t.a term ordering < of an ideal

I ⊂ k[X1, . . . , Xn] := P

given by a basis F in terms of

• n, the number of variables,• D := maxdeg( f ) : f ∈ F, the maximal degree of the elements of the input

basis,• d := dim(I), the dimension,• r := n − d, the rank,• λ := depth(I), the depth of I.

Giusti’s result relates G<(I) with Macaulay’s index of regularity and (Castel-nuovo–Mumford) regularity and proves that for a homogeneous ideal I ⊂ hPin generic position and for the degrevlex ordering < the double-exponentialbound

G<(I) ≤ (D(I) + 1

)r2d−λ

holds (Corollary 38.3.3). The strictness of the result is proved by Mayr–Meyer’s examples.

725

726 Giusti

Section 38.1 introduces the notation and states the relations betweenGrobner bound, index of regularity and (Castelnuovo–Mumford) regularity;Section 38.2 introduces the argument behind Giusti’s bound, which is provedin Section 38.3; Mayr–Meyer’s examples are proved in Section 38.4.

Section 38.5 presents a proof of the Bayer–Stillman result, reported inFact 24.9.12, on the optimality of degrevlex.

38.1 The Complexity of an Ideal

Let

k be a field of characteristic zero,I ⊂ k[X0, . . . , Xn] =: P be a homogeneous ideal,M := (

ci j) ∈ GL(n + 1, k) be a matrix,

Y0, Y1, . . . , Yn be the system of coordinates for k[X0, . . . , Xn] defined byYi := M(Xi ) = ∑

j ci j X j ,

G< be the Grobner basis of I ⊂ k[X0, . . . , Xn] w.r.t. the term ordering <,G<,M be the Grobner basis of M(I) ⊂ k[X0, . . . , Xn] w.r.t. the term order-

ing <,≺ be the degrevlex ordering induced by X1 ≺ · · · ≺ Xn ,

0 → Prρδρ−→ Prρ−1

δρ−1−→ · · ·Pri+1δi+1−→ Pri

δi−→ Pri−1 · · ·Pr1δ1−→

Pr0δ0−→ I be a minimal homogeneous resolution of I,

e(i)1 , . . . , e(i)

ri be the canonical basis of Pri , for each i .

Note that there is a non-empty Zariski open set U ⊂ GL(n +1, k) such that,for each M ∈ U,

• Y0, . . . , Yλ−1, λ := depth(I), is a regular sequence,• Y0, . . . , Yd−1, d := dim(I),1 is a maximal set of independent variables,• and, since char(k) = 0, the results of Chapter 37 hold so that there is a

monomial ideal ε(I) such that ε(I) = T≺(M(I)), for each M ∈ U.

1 In order to avoid ambiguities let me stress that for an affine ideal I ⊂ k[X1, . . . , Xn ] and ahomogeneous ideal J ⊂ k[X0, X1, . . . , Xn ] related by J = h I, I = aJ, I consider the followingrelation to be valid:

dim(I) = dim(J) − 1, r(I) = r(J).

38.1 The Complexity of an Ideal 727

Let us consider the following values:

γ (I), the index of regularity;Si (I) := max j deg(e(i)

j ), 1 ≤ j ≤ ri − i;S(I) := reg(I) := maxi Si , the regularity of I;G<(I) := maxdeg(g) : g ∈ G<;G<,M(I) := maxdeg(g) : g ∈ G<,M;G(I) := maxdeg(τ ) : τ ∈ G(ε(I)) = G≺,M(I), M ∈ U, where G(ε(I)) is the

minimal basis of ε(I);

and let us evaluate them in terms of

n, the number of variables,λ := depth(I),d := dim(I),D(I) := maxdeg( f ) : f ∈ F where F is a generating basis of I.

Note that, as a direct consequence of Macaulay’s results, the following triv-ially holds

Lemma 38.1.1. Let

I ⊂ k[X0, . . . , Xn] be a homogeneous ideal,λ := depth(I),Y0, Y1, . . . , Yn be a system of coordinates for k[X0, . . . , Xn] such thatY0, . . . , Yλ−1 is a regular sequence,J := I + (Y0, . . . , Yλ−1).

Then

J ∩ k[Yλ, . . . , Yn] = I ∩ k[Yλ, . . . , Yn] =: L ⊂ k[Yλ, . . . , Yn],γ (L) = γ (J) = γ (I) − λ,depth(L) = depth(J) = 0,dim(L) = dim(J) = dim(I) − λ,r(L) = r(J) − λ = r(I),G(L) = G(I) = G(J).

Theorem 38.1.2. With these assumptions and notation we have

(1) S(I)/(n + 1) < G(I),(2) S1(I) ≤ G(I),(3) S(I) = G(I),(4) G(I) = γ (I) + depth(I),(5) γ (I) ≤ S(I).

728 Giusti

Proof.

(1) As a consequence of Taylor’s resolution (Lemma 23.4.1) we have

Si (I) + i ≤ (i + 1)G(I)

whence the claim.(2) This is a direct consequence of Galligo’s results (Section 37.4) which

imply deg(e(1)j ) ≤ G(I) + 1 for each j .

(3) See Theorem 38.5.11 below.(4) By Lemma 38.1.1 we are reduced to the case of an ideal I such that

depth(I) = 0 for which G(I) = γ (I) is a direct consequence of Galligo’sresult.

(5) Hilbert’s formula (Corollary 20.7.1) gives that

γ (I) ≤ maxj

deg(e(i)j , 1 ≤ j ≤ ri , 1 ≤ i ≤ ρ − n

≤ maxi

Si + i − n ≤ S(I).

38.2 Toward Giusti’s Bound

Using the same notation as in the section above, our aim is to evaluate G(I).Writing I( j) := I + (Y0, . . . , Y j−1), our strategy is to evaluate each

value G(I( j)) by decreasing induction on j = d, . . . , λ, noting that, byLemma 38.1.1, we have G(I) = G(I(λ)).

The basic idea behind this iterative evaluation is the following: let us assumethat we begin with an input basis G := F = f0, . . . , fs of homogenous ele-ments all having degree δ := D(I).

Since we assume that we are in generic position, the Borel condition impliesdirectly that

TF = Y δn , Yn−1Y δ−1

n , . . . , Y in−1Y δ−i

n , . . . , Y sn−1Y δ−s

n and that each S-pair element has exactly degree D(I) + 1; if we perform thereduction of such S-pairs we therefore obtain a set of polynomials of degreeD(I) + 1 to be added to the basis.

Since we want to analyse the degree obtained when T(I) is increasing asslowly as possible, we will assume that all such S-pairs except one reduceto zero. Therefore we upgrade the basis G by adding a single polynomialg1, deg(g1) = D(I) + 1 obtaining G1 := G ∪ g1; the Borel condition alsoforces the value of T(g1) which necessarily is

T(g1) := min≺ τ ∈ N(G), deg(τ ) = D(I) + 1.We now have to consider all useful S-pairs S(g1, f ), f ∈ G, combining g1

with all the polynomials in G; Borel again forces deg(S(g1, f )) = D(I) + 2;

38.2 Toward Giusti’s Bound 729

again we will assume that just a single S-pair does not reduce to zero, thusproducing a single polynomial g2:

deg(g2) = D(I) + 2, T(g2) := min≺ τ ∈ N(G1), deg(τ ) = D(I) + 2,which is added to G1, giving G2 := G1 ∪ g2.

We therefore assume that in each loop the algorithm produces all useful S-pairs S(gi , f ), f ∈ Gi , all having degree D(I) + i + 1 and such that at mostone of them is reduced to a non-zero polynomial gi+1:

deg(gi+1) = D(I)+ i +1, T(gi+1) := min≺ τ ∈ N(Gi ), deg(τ ) = D(I)+1,which is added to Gi , giving Gi+1 := Gi ∪ gi+1.

So, approximately, we can expect that in each degree just a single polyno-mial of that degree is added to the basis.

This expectation is quite pessimistic as proved by the following:

Example 38.2.1. Let us consider an ideal generated in degree 3 by 2 polyno-mials in k[X, Y, Z ] and the degrevlex ordering < induced by X < Y < Z .In accordance with the scenario described above we start with X3, X2Y and,at any degree, we add the least monomial not included in the monomial idealunder construction. The result is:

X3, X2Y, XY 3, Y 5∪ Y 4 Z2, XY 2 Z4, Y 3 Z5, X2 Z7, XY Z8, Y 2 Z9, X Z11, Y Z12 Z14

which defines a monomial ideal M which is not Borel; for instance X Z11 ∈ Mbut Z12 /∈ M.

Therefore a deeper analysis of the Borel structure is required.


(1) N(I( j)) = N(I) ∩ T [ j, n],(2) N(I( j)) ⊂ Y a j

j τ, τ ∈ N(I( j+1)).

Proof.

(1) Since I( j) = I + (Y0, . . . , Y j−1).(2) We have

N(I( j)) = t ∈ T [ j, n] : t ∈ N(I)= Y a j

j τ ∈ N(I) : τ ∈ T [ j + 1, n], a j ∈ N)⊂ Y a j

j τ : τ ∈ N(I) ∩ T [ j + 1, n], a j ∈ N)= Y a j

j τ, τ ∈ N(I( j+1)).

730 Giusti

Let us perform, for each j, d > j ≥ λ, a partition of N(I( j+1)) as

N(I( j+1)) = N∞(I( j+1)) Nfin(I( j+1))

where

N∞(I( j+1)) := τ ∈ N(I( j+1)) : for each a ∈ N, Y aj τ ∈ N(I),

Nfin(I( j+1)) := τ ∈ N(I( j+1)) : there exists a ∈ N, a = 0, Y aj τ ∈ T(I),

and for each τ ∈ Nfin(I( j+1)) denote aτ ∈ N by the value such that Y aτ

j τ ∈T(I), Y aτ −1

j τ /∈ T(I).The structure of both subsets is partially determined by the fact that N(I) is

Borel; in particular, since we have Y0 ≺ · · · ≺ Yn we have

τ1 ≺ τ2, τ1 ∈ T(I) ⇒ τ2 ∈ T(I),

and for each τ ∈ T [ j + 1, n], Y deg(τ )

j ≺ τ.

Lemma 38.2.3. For each j, d > j ≥ λ, Nfin(I( j+1)) is finite.

Proof. The statement is true for j = d: in fact Nfin(I(d+1)) = N(I(d+1)) isfinite since I(d+1) is irrelevant.

By decreasing induction, the finiteness of Nfin(I( j+1)) implies

Nfin(I( j)) = Y a jj τ, τ ∈ Nfin(I( j+1)), a j < aτ .

In fact if there are τ ∈ N∞(I( j+1)) and a j ∈ N such that Ya jj τ ∈ Nfin(I( j))

then for some a j−1 ∈ N we have Ya j−1j−1 Y

a jj τ ∈ T(I) but this would imply

Ya j−1+a jj τ ∈ T(I).

Lemma 38.2.4. With the notation above, for each j, j ≥ λ, each τ ∈Nfin(I( j+1)) and each ω ∈ T [ j + 1, n], writing δ := deg(ω), we have

(1) aτ ≤ δ ⇒ τω ∈ T(I( j)),

(2) aτ > δ ⇒ τω ∈ Nfin(I( j+1)), aτω ≤ aτ − δ.

Proof. By assumption Y aτ

j τ ∈ T(I) and Y δj τ ≺ τω. Therefore:

(1) if aτ ≤ δ, let ω1, ω2 ∈ T [ j + 1, n] be such that ω1ω2 = ω anddeg(ω1) = aτ ; then

Y aτ

j τ ∈ T(I) ⇒ τω1 ∈ T(I( j)) ⇒ τω ∈ T(I( j));

(2) aτ > δ ⇒ Y aτ −δj τω ∈ T(I( j)).

38.2 Toward Giusti’s Bound 731

This apparently allows us to upperbound the growth of G(I( j)); in fact weassume that in each degree at least one polynomial of that degree is added tothe basis, and the elements to be added to the Grobner basis of I( j+1) in orderto obtain that of I( j) have the form Y a

j τ, τ ∈ Nfin(I( j+1)), therefore we candeduce

G(I( j)) ≤ G(I( j+1)) + #(Nfin(I( j+1))).

We cannot easily evaluate 2 #(Nfin(I( j+1))) but, since N(I) is Borel, we have

Nfin(I( j+1)) ⊂ τ ∈ N(I( j+1)) : deg(τ ) < G(I( j+1)) =: F ( j).

We therefore need to investigate the relation between the Grobner bases ofI( j+1) and I( j):

Lemma 38.2.5. We have

T≺(I) ∩ k[Y j , . . . , Yn] = T≺(I( j)) ∩ k[Y j , . . . , Yn].

Proof. Note that for any polynomial g ∈ k[Y0, . . . , Yn]

T≺(g) = Ya jj . . . Y an

n , a j > 0 ⇒ g ∈ k[Y j , . . . , Yn], Ya jj | g.

Therefore for any τ ∈ T≺(I) ∩ k[Y j , . . . , Yn], there is g ∈ I ⊂ I( j) ∩k[Y j , . . . , Yn] such that τ ∈ T≺(I( j)) ∩ k[Y j , . . . , Yn].

Conversely if τ ∈ T≺(I( j)) ∩ k[Y j , . . . , Yn], then there is g ∈ I( j) suchthat T≺(g) = τ ∈ k[Y j , . . . , Yn]. This implies the existence of h ∈ I and

h0, . . . , h j−1 ∈ k[Y0, . . . , Yn] such that g = h + ∑ j−1i=0 Yi hi . Then either

T≺(h) ∈ (Y0, . . . , Y j−1), h ∈ (Y0, . . . , Y j−1) and we get the contradictiong ∈ (Y0, . . . , Y j−1)

or τ = T≺(h) ∈ T≺(I) ∩ k[Y j , . . . , Yn].

Corollary 38.2.6. Let G be the Grobner basis of I w.r.t. ≺ and for each j ,G( j+1) ⊂ k[Y j+1, . . . , Yn] be a set such that G( j+1) ∪ Y0, . . . , Y j is aGrobner basis of I( j+1).

Then, writing

G∗ := g ∈ G, T≺(g) ∈ T [ j, n],G ′ := g ∈ G, T≺(g) = Y

a jj . . . Y an

n , a j > 0,G ′′ := g ∈ G, T≺(g) ∈ (Y0, . . . , Y j−1),

2 As shown by Example 38.2.1.

732 Giusti

we have

(1) G∗ ∪ Y0, . . . , Y j−1 is a Grobner basis of I( j),(2) G( j+1) ∪ G ′ is a Grobner basis of I( j),(3) G( j+1) ∪ G ′ ∪ G ′′ is a Grobner basis of I w.r.t. ≺.

Proof.

(1) Let h ∈ I( j) ∩ k[Y j , . . . , Yn], so that

T≺(h) ∈ T≺(I( j)) ∩ k[Y j , . . . , Yn] = T≺(I) ∩ k[Y j , . . . , Yn],

and let g ∈ G ⊂ I be such that T≺(g) | T≺(h); then T≺(g) ∈k[Y j , . . . , Yn], and g ∈ G∗.

(2) Let h ∈ I( j) ∩ k[Y j , . . . , Yn], so that

T≺(h) ∈ T≺(I( j)) ∩ k[Y j , . . . , Yn] = T≺(I) ∩ k[Y j , . . . , Yn],

and let g ∈ G ⊂ I be such that T≺(g) | T≺(h); then either

T≺(g) ∈ (Y j ) and g ∈ G ′ orT≺(g) = Y ai

i . . . Y ann , ai > 0 with i > j and

T≺(g) ∈ T≺(I) ∩ k[Y j+1, . . . , Yn] = T≺(I( j+1)) ∩ k[Y j+1, . . . , Yn],

so there is g′ ∈ G( j+1) such that T≺(g′) | T≺(h).

(3) Let h ∈ I ⊂ I( j+1), and let g ∈ G ⊂ I be such that T≺(g) | T≺(h).Then either

T≺(g) ∈ (Y0, . . . , Y j ) and g ∈ G ′ ∪ G ′′ orT≺(g) = Y ai

i . . . Y ann , ai > 0 with i > j and

T≺(g) ∈ T≺(I) ∩ k[Y j+1, . . . , Yn] = T≺(I( j+1)) ∩ k[Y j+1, . . . , Yn],

so there is g′ ∈ G( j+1) such that T≺(g′) | T≺(h).

On the basis of this discussion we can conclude that our aim, to itera-tively evaluate the values G(I( j)) and F ( j), requires us to deduce G( j) ⊂k[Y j , . . . , Yn] from G( j+1) ⊂ k[Y j+1, . . . , Yn].

So we can wlog assume we have an ideal I ⊂ k[Y j , . . . , Yn] and the Grobnerbasis G w.r.t. ≺ of I1 := I + Y j and our aim is to evaluate G(I) and thecardinality of

F(I) := τ ∈ N(I), deg(τ ) < G(I)= Y a

j τ ∈ N(I), τ ∈ N(I1), a ∈ N, deg(Y aj τ) < G(I)

in terms of G(I1) and of F(I1) := τ ∈ N(I1), deg(τ ) < G(I1).

38.3 Giusti’s Bound 733

Theorem 38.2.7 (Giusti). We have

(1) G(I) ≤ G(I1) + #(F(I1)),(2) #F(I) ≤ (#F(I1))2.

Proof. Our previous discussion tells us that F(I1) = Ffin(I1) F∞(I1) where

F∞(I1) := τ ∈ F(I1) : for each a ∈ N, Y aj τ ∈ N(I),

Ffin(I1) := τ ∈ F(I1) : there exists a ∈ N, a = 0, Y aj τ ∈ T(I)

and that

G(I) ≤ G(I1) + #Ffin(I1) ≤ G(I1) + #F(I1).

We can now partition F(I) as F(I) = ⊔δ Fδ(I) where

Fδ(I) := Y δj τ ∈ N(I), τ ∈ N(I1), deg(Y δ

j τ) < G(I).The Borel condition gives that, for each τ ∈ T [ j + 1, n] and each l > j ,

Y δ−1j τYl /∈ F ( j)

δ−1(I) ⇒ Y δj τ /∈ F ( j)

δ (I).

We therefore have

F0(I) = F(I1),

Fδ(I) = Y δj τ : Y δ−1

j τ ∈ Fδ−1(I), Ylτ /∈ T(I) for each l > j= Y δ

j τ : τ ∈ F0(I) : ωτ ∈ T(I), ∀ω ∈ T [ j + 1, n], deg(ω) = δ,and #Fδ(I) ≤ #Fδ−1(I) − 1, whence

#Fδ(I) ≤ #Fδ−1(I) − 1

≤ #Fδ−i (I) − i

≤ #F0(I) − δ

= #F(I1) − δ.

Therefore we obtain

#F(I) ≤#F(I1)∑δ=0

(#F(I1) − δ

)<

#F(I1)∑δ=0

#F(I1) ≤ (#F(I1))2 .

38.3 Giusti’s Bound

Using the same notation as in the last sections, let us now apply Theo-rem 38.2.7 in order to evaluate G(I) = γ (I) + depth(I).

Let us begin by recording this reformulation of Corollary 36.1.6:

734 Giusti

Corollary 38.3.1. For any homogeneous ideal

I ⊂ k[X0, . . . , Xn], depth(I) = dim(I) = 0, r(I) = n + 1,

we have

• G(I) ≤ (n + 1)(D(I) − 1) + 1,

• #F(I) ≤ D(I)n+1 .

Proposition 38.3.2 (Giusti). For any homogeneous ideal

I ⊂ k[X0, . . . , Xn], depth(I) = 0, dim(I) = d > 0, r(I) = r = n + 1 − d,

we have

• G(I) ≤ (D(I) + 1

)r2d

,

• #F(I) ≤ D(I)r2d+1.

Proof. We will directly apply the result of Theorem 38.2.7 using freely thenotation set out there. So we can consider the ideal I1 := I + Y j for whichwe have D(I1) = D(I) and

depth(I1) = 0 = depth(I),dim(I1) = d − 1 = dim(I) − 1,

r(I1) = n − d + 1 = r(I),

and we can deduce the values for I from those for I1 by induction ondim(I).

For dim(I) = 0, Corollary 36.1.6 gives G(I1) ≤ 1 + r(D(I1) − 1), whence

#F(I) ≤ (#F(I1)

)2

= D(I)r2,

G(I) ≤ G(I1) + #F(I1)

≤ 1 + r(D(I1) − 1) + (D(I1)

)r

≤ D(I + 1)r .

Then inductively for dim(I) = d

#F(I) ≤ (#F(I1)

)2

≤(

D(I)r2d)2

= D(I)r2d+1,

G(I) ≤ G(I1) + #F(I1)

≤ (D(I) + 1

)r2d−1 + D(I)r2d

38.4 Mayr and Meyer’s Example 735

≤ (D(I)2 + D(I) + 1

)r2d−1

≤ (D(I) + 1

)r2d

.

Corollary 38.3.3 (Giusti). For any homogeneous ideal

I ⊂ k[X0, . . . , Xn], depth(I) = λ, dim(I) = d, r(I) = r = n + 1 − d,

we have

• if d − λ = 0 then

• G(I) ≤ r(D(I) − 1) + 1,

• #F(I) ≤ D(I)r

• if d − λ > 0 then

• G(I) ≤ (D(I) + 1

)r2d−λ

,

• #F(I) ≤ D(I)r2d−λ+1.

Proof. By Lemma 38.1.1 for L := I ∩ k[Yλ, . . . , Yn] we have

G(L) = G(I),

#F(L) = #F(I),

D(L) = D(I),

depth(L) = 0,

dim(L) = d − λ,

r(L) = r.

38.4 Mayr and Meyer’s Example

Fix an integer d ≤ 2 and define:

for each n ∈ N, en := d2n, so that, in particular en = e2

n−1;P0 := k[S0, F0, C10, C20, C30, C40, B10, B20, B30, B40];Pi := Pi−1[Si , Fi , C1i , C2i , C3i , C4i , B1i , B2i , B3i , B4i ] for each i > 0;for each i ∈ N, Ti , the monomial k-basis of Pi ;I0 ⊂ P0, the ideal generated by

S0Ci0 − F0Ci0 Bdi0, 1 ≤ i ≤ 4;

for each i > 0, Ii ⊂ Pi the ideal generated by Ii−1 and by the following ten

736 Giusti

new generators

(a) Si − Si−1C1i−1,(b) Fi−1C1i−1 B1i−1 − Si−1C2i−1,(c) Fi−1C2i−1 − Fi−1C3i−1,(d) Si−1C3i−1 B1i−1 − Si−1C2i−1 B4i−1,(e) Si−1C3i−1 − Fi−1C4i−1 B4i−1,(f) Si−1C4i−1 − Fi ,(g) C1i Fi−1 B2i−1 − C1i B1i Fi−1 B3i−1,(h) C2i Fi−1 B2i−1 − C2i B2i Fi−1 B3i−1,(i) C3i Fi−1 B2i−1 − C3i B3i Fi−1 B3i−1,(j) C4i Fi−1 B2i−1 − C4i B4i Fi−1 B3i−1;

Bi , the basis of Ii consisting of the 4 + 10i generators listed here.

Since each ideal Ii is a binomial ideal, it defines on Ti the equivalencerelation ∼,

α ∼ β ⇐⇒ α − β ∈ Ii

which is generated by the antisymmetric relation → which is defined by

α → β ⇐⇒ there exists τ ∈ Ti , α′ − β ′ ∈ Bi : α = τα′, β = τβ ′.

We will also denote by ↔ the symmetric relation generated by →.

Theorem 38.4.1 (Mayr–Meyer). If α ∈ (Sn, Fn), then

SnCin ∼ α ⇐⇒ either α = SnCin or α = FnCin Benin

Proof. The proof

produces a finite, repetition-free, derivation

SnCin = γ0 ↔ γ1 ↔ · · · ↔ γr

where γr = FnCin Benin and, at the same time,

proves that such derivation is the single repetition-free derivation such thatγr ∈ (Sn, Fn).

The statement being trivial for n = 0, the proof will be performed by induc-tion. In order to simplify the notation, we will denote by X (respectively x)the variable Xn (respectively Xn−1) so that, for example, Ci f b2 − Ci Bi f b3

represents Cin Fn−1 B2n−1 − Cin Bin Fn−1 B3n−1.


We have

SCi ↔ Ci sc1 (38.1)

↔ · · ·↔ Ci f c1ben−1

1 (38.2)

↔ Ci sc2ben−1−11 (38.3)

↔ · · ·↔ Ci f c2ben−1

2 ben−1−11 (38.4)

↔ · · ·↔ Ci Ben−1

i f c2ben−13 ben−1−1

1 (38.5)

↔ Ci Ben−1i f c3ben−1

3 ben−1−11 (38.6)

↔ · · ·↔ Ci Ben−1

i sc3ben−1−11 (38.7)

↔ Ci Ben−1i sc2ben−1−2

1 b4 (38.8)

↔ · · ·↔ Ci Ben−1en−1

i sc3ben−1−14 (38.9)

↔ Ci Beni f c4ben−1

4 (38.10)

↔ · · ·↔ Ci Ben

i sc4 (38.11)

↔ FCi Beni . (38.12)

Let us begin by noting that except for j = 0 and j = r , γ j /∈ (S, F): in facteach appearance of S (respectively F) can only be obtained by performing

γ j−1 = τ sc1 ↔ τ S = γ j (respectively γ j−1 = τ sc4 ↔ τ F = γ j )

and the next reduction necessarily is

γ j = τ S ↔ τ sc1 = γ j+1(respectively γ j = τ F ↔ τ sc4 = γ j+1)

implying γ j−1 = γ j+1 and contradicting the assumption that the derivation isrepetition-free.

Then we have:

(38.1) By (a), which is the only applicable relation.(38.2) By induction assumption. Note that the only applicable relation on

Ci sc1 is Sn−1 − Sn−2C1n−2; denoting j1 the minimal value j1 suchthat γ j1 ∈ (s, f ), in the segment of reduction

Ci c1Sn−2C1n−2 = γ2 ↔ · · · ↔ γ j ↔ · · · ↔ γ j1

738 Giusti

we necessarily have, for each j, 2 ≤ j ≤ j1, γ j = Ciγ′j for some

γ ′j ∈ Tn−1 and a derivation

c1Sn−2C1n−2 = γ ′2 ↔ · · · ↔ γ ′

j ↔ · · · ↔ γ ′j1 .

The inductive assumption implies that there is a single such derivationand that either

γ ′j1

= sc1 and γ j1 = Ci sc1 = γ1, which is impossible since by as-sumption the derivation is repetition-free, or

γ ′j1

= f c1ben−11 and γ j1 = Ci f c1ben−1

1 .

(38.3) Since we assume that the derivation is repetition-free, (b) is the onlyapplicable relation and returns γ j1+1 = Ci sc2ben−1−1

1 .(38.4) As in (38.2) we can apply only (a) necessarily followed by a single

repetition-free reduction

Ci ben−1−11 c2Sn−2C1n−2 = γ j1+2 ↔ · · · ↔ γ j ↔ · · · ↔ γ j2

where

γ j2 ∈ (s, f ), γ j /∈ (s, f ) for each j, j1 + 1 < j < j2,

so that, for each such j ,

γ j = Ciγ′j , γ ′

j ∈ Tn−1, γ′j+1 − γ ′

j ∈ In−1.

This also implies the more important fact that c2 | γ ′j for each j since

no relation in In−1 can change it.So, by the same argument as in (38.2), we conclude that

γ j2 = Ci ben−1−11 f c2ben−1

2 .

(38.5) Here we can iteratively apply k times,3 0 ≤ k ≤ en−1, the properrelation among (g)–(j) obtaining γ j3 = Ci Bk

i ben−1−11 ben−1−k

2 f c2bk3.

(38.6) This is followed by (c), giving γ j3+1 = Ci Bki ben−1−1

1 ben−1−k2 f c3bk

3.

(38.7) Here we need again to have recourse to the induction assumption, sincethe only applicable relation is Fn−1 − Sn−2C4n−2; therefore there area minimal value j4 and elements γ ′

j ∈ Tn−1, j3 + 1 < j ≤ j4, suchthat

Ci Bki γ ′

j4 = γ j4 ∈ (s, f )

and

Ci Bki γ ′

j := γ j /∈ (s, f ) for each j, j3 + 1 < j < j4.

3 We will prove in (38.7) that necessarily k = en−1.


As in (38.4) we can deduce that c3 | γ ′j for each j , and this implies

that there are elements γ ′′j ∈ Tn−1, j3 + 1 ≤ j ≤ j4, such that c3 | γ ′′

jand

γ j = Ci Bki ben−1−1

1 ben−1−k2 γ ′′

j for each j, j3 + 1 ≤ j ≤ j4.

Since γ ′′j4

∈ (s, f ) we can refine the same argument as in 38.2:

if γ ′′j4

= f c3η, η ∈ Tn−1, then, the assumption that the derivation is

repetition-free implies η = bk3. Then the derivation

sc3 ↔ · · · ↔ f c3ben−13 = f c3bk

3ben−1−k3 = γ ′′

j3+1ben−1−k3

↔ · · · ↔ γ ′′j4 ben−1−k

3 = f c3ηben−1−k3 = f c3ben−1

3

contradicts the inductive assumption;if γ ′′

j4= sc3η, η ∈ Tn−1, we have the derivation

sc3 ↔ · · · ↔ f c3ben−13 = f c3bk

3ben−1−k3 = γ ′′

j3+1ben−1−k3

↔ · · · ↔ γ ′′j4 ben−1−k

3 = sc3ηben−1−k3 ;

then the inductive assumption implies sc3 = sc3ηben−1−k3 ,

that is η = 1 and en−1 = k.

In conclusion we know that

γ j3 = Ci Ben−1i f c2ben−1−1

1 ben−13 ,

γ j3+1 = Ci Ben−1i f c3ben−1−1

1 ben−13 ,

γ j4 = Ci Ben−1i sc3ben−1−1

1 .

(38.8) An application of (e) would lead to a series of reductions in Tn−1

sc3 ↔ f c4b4 ↔ · · · ↔ f c4b4η

and to the (impossible) relation sc3 = f c4b4η. So the only applicablerelation is (d) which leads to

γ j4+1 := Ci Ben−1i ben−1−2

1 b4sc2.

(38.9) In the same way, if we now apply (b), we obtain

sc2 ↔ f c1b1 ↔ · · · ↔ f c1b1η

and a contradiction.Therefore, we can only iterate en−1 times the same argument which

deduced the reduction

γ j1+1 = Ci sc2ben−1−11 ↔ · · · ↔ γ j4+1 := Ci Ben−1

i sc2ben−1−21 b4,

740 Giusti

thus finally obtaining

γ j5 := Ci Ben−1en−1i sc3ben−1−1

4 .

(38.10) Here we can only apply (e).(38.11) As in 38.7 the only applicable relation is Fn−1 − Sn−2C4n−2 and a

similar argument gives the required result.(38.12) Finally the only applicable relation (f) allows us to conclude.

Corollary 38.4.2 (Lazard). For each integer d ≥ 2 and each n ∈ N,

(1) there is an ideal Idn generated by 10n + 3 polynomials in 10n + 4variables and degree bounded by d + 2, which has S1(I) ≥ en−1 =d2n−1

, and(2) there are an ideal Jdn generated by 10n + 2 polynomials

p1, . . . , p10n+2 in 10n +2 variables and degree bounded by d +2, anda polynomial p ∈ Jdn for which each representation p = ∑10n+2

i=1 gi pi

satisfies

deg(gi ) + deg(pi ) ≥ en−1 = d2n−1.

Proof. Let us enumerate Bn as

Bn := f1, . . . , f10n+4.Consider the projection

πn : Pn → Pn−1[Sn, Fn]

defined by πn(Cin) = πn(Bin) = 1, 1 ≤ i ≤ 4, so that 4

πn( f10n+i ) = πn(Ci f b2 − Ci Bi f b3) = f b2 − f b3, 1 ≤ i ≤ 4,

and let

(1) Idn ⊂ Pn−1[Sn, Fn, Bin, Cin] be the ideal generated by

f j 1 ≤ j ≤ 10n ∪ f10n+i , Sn, Fn(2) Jdn ⊂ Pn−1[Sn, Fn] be the ideal generated by

πn(Bn) ∪ Fn = πn( fi ), 1 ≤ i ≤ 10n ∪ f b2 − f b3, Fand p := Sn .

Then, the single repetition-free derivation

SnCin = γ0 ↔ γ1 ↔ · · · ↔ FnCin Benin

4 Using the same shorthand as in the proof of the theorem.

38.5 Optimality of Revlex 741

returns

(1) a syzygy

SnCin −10n∑i=1

gi fi − g10n+1 f10n+i − FnCin Benin = 0,

where, necessarily,

max(deg(gi fi ) ≥ deg(Ci Ben−1i f c2ben−1

3 ben−1−11 )

(2) a polynomial representation

S =10n∑i=1

πn(gi )πn( fi ) + πn(g10n+1)( f b2 − f b3) + F

where, necessarily,

maxdeg(πn(gi fi )) ≥ deg( f c2ben−13 ben−1−1

1 ) ≥ en−1.

Compare this result with the Nullstellensatz (Corollary 23.10.6) which givesthe existence of elements gi : 0 ≤ i ≤ 10n + 2 such that

Sen = g0 Fn + ∑

i gi fi

deg(gi ) + deg( fi ) ≤ e

with e ≤ max(310n+2, d10n+2).

38.5 Optimality of Revlex

Let I ⊂ k[X0, . . . , Xn] =: P be a homogeneous ideal. We need to state acharacterization of regularity, whose cohomological proof is out side the scopeof the book:5

Definition 38.5.1. A linear form Y ∈ P is called generic for a homogeneousideal J ⊂ P , dim(J) > 0, if Y is not a zero-divisor on P/Jsat; with an abuseof notation, we consider any linear form as generic for an irrelevant homoge-neous ideal.

For each j ≥ 0 denote by U j (I) the set of all sequences (Y0, . . . , Y j−1)

of linear forms such that, for each i, 0 ≤ i < j , Yi is generic forI + (Y0, . . . , Yi−1).

5 For a proof see D. Bayer, and M. Stillman, A criterion for detecting m-regularity, Invent. Math.87 (1987), 1–11.

742 Giusti

Note that, k being infinite, the set of all generic elements for any homoge-neous ideal J is a non-empty Zariski open subset of P1; as a consequence eachU j (I) is a non-empty Zariski open subset.

Fact 38.5.2. Assuming that I is generated in degree bounded by m, settingd := dim(I), the following conditions are equivalent:

(1) reg(I) ≤ m;(2) there are linear forms Y0, . . . , Y j−1, for some j ≥ 0, such that(

(I + (Y0, . . . , Yi−1)) : Yi)

m = (I + (Y0, . . . , Yi−1))m for i, 1≤ i < j ,(I + (Y0, . . . , Y j−1)

)m = Pm;

(3) for any (Y0, . . . , Yd−1) ∈ Ud(I) and any p ≥ m((I + (Y0, . . . , Yi−1)) : Yi

)p = (I + (Y0, . . . , Yi−1))p for i, 0 ≤ i < d,

(I + (Y0, . . . , Yd−1))p = Pp.

Furthermore, any sequence (Y0, . . . , Y j−1) satisfying (2) is a member of U j (I).

Corollary 38.5.3. For any term ordering <, reg(I) ≤ reg(T<(I)).

Proof. It is sufficient to apply Algorithm 23.8.3.

Proposition 38.5.4 (Bayer–Stillman). Let E ⊂ k[X0, . . . , Xn] be a Borelmonomial ideal, generated by monomials of degree bounded by m and hav-ing a minimal generator of degree m. Then reg(E) = m.

Proof. Since E is Borel and contains a monomial of degree m, then Xmn ∈ E.

Let d ≥ 0 be the value such that

X δd−1 /∈ E for each δ ∈ N and

Xµd ∈ E for some µ ∈ N.

Since E is generated in degree bounded by m we have µ ≤ m and Xmd ∈ E.

In order to show that reg(E) = m, it is sufficient to prove, by Fact 38.5.2,that(

(E + (X0, . . . , Xi−1)) : Xi)

m = (E + (X0, . . . , Xi−1))m for i, 1 ≤ i < d,(E + (X0, . . . , Xd−1))m = Pm .

Since E is Borel,

Xmd ∈ E ⇒ T [d, n] ∩ Tm ⊂ E


so that

(E + (X0, . . . , Xd−1))m = Pm .

Fix i, 0 ≤ i < d and write J := E + (X0, . . . , Xi−1); for any term τ ∈ Tm

such that Xiτ ∈ J, either

τ is divided by some X0, . . . , Xi−1 and so τ ∈ J, orXiτ ∈ E; since deg(Xiτ) = m + 1, Xiτ is not a minimal generator of E and

Xiτ = X jω for some j ≥ i and ω ∈ E. Either

j = i and τ = ω ∈ E ⊂ J, orj > i and ω = Xiυ for a suitable υ ∈ T so that τ = X jυ. Since E

is Borel,

ω = Xiυ ∈ E ⇒ τ = X jυ ∈ E ⊂ J.

Therefore(J : Xi

)m = Jm .

Corollary 38.5.5. For any term ordering <, there is a non-empty Zariski openset U ⊂ GL(n + 1, k) for which

G<,M(I) ≥ reg(I) for each M ∈ U.

Proof. Since char(k) = 0, the results of Chapter 37 hold and there are a non-empty Zariski open set U ⊂ GL(n + 1, k) and a Borel ideal E such that E =T<(M(I)), for each M ∈ U.

Corollary 38.5.4 then implies, for each M ∈ U,

reg(I) = reg(M(I)) ≤ reg(T<(M(I))) = reg(E) = G<(E) = G<,M(I).

We show now that, if we restrict < to the rev-lex ordering induced by X0 <

· · · < Xn , the inequality becomes an equality.Let us begin by recalling (Lemma 26.3.12) that for each i ≤ n and each

f ∈ k[Xi , . . . , Xn]

Xi | f ⇐⇒ Xi | T<( f ).

Lemma 38.5.6. For the rev-lex ordering < induced by X0 < · · ·< Xn and anyi, 0 ≤ i ≤ n, the following hold:

(1) T< (I + (X0, . . . , Xi )) = T<(I) + (X0, . . . , Xi );(2) if X0, . . . , Xi−1 ∈ I and m ≥ 0, then

(I : Xi )m = Im ⇐⇒ (T<(I) : Xi )m = (T<(I))m ;

744 Giusti

(3) if

X0, . . . , Xi−1 ∈ I,m ≥ 0,(I : Xi )p = Ip for each p ≥ m,(T<(I) + (Xi )) is generated in degree bounded by m,

then T<(I) is generated in degree bounded by m.

Proof.

(1) T< (I + (X0, . . . , Xi )) ⊇ T<(I)+ (X0, . . . , Xi ) for any term ordering.Let f ∈ I + (X0, . . . , Xi ): if X j | T<( f ) for some j T<(h0 X0 + · · · + hi Xi ), we have T<( f ) = T<(g) ∈T<(I).

(2) Assume (I : Xi )m = Im and let τ be a term, deg(τ ) = m: if Xiτ ∈T<(I), then Xiτ = T<( f ) for some f ∈ Im+1. Either

X j | τ for some j < i and τ ∈ (T<(I))m ,or we can express f as

f = g + h0 X0 + · · · + hi−1 Xi−1 with g ∈ Im+1 ∩ k[Xi , . . . , Xn],

and Xiτ = T<( f ) = T<(g).

In the second case g = Xi h for some h ∈ k[Xi , . . . , Xn], deg(h) = mand T<(h) = τ. Since g = Xi h ∈ Im+1, then h ∈ (I : Xi )m = Im andT<(h) = τ ∈ (T<(I))m .

Conversely let us assume that (T<(I) : Xi )m = (T<(I))m . Let us con-sider an element Xi f ∈ Im+1 and let us inductively assume that, foreach g ∈ Pm,

T<(g) < T<( f ), Xi g ∈ I ⇒ g ∈ I.

Since Xi T<( f ) = T<(Xi f ) ∈ (T<(I))m+1 then T<( f ) ∈ (T<(I))m

and T<( f ) = T<(g) for some g ∈ Im . Thus Xi ( f − g) ∈ Im+1 andT<( f − g) < T<( f ) implies by induction f − g ∈ Im so that f ∈ Im .

(3) Let f ∈ I, deg( f ) > m.

Either T<( f ) is divided by some X0, . . . , Xi−1 and so it is not a min-imal generator of T<(I)

or we can express f as f =g+∑i−1j=0 X j h j with g ∈ I∩k[Xi , . . . , Xn]

and T<( f ) = T<(g).


Again,

either Xi | T<( f ) = T<(g) so that g = Xi h for some h ∈Pm−1+p, p ≥ m and h ∈ (I : Xi )m−1+p = Im−1+p, T<(h) ∈T<(I) and T<( f ) = Xi T<(h) is not a mininal generator ofT<(I);

or none of X0, . . . , Xi divide T<( f ). Now f ∈ I + (Xi ) but is not aminimal generator of I + (Xi ) since deg( f ) > m. Thereforethere are a term τ = 1 and an element g ∈ I + (Xi ) suchthat T<( f ) = τT<(g). Expressing g as g = g1 + Xi g2 withg1 ∈ I, necessarily we have T<(Xi g2) < T<(g1) = T<(g) sothat T<( f ) = τT<(g1) is not a minimal generator of T<(I).

Corollary 38.5.7. For the rev-lex ordering < induced by X0 < · · · < Xn andfor d ≥ 0, m ≥ 0, the following conditions are equivalent.

(1) We have((I + (X0, . . . , Xi−1)) : Xi

)m = (I + (X0, . . . , Xi−1))m for i, 0 ≤

i < d,(I + (X0, . . . , Xd−1))m = Pm.

(2) We have((T<(I) + (X0, . . . , Xi−1)) : Xi

)m = (T<(I) + (X0, . . . , Xi−1))m

for i, 0 ≤ i < d,(T<(I) + (X0, . . . , Xd−1))m = Pm .

Theorem 38.5.8 (Bayer–Stillman). Let I ⊂ k[X0, . . . , Xn] =: P be a ho-mogeneous ideal, dim(I) = d, and let < be the revlex ordering induced byX0 < X1 < · · · < Xn. Then

(1) (X0, . . . , Xd−1) ∈ Ud(I) ⇐⇒ (X0, . . . , Xd−1) ∈ Ud(T<(I)),(2) (X0, . . . , Xd−1) ∈ Ud(I) ⇒ reg(I) = reg(T<(I)).

Proof. Note that by Corollary 23.3.2, dim(T<(I)) = d; let m := reg(I)and assume that (X0, . . . , Xd−1) ∈ Ud(I). Then (X0, . . . , Xd−1) satisfiesFact 38.5.2(3).

Since (I + (X0, . . . , Xd−1))m = Pm , then T< (I + (X0, . . . , Xd−1)) is gen-erated in degree bounded by m.

Therefore, Lemma 38.5.6(3) allows us to conclude inductively that each

T< (I + (X0, . . . , Xi−1))

is generated in degree bounded by m. In particular T<(I) is generated in degreebounded by m.

746 Giusti

By Corollary 38.5.7, (X0, . . . , Xd−1) also satisfies Fact 38.5.2(2) for T<(I)so that, by Fact 38.5.2, (X0, . . . , Xd−1) ∈ Ud(T<(I)) and reg(T<(I)) ≤ m =reg(I).

Conversely, let us assume that (X0, . . . , Xd−1) ∈ Ud(T<(I)) and let µ :=reg(T<(I)); let us consider a minimal generator f of I; by Buchberger reduc-tion we can wlog assume that T<( f ) is a minimal generator of T<(I); sinceT<(I) is generated in degree bounded by µ, we can deduce that deg( f ) =deg(T<( f )) ≤ µ so that I is generated in degree bounded by µ.

As above, by Corollary 38.5.7, (X0, . . . , Xd−1) satisfies Fact 38.5.2(2) for Iso that, by Fact 38.5.2, (X0, . . . , Xd−1) ∈ Ud(I) and reg(I) ≤ µ = reg(T<(I)).

Note that Theorem 38.5.8 does not state the false equality Ud(I) =Ud(T<(I)).

Lemma 38.5.9. Let E ⊂ k[X0, . . . , Xn] be a Borel monomial ideal. Its asso-ciated primes are all of the form (X j , . . . , Xn).

Corollary 38.5.10. Let

I ⊂ k[X0, . . . , Xn] =: P be a homogeneous ideal,d := dim(I),< be any term ordering for which X0 < X1< · · · < Xn,U ⊂ GL(n + 1, k) be the non-empty Zariski open set, andE be the Borel ideal such that

E = T<(M(I)), for each M ∈ U.

Then

(X0, . . . , Xd−1) ∈ Ud(T<(M(I))), for each M ∈ U.

Proof. For each i, 1 ≤ i < d , by Lemma 38.5.9, the associated primes of

Ji := E + (X0, . . . , Xi−1)

are all of the form p j := (X0, . . . , Xi−1, X j , . . . , Xn) with j ≥ i . Since pi isassociated only to non-saturated ideals, and Xi can be contained only in pi , wecan conclude that Xi is not a zero-divisor on P/(Ji )sat.

Then by definition (X0, . . . , Xd−1) ∈ Ud(E).

Theorem 38.5.11 (Bayer–Stillman). The equality S(I) = reg(I) = G(I)holds for any homogeneous ideal I ⊂ k[X0, . . . , Xn] =: P .


Proof. As in Corollary 38.5.5, let

U ⊂ GL(n + 1, k) be the non-empty Zariski open set, andE the Borel ideal

such that E = T<(M(I)), for each M ∈ U where < is the rev-lex orderinginduced by X0 < · · · < Xn .

Setting d := dim(I), we have for each M ∈ U

(X0, . . . , Xd−1) ∈ Ud(E) = Ud(T<(M(I)))

by Corollary 38.5.9, whence, by Theorem 38.5.8,

(X0, . . . , Xd−1) ∈ Ud(M(I)), andreg(M(I)) = reg(T<(M(I)).

While the former is reg(I), the latter is G(I) by Proposition 38.5.4.

Let us now consider

a weight function w := (w0, . . . , wn) ∈ Rn+1 \ 0 satisfying 6

w0 ≤ w1 ≤ · · · ≤ wn,

vw : P → R be the valuation induced by vw(Xi ) = wi for each i ,< be any term ordering on T ,≺ the refinement of vw with <.

Then:

Theorem 38.5.12 (Bayer–Stillman). For any homogeneous ideal

I ⊂ k[X0, . . . , Xn] =: P

and any matrix M ∈ GL(n + 1, k), with the notation above we have

reg(T≺(M(I))) ≥ reg(Lw(M(I))).

If moreover, < is the revlex ordering induced by X0 < X1 < · · · < Xn, thenthere is a non-empty Zariski open set U ⊂ GL(n + 1, k) such that

reg(T≺(M(I))) = reg(Lw(M(I)), for each M ∈ U.

Proof. The equation reg(T≺(M(I))) ≥ reg(Lw(M(I))) being trivial, let us as-sume that < is the revlex ordering induced by X0 < X1 < · · · < Xn , andlet

6 This assumption has the only effect of requiring a renumbering of the variables such that

vw(X0) ≤ vw(X2) ≤ · · · ≤ vw(Xn)

and X0 ≺ · · · ≺ Xn .

748 Giusti

U ⊂ GL(n + 1, k) be the non-empty Zariski open set, andE be the Borel ideal such that

E = T≺(M(I)), for each M ∈ U.

By the lemma above, (X0, . . . , Xd−1) ∈ Ud(T≺(M(I))), for each M ∈ U.

Since, by Corollary 24.10.2, we have E = T≺(M(I)) = T<(Lw(M(I))),Theorem 38.5.8 implies that

(X0, . . . , Xd−1) ∈ Ud(Lw(M(I))) andreg(Lw(M(I))) = reg(T≺(M(I)))

for each M ∈ U.

Bibliography

Abhiankar, S. S. and Li, W., On the Jacobian Conjecture: A New Approach viaGrobner Bases, J. Pure Appl. Alg. 61 (1989), 211–222.

Adams, W. W. and Loustaunau, P., An Introduction to Grobner Bases, AMS (1994).Adams, W. W., Boyle, A. and Loustaunau, P., Transitivity for Weak and Strong

Grobner Bases, J. Symb. Comp. 15 (1993), 49–65.Agnarsson, G., The Number of Outside Corner of Monomial Ideals, J. Pure Appl. Alg.

117–8 (1997), 3–22.Albano, G. and La Scala, R., A Koszul Decomposition for the Computation of Linear

Syzygies, J. AAECC 11 (2001), 181–202.Apel, J., Division of Entire Functions by Polynomial ideals, L. N. Comp. Sci. 948

(1995), 82–958, Springer.Assi, A., Standard Bases, Criticals Tropisms and Flatness, J. AAECC 4 (1993),

197–215.Assi, A., On Flatness of Generic Projections, J. Symb. Comp. 18 (1994), 447–462.Ayoub, C. W., The Decomposition Theorem for Ideals in Polynomial Rings of

Domain, J. Alg. 76 (1982), 99–110.Ayoub, C. W., On Constructing Bases for Ideals in Polynomial Rings over the

Integers, J. Number Th. 17 (1983), 204–225.Backelin, J. and Froberg, R., How we proved that there are exactly 924 cyclic 7-roots,

Proc. ISSAC’91 (1991), 103–111, ACM.Barkee, B., Grobner Bases. The Ancient Secret Mystic Power of the Algu

Compubraicus. A Revelation Whose Simplicity Will Make Ladies Swoon andGrown Men Cry, Technical Report (1988), Cornell.

Bayer, D., The Division Algorithm and the Hilbert Scheme, Ph.D. thesis, Harvard(1981).

Bayer, D., An introduction to the Division Algorithm, Lecture Notes MeetingGeometria Algebrica e Informatica (1985).

Bayer, D. and Morrison, I., Standard Bases and Geometric Invariant Theory I. InitialIdeals and State Polytopes, J. Symb. Comp. 6 (1988), 209–217.

Bayer, D. and Mumford, D., What Can Be Computed in Algebraic Geometry,Symposia Mathematica 34 (1993), 1–48, Cambridge University Press.

Bayer, D. and Stillman, M., The Designs of Macaulay: A System for Computing inAlgebraic Geometry and Commutative Algebra, Proc. SYMSAC’86 (1986),157–162, ACM.

Bayer, D. and Stillman, M., A Theorem on Refining Division Orders by the ReverseLexicographic Order, Duke J. Math. 55 (1987), 321–328. .

749

750 Bibliography

Bayer, D. and Stillman, M., On the Complexity of Computing Syzygies, J. Symb.Comp. 6 (1988), 135–147.

Bayer, D. and Stillman, M., Computation of Hilbert Functions, J. Symb. Comp.14 (1992), 31–50.

Bayer, D. and Stillman, M., A Criterion for Detecting m-regularity, Invent. Math.87 (1998), 1–11.

Bayer, D., Stillman, M. and Galligo, A., Primary Decompositions, (1989).Bayer, D., Galligo, A. and Stillman, M., Grobner bases and extension of scalars,

Symposia Mathematica 34 (1993), 198–215, Cambridge University Press.Beck, S. and Kreuzer, M., How to Compute the Canonical Module of a Set of Points,

Progress in Mathematics 143 (1996), 51–78, Birkhauser.Becker, T. and Weispfenning, V., The Chinese Remainder Problem, Multivariate

Interpolation, and Grobner Bases, Proc. ISSAC’91 (1991), 64–69, ACM.Becker, T. and Weispfenning, V., Grobner Bases, Springer (1982).Bergman G. M., The Diamond Lemma for Ring Theory, Adv. Math. 29 (1978),

178–218.Bigatti, A. M., Computation of Hilbert-Poincare Series. J. Pure Appl. Alg. 119, (1997)

237–253.Bigatti, A. M., Caboara, M. and Robbiano, L., On the Computation of

Hilbert–Poincare Series, J. AAECC 2 (1991), 21–33.Boege, W., Gebauer, R. and Kredel, H., Some Examples for Solving Systems of

Algebraic Equations by Calculating Grobner Bases, J. Symb. Comp. 2 (1986),83–98.

Brennan, J. P. and Vascocelos, W. V., Effective Computation of the Integral Closure ofa Morphism, J. Pure App. Alg. 86 (1993), 125–134.

Bresinsky, H. and Renschuch, B., Basisbestimmung Veronescher Projecktionsidealemit allgemeiner Nullstelle (tm

0 , tm−r0 tr

1 , tm−s0 ts

1), Math. Nachr. 96 (1980),257–269.

Briancon, J. and Galligo, A., Deformations distinguees d’un point de C2 ou R

2,Asterisque 7–8 (1973), 129–138.

Brownawell, W. D., Bounds for the Degree in the Nullstellensatz, Ann. Math.126 (1987), 577–591.

Brownawell, W. D., Borne effective pour l’exposant dans le theoreme des zeros, C. R.Acad. Sci. Paris 305 (1987), 287–290.

Buchberger, B., Ein Algorithmus zum Auffinden der Basiselemente desRestklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis,Innsbruck (1965).

Buchberger, B., Ein algorithmisches Kriterium fur die Losbarkeit eines algebraischenGleischunssystem, Aeq. Math. 4 (1970), 374–383.

Buchberger, B., A Theoretical Basis for the Reduction of Polynomials to CanonicalForms, SIGSAM Bull. 10, 3 (1976), 19–29.

Buchberger, B., Some Properties of Grobner Bases, SIGSAM Bull. 10, 4 (1976),19–24.

Buchberger, B., A Criterion for Detecting Unnecessary Reduction in the Constructionof Grobner Bases, L. N. Comp. Sci 72 (1979), 3–21, Springer.

Buchberger, B., Grobner Bases: An Algorithmic Method in Polynomial Ideal Theory,in Bose, N. K. (ed.), Multidimensional Systems Theory (1985), 184–232,Reider.

Buchberger, B., Applications of Grobner-Bases in Non-Linear ComputationalGeometry, L. N. Comp. Sci. 296 (1987), 52–80, Springer.

Bibliography 751

Buchberger, B., Introduction to Grobner Bases, in Buchberger, B. and Winkler, F.(eds) Grobner Bases and Application (1998) 3–31, Cambridge UniversityPress.

Buchberger, B. and Loos, R., Algebraic Simplification, in Buchberger et al. (1982),11–44.

Buchberger, B. and Winkler, F., Miscellaneous Results on the Construction ofGrobner-Bases for Polynomial Ideals, Bericht 137, Linz (1979).

Buchberger, B. and Winkler, F. (eds), Grobner Bases and Application (1998)Cambridge University Press.

Buchberger, B., Collins, G. E. and Loos, R. (eds), Computer Algebra. Symbolic andAlgebraic Computation, Springer (1982).

Caboara, M., A Modified Algorithm for Resolution (2001), unpublished.Caboara, M., Conte, P. and Traverso, C., Yet Another Ideal Decomposition Algorithm,

L. N. Comp. Sci. 1255 (1995), 39–54, Springer.Caniglia, L., Galligo, A. and Heintz, J., Borne simple exponentielle pour les degres

dans le theoreme des zeros sur un corps de caracteristique quelconque, C. R.Acad. Sci. Paris 307 (1988), 255–258.

Caniglia, L., Galligo, A. and Heintz, J., Some New Effective Bounds in ComputationalGeometry, L. N. Math. 357 (1989), 131–151, Springer.

Cerlienco, L. and Mureddu, M., From algebraic sets to monomial linear bases bymeans of combinatorial algorithms, Discrete Math. 139 (1995), 73–87.

Cerlienco, L. and Mureddu, M., Multivariate Interpolation and Standard Bases forMacaulay Modules, J. Alg. 251 (2002), 686–726.

Chardin, M. and Moreno-Socias, G., Regularity of Lex-Segment Ideals: Some ClosedFormulas and Applications Prepublication 292, Inst. Math. Jussieu (2001).

Collard, S. and Mall, D., The Ideal Structure of Grobner Base Computations, L. N.Comp. Sci. 958 (1994), Springer.

Collard, S., Mall, D. and Kalkbrener, M., The Grobner Walk (1993).Conti, P. and Traverso, C., Computing the Conductor of an Integer Extension, Disc.

Appl. Math. 33 (1991), 43–60.Czapur, S. R., Solving Algebraic Equations via Buchberger’s Algorithm, L. N. Comp.

Sci. 378 (1987), 260–269, Springer.Czapur, S. R. and Gedder, K. O., On Implementing Buchberger’s Algorithm for

Grobner Bases, Proc. SYMSAC’86 (1986), 233–238, ACM.Czapur, S. R. and Gedder, K. O., A Heuristic Strategy for Lexicographic Grobner

Bases, Proc. ISSAC’91 (1991), 39–48, ACM.Decker, W., Greuel, G.-M. and Pfister, G., Primary Decomposition: Algorithms and

Comparisons, in Greuel, G.-M., Matzat, B. H. and Hiss, G. (eds), AlgorithmicAlgebra and Number Theory (1998), 187–220, Springer.

De Dominicis, G., Algoritmi di decomposizione primaria in anelli polinomiali, Tesi,Genova (1995).

Demazoure, M., Notes informelles de calcul formel I. Functions d’Hilbert Samueld’aprs Macaulay, Stanley et Bayer, Publication M 645. 0784 Ecole PolytechniquePalaiseau, (1984).

Demazoure, M., Notes informelles de calcul formel II. Une definition constructive duresultant, Prepublication M660. 0584 Ecole Polytechnique Palaiseau (1984).

Dickenstein, A. M. and Sessa C. Duality Methods for the Membership Problem,Progress in Mathematics 94 (1990), 89–104, Birkhauser.

Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry,Springer (1998).

752 Bibliography

Eisenbud, D. and Sturmfels, B., Finding Sparse Systems of Parameters, J. Pure Appl.Alg. 94 (1994), 143–157.

Eisenbud, D. and Sturmfels, B., Binomial ideals, Duke Math. J. 84 (1996), 1–45.Eisenbud, D., Huneke, C. and Vasconcelos, W., Direct Methods for Primary

Decomposition, Inventiones Math. 110 (1992), 207–235.Eliahou, S. and Kervaire, M., Minimal Resolutions of Some Monomial Ideals, J. Alg.

129 (1990), 1–25.Erdos, J., On the Structure of Ordered Real Vector Spaces, Publ. Math. Debrecen

4 (1956), 334–343.Faugere, J.-C., A New Efficient Algorithm for Computating Grobner Bases without

Reduction to Zero (F5), Proc. ISSAC 2002 (2002), 75–83, ACM.Fitchas, N. and Galligo, A., Nullstellensatz effectif et conjecture de Serre (theoreme de

Quillen–Suslin) pour le Calcul Formel, Math. Nachr. 149 (1990), 231–253.Fortuna, E., Gianni, P. and Trager, B., Derivations and Radicals of Polynomial Ideals

over Fields of Arbitrary Characteristic, J. Symb. Comp. 33 (2002), 609–625.Fortuna, E., Gianni, P. and Parenti, P., Some Constructions for Real Algebraic Curves,

J. Symb. Comp., to appear (2003).Froberg, R. and Hollman, J., Some Comments on a Paper by Moreno, J. Symb. Comp.

11 (1993).Galligo, A., A propos du theorem de preparation de Weierstrass, L. N. Math.

409 (1974), 543–579, Springer.Galligo, A., Theoreme de division et stabilite en geometrie analytique, Ann. Inst.

Fourier Grenoble 29 (1979), 107–184.Galligo, A., Algorithmes de calcul de bases standard, Nice (1982).Galligo, A., Examples d’ensembles de Point en Position Uniforme, Progress in

Mathematics 94 (1990), 105–117, Birkhauser.Galligo, A., Poittier, L. and Traverso, C., Greatest Easy Common Divisor and

Standard Bases Completion Algorithms, L. N. Comp. Sci. 358 (1988), 162–176,Springer.

Gallo, G., Complexity Issues in Computational Algebra, Ph.D. thesis, New York(1992).

Gallo, G. and Mishra, B., A Solution to Kronecker’s Problem, J. AAECC 5 (1994),343–370.

Gebauer, R. and Moller, H. M., A fast Variant of Buchberger’s Algorithm, (1985).Gebauer, R. and Moller, H. M., Buchberger’s Algorithm and Staggered Linear Bases,

Proc. SYMSAC’86 (1986), 218–221, ACM.Gebauer, R. and Moller, H. M., On an Installation of Buchberger’s Algorithm, J. Symb.

Comp. 6 (1988), 275–286.Gianni, P., Properties of Grobner Bases under Specialization, L. N. Comp. Sci. 378

(1987), 293–297, Springer.Gianni, P., Trager, B. and Zacharias, G., Grobner Bases and Primary Decomposition of

Polynomial Ideals, J. Symb. Comp. 6 (1988), 149–167.Giovini, A. et al., ‘One sugar cube, please’ OR Selection Strategies in the Buchberger

Algorithm, Proc. ISSAC ’91 (1991), 49–54, ACM.Giusti, M., Some Effectivity Problems in Polynomial Ideal Theory, L. N. Comp. Sci.

174 (1984), 159–171, Springer.Giusti, M., Combinatorial Dimension Theory of Algebraic Varieties, J. Symb. Comp.

6, (1988), 249–267.Giusti, M. and Heintz, J., Algorithmes – disons rapides – pour la decomposition d’une

variete algebrique en composantes irreductibles, Progress in Mathematics94 (1990), 169–194, Birkhauser.

Bibliography 753

Giusti, M. and Heintz, J., La determination des points isoles et de la dimension d’unevariete algebrique peut se faire en temps polynomial, Symposia Mathematica34 (1993), 216–256, Cambridge University Press.

Giusti, M. and Lazard, D., Complexity of Standard Basis Computations, RelatedAlgebraic Problems and their Common Double Exponential Behaviour(1985).

Giusti, M., Heintz, J., Morais, J. E. and Pardo, L. M., Le role des structures de donneesdans les problemes d’elimination, C. R. Acad. Sci. Paris 325 (1997),1223–1228.

Gjunter, N., Sur les modules des formes algebriques, Trudy Tbilis. Mat. Inst. 9 (1941),97–206.

Gordan, P., Neuer Beweis des Hilbertschen Satzes uber homogene Funktionen,Gottingen Nachr. (1899), 240–242.

Gordan, P., Les invariants des formes binaries, J. Math. Pure Appl. (5e series) 6(1900), 141–156.

Grobner, W., Uber das Macaulaysche inverse System und desser Bedeutung fur dieTheorie der lineren Differentialgleichungen mit konstanten Koeffizienten, MonatMath. Phys. 47 (1939), 247–284.

Grobner, W., Uber die algebraischen Eigenschaften der Integrale von linearenDifferentialgleichungen mit konstanten Koeffizienten, Monat. Math. Phis. 47(1939), 247–284.

Grobner, W., Moderne Algebraische Geometrie, Springer (1949).Grobner, W., Uber die eliminationstheorie, Monat. Math. 54 (1950), 71–78.Grobner, W., Teoria degli ideali e geometria algebrica, Seminari INDAM 1962–63

(1963), 1–97.Grobner, W., Algebraische Geometrie I, (1968) Bibliographische Institut.Grobner, W., Algebraische Geometrie II, (1970) Bibliographische Institut.Grobner, W., Il concetto di molteplicita nella geometia algebrica, Rend. Sem. Mat. Fis.

Milano 40 (1970), 3–10.Grobner, W., Teoria degli ideali e geometria algebrica. Rend. Sem. Mat. Fis. Milano 46

(1971), 171–242.Hartshorne, R., Connectedness of the Hilbert scheme, Publ. Math. I. H. E. S.

29 (1966), 261–304.Heintz, J. and Morgenstern J., On the Intrinsic Complexity of Elimination Theory,

J. Complexity 9 (1993), 471–489.Heiß, W., Oberst, U. and Pauer, F. On Inverse Systems and Squarefree

Decomposition of Zero-Dimensional Polynomial Ideals J. Symb. Comp., toappear (2003).

Hermann, G., Die Frage der endlich vielen Schritte in die Theorie der Polynomideale,Math. Ann. 95 (1926), 736–788.

Hilbert, D., Uber die Theorie der algebraischen Formen, Math. Ann. 36 (1890),473–534.

Hilbert, D., Theory of Algebraic Invariant (1993), Cambridge University Press.Hollman, J., On the computation of the Hilbert Series L. N. Comp. Sci. 583 (1992),

272–280, Springer.Janet, M., Sur les systemes d’equations aux derivees partielles, J. Math. Pure Appl.

3 (1920), 65–151.Kalkbrener, M., Solving Systems of Algebraic Equations by Using Grobner Bases,

L. N. Comp. Sci. 378 (1987), 282–292, Springer.Kalkbrener, M., On the Stability of Grobner Bases under Specialization, J. Symb.

Comp. 24 (1997), 51–58.

754 Bibliography

Kandri-Rody, A., Radical of ideals in polynomial rings.Kandri-Rody, A., Kapur, D. and Winkler, F., Knuth–Bendix Procedure and Buchberger

Algorithm – A Synthesis, Proc. ISSAC’89 (1989), 55–67, ACM.Kollar, J., Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), 963–975.Kollreider, C., Polynomial reduction: The Influence of the Ordering of Terms on a

Reduction Algorithm, Bericht 124, Linz (1978).Kredel, H., Primary Ideal Decomposition, L. N. Comp. Sci. 378 (1987), 270–281,

Springer.Kredel, H. and Weispfenning, V., Computing the Dimension and Independent Sets of a

Polynomial Ideal J. Symb. Comp. 6 (1988), 231–247.Krick, T. and Logar, A., Membership Problem, Representation Problem and the

Computation of the Radical for One-dimensional Ideal, Progress in Mathematics94 (1990), 203–216, Birkhauser.

Krick, T. and Logar, A., An algorithm for the Computation of the Radical of anIdeal in the Ring of Polynomials, L. N. Comp. Sci. 539 (1991), 195–205,Springer.

Lakshman, Y. N., A Single Exponential Bound on the Complexity of ComputingGrobner Bases of Zero Dimensional Ideals, Progress in Mathematics 94 (1990),227–234, Birkhauser.

La Scala, R., An algorithm for Complexes, Proc. ISSAC’94 (1994), 264–268,ACM.

La Scala, R. and Stillman, M., Strategies for Computing Minimal Free Resolutions,J. Symb. Comp. 26 (1998), 409–431.

Lazard, D., Calculs sur les modules projectifs, Publ. Sem. Math. Univ. Rennes (1972).Lazard, D., Algorithmes fondamentaux en Algebre Commutative, Asterisque

38–39 (1976), 131–138.Lazard, D., Algebre lineaire sur K [X1, . . . , Xn] et elimination, Bull. Soc. Math.

France 105 (1977), 165–190.Lazard, D., Resolution des systemes d’equations algebriques, Theor. Comp. Sci. 15

(1981), 71–110.Lazard, D., Commutative Algebra and Computer Algebra, L. N. Comp. Sci. 144

(1982), 40–48, Springer.Lazard, D., Grobner bases, Gaussian Elimination and Resolution of Systems of

algebraic equations, L. N. Comp. Sci. 162 (1983), 146–156, Springer.Lazard, D., Ideal Bases and Polynomial Decomposition: Case of Two Variables,

J. Symb. Comp. 1 (1985), 261–270.Lazard, D., A Note on Upper Bounds for Ideal-theoretical Problems, J. Symb. Comp.

13 (1992), 231–233.Logar, A., Constructions over Localizations of Rings, La Matematiche 42 (1987),

131–150.Logar, A., A Computational Proof of the Noether Normalization Lemma, L. N. Comp.

Sci. 357 (1988), 259–273, Springer.Logar, A. Computational Aspects of the Coordinate Ring of an Algebraic Variety,

Comm. Algebra 18 (1990), 2641–2662.Madlener, K. and Reinert, B., Computing Grobner Bases in Monoid and Group Rings,

Proc. ISSAC ’93 (1993), 254–263, ACM.Macaulay, F. S., On the Resolution of a given Modular System into Primary Systems

including Some Properties of Hilbert Numbers, Math. Ann. 74 (1913),66–121.

Macaulay, F. S., The Algebraic Theory of Modular Systems, Cambridge UniversityPress (1916).

Bibliography 755

Macaulay, F. S., Some Properties of Enumeration in the Theory of Modular Systems,Proc. London Math. Soc. 26 (1927), 531–555.

Macaulay, F. S., Modern Algebra and Polynomial Ideals, Proc. Cambridge Philos.Soc. 30 (1934), 27–46.

Marinari, M. G., Sugli ideali di Borel, Boll. UMI 4 (2001), 207–237.Marinari, M. G. and Ramella, L. Some Properties of Borel Ideals, J. Pure Appl. Alg.

139 (1999), 833–200.Masser, D. W. and Wustholz, G., Fields of Large Transcendence Degree Generated by

Values of Elliptic Functions, Invent. Math. 72 (1983), 407–464.Matsumoto, R., Computing the Radical of an Ideal in Positive Characteristic, J. Symb.

Comp. 32 (2001), 263–271.Mayr, E. W. and Meyer, A. R., The Complexity of the Word Problem for Commutative

Semigroups and Polynomial Ideals, Adv. Math. 46 (1982), 305–329.Moller, H. M., A Reduction Strategy for the Taylor Resolution, L. N. Comp. Sci.

204 (1985), 526–534, Springer.Moller, H. M., On the Construction of Grobner Bases Using Syzygies, J. Symb. Comp.

6 (1988), 345–359.Moller, H. M., Computing Syzygies a la Gauß-Jordan, Progress in Mathematics

94 (1990), 335–346, Birkhauser.Moller, H. M., Systems of Algebraic Equations Solved by Means of Endomorphisms,

L. N. Comp. Sci. 673 (1993), 43–56, SpringerMoller, H. M. and Buchberger B., The construction of multivariate polynomials with

preassigned zeros, L. N. Comp. Sci. 144 (1982), 24–31, Springer.Moreno Socias, G., An Ackermannian Polynomial Ideal, L. N. Comp. Sci. 539 (1991),

269–280, Springer.Noether, E., Idealtheorie in Ringbereichen, Math. Annales 83 (1921), 25–66.Northcott, D. G. and Rees, D., Principal Systems, Quart. J. Math. Oxford 2 (1957),

119–27.Pohst, M. and Yun, D., On Solving Systems of Algebraic Equations via Ideal Bases

and Elimination, Proc. SYMSAC’81 (1981), 206–211, ACM.Renschuch, B., Elementare und prachtische Idealtheorie, (1976) VEB Deutscher

Verlag der Wissenschaften.Renschuch, B., Beitrage zur konstructiven Theorie des Polynomideal, Wiss. Z.

Padagogische Hochschule Karl Liebknecht, Postdam, 17-31 (1973-87).Richman, F., Constructive Aspects of Noetherian Rings, Proc. AMS 44 (1974),

436–441.Ritter, G. and Weispfenning, V., On the Number of Term Orders, J. AAECC 2 (1991),

55–79Rosenmann, A., An Algorithm for Constructing Grobner and Free Schreier Bases in

Free Group Algebras, J. Symb. Comp. 16 (1993), 523–549.Schreyer, F. O., Die Berechnung von Syzygien mit dem verallgemeinerten

Weierstrass’schen Divisionsatz, Diplomarbait, Hamburg (1980).Schreyer, F. O., A standard Basis Approach to Syzygies of Canonical Curves, J. Reine

angew. Math. 421 (1991), 83–123.Seidenberg, A., Constructive Proof of Hilbert’s Theorem on Ascending Chains, Trans.

A. M. S. 174 (1972), 305–312.Seidenberg, A., Constructions in a Polynomial Ring over the Rings of Integers, Amer.

J. Math 100 (1978), 685–703.Seidenberg, A., What is Noetherian, Rend. Sem. Mat. Fis. Milano 44 (1974), 55–61.Seidenberg, A., Constructions in Algebra, Trans. Amer. Math. Soc. 197 (1974),

273–313.

756 Bibliography

Shannon, D. and Sweedler, M., Using Grobner Bases to Determine AlgebraMembership, Splitting Surjective Algebra Homomorphisls and DetermineBirational Equivalence J. Symb. Comp. 6 (1988), 267–273

Shimoyama, T. and Yokohama, K., Localization and Primary Decomposition ofPolynomial Ideals, J. Symb. Comp. 22 (1996), 247–277

Siebert, T., Recursive Computation of Free Resolutions and a Generalized KoszulComplex, J. AAECC 14 (2003), 133–149.

Spear, D. A., A Constructive Approach to Commutative Ring Theory, in Proc.of the 1977 MACSYMA Users’ Conference, NASA CP-2012 (1977),369–376.

Sperner, E., Uber einen kombinatorishen Satz von Macaulay und seine Anwerdungenauf die Theorie der Polynomideale, Abh. Math. Sem. Univ. Hamburg 7 (1930),149–163.

Stetter, H. J. and Moller, H. M., Multivariate Polynomial Equations With MultipleZeros Solved by Matrix Eigenproblems Numer. Math. 70 (1995), 311–329.

Sturmfels B., Grobner Bases and Convex Polytopes, (1996) A. M. S.Sturmfels, B. and White, N., Grobner Bases and Invariant Theory Adv. Math. 76

(1989), 245–259.Sweedler, M., Using Grobner Bases to Determine the Algebraic and Trascendental

Nature of Field Extensions: Return of the Killer Tag Variables, L. N. Comp. Sci.673 (1993), 43–56, Springer.

Taylor, D., Ideals Generated by Monomials in an R-sequence, Ph. D. Thesis, Chicago(1960).

Traverso, C. and Donato, L., Experimenting the Grobner Basis Algorithm with AlPISystem, Proc. ISSAC ’89, (1989), 192–198, ACM.

Traverso, C., Hilbert function and the Buchberger algorithm, J. Symb. Comp.22 (1996), 355–376.

Traverso, C., Metodi costruttivi e calcolo automatico in algebra commutativa, Boll. U.M I.

Traverso, C., Grobner Trace Algorithm, L. N. Comp. Sci. 358 (1988), 125–138,Springer.

Traverso, C. and Caboara, M., Efficient algorithms for Module Operation. Proc.IS-SAC’98, (1998), 147–152. ACM.

van der Waerden, B. L., Modern Algebra, (1949) Ungar.Vasconcelos, W. V., What is a Prime Ideal?, Atas IX Escola de Algebra (1986),

141–149, IMPA.Vasconcelos, W. V., Jacobian Matrices and Constructions in Algebra, L. N. Comp. Sci.

539 (1991), 48–64, Springer.Vasconcelos, W. V., Computational Methods in Commutative Algebra and Algebraic

Geometry, (1998) Springer.Vasconcelos, W. V., The Top of a System of Equations, Bol. Soc. Mat. Mexical 37

(1992), 549–226.Vasconcelos, W. V., Computing the Integral Closure of an Affine Domain, Proc. Amer.

Math. Soc. 113 (1991), 633–638Yap, C. K., A New Lower Bound Construction for the Word Problem for Commutative

Thue Systems, J. Symb. Comp. 12 (1991), 1–27Weispfenning, V., Some Bounds for the Construction of Grobner Bases L. N. Comp.

Sci. 307 (1988), 125–138, Springer.Weispfenning, V., Constructing Universal Grobner Bases, L. N. Comp. Sci 356 (1987),

95–201, Springer.

Bibliography 757

Winkler, F., The Chuch-Rosser Property in Computer Algebra and Special TheoremProving: An Investigation of Critical-Pair/Completion Algorithms, Thesis, Linz(1984).

Winkler, F., A Theorem on the Headterms of a Grobner Basis, Report, Linz (1982).Winkler, F., Knuth–Bendix Procedure and Buchberger Algorithm – A Syntesis, Proc.

ISSAC’86 (1986), 55–67, ACM.Zacharias, G., Generalized Grobner Bases in Commutative Polynomial Rings,

Bachelor’s thesis, M. I. T. (1978).Zarishi, O. and Samuel, P., Commutative Algebra (1958), Van Nostrand.

Index

affine algebraic variety 5affine space 4allgemeine basis 595, 601allgemeine coordinate 604allgemeine position 603ARGH-decomposition 629associated graded module 208associated graded ring 207associated prime ideal 344, 351autoreduced 104

block ordering 240border basis 428border set 427Borel ideal 703Borel relation 704Buchberger Canonical Form Algorithm

81Buchberger Normal Form Algorithm

78

canonical echelon set 64canonical form 82Cauchy sequence 225Cauchy standard representation 210CCT-decomposition 659CCT-scheme 659characteristic number 344, 355Church–Rosser property 178complete intersection 666conductor 328continuation 537contraction 357corner set 427, 537

degree 35, 389degree-compatible 107degrevlex ordering 119depth 666dialytic equation 455

Dickson’s Lemma 38dimension 35, 376, 378

echelon set 63Eliahou–Kervaire resolution 723embedded prime 351equidimensional representation 391extension 357

FGLM problem 416form 111formal term 189

Gauss basis 51Gauss representation 55Gebauer–Moller linear basis 275Gebauer–Moller set 258general linear group xix, 371, 687generic escalier 697generic initial ideal 697Gianni–Kalkbrener’s Theorem 15, 610Giusti–Heintz coordinate 648Gordan’s Lemma 38graded module 195graded ring 195Grobner basis 78, 141, 189, 196, 292Grobner description 428Grobner fan 252Grobner representation 78, 189, 428

weak 96, 190GTZ-decomposition 626GTZ-scheme 626

H-basis 114, 139Hermann bound 162Hilbert function 29Hilbert polynomial 35Hilbert series 35Hilbert’s Basissatz 5, 23, 36, 40Hilbert’s Nullstellensatz 6, 7, 13, 19, 25

758

Index 759

homogeneous ideal 23homogeneous polynomial 23Horner complexity 429

independent variables 382index of regularity 35, 666inf-limited 200inverse system 456irreducible ideal 346irredundant primary representation 348irrelevant ideal 25isolated prime 351

Kredel–Weispfenning’s algorithm 384Kronecker-module 116

leading form 23, 195Leibniz Formula 515leitideal 196length 379lexicographical ordering 43, 47lift 202linear representation 428

Macaulay basis 520Macaulay representation 555Mayr–Meyer Examples 108multiplicative system 356multiplicity 389

Noether position 377Noetherian equation 466Noetherian ring 291, 338normal form 79, 190normal selection strategy 274

perfect 678Primarbasis 591primary component 351primary ideal 342Primbasis 589, 600prime ideal 341projective space 22projective variety 26

Rabinowitch Trick 7radical ideal 6, 340

radical of an ideal 7Radikalbasis 593rank 379recursive Horner representation

429reduced Grobner basis 83, 191regular sequence 666regularity 243, 727resolution 33reverse lexicographical ordering

121

S-polynomial 96, 142, 189redundant 261useless 93

saturated 367saturation 28, 366Seidenberg Algorithm 618Seidenberg Lemma 617semigroup ordering 75Shape Lemma 42squarefree decomposition 660stable 715staggered linear basis 95, 274standard basis 106, 196, 204standard representation 196, 210state polytope 253subalgebra 316syzygy 32

T-basis 140tangent cone algorithm 106Taylor minimal resolution 129Taylor resolution 124term-order homogenization xviii,

117term ordering 75Trink’s Algorithm 15, 610truncated standard representation

210

unmixed 391universal Grobner basis 252

weight 202

Zacharias ring 291

Documents

the-eye.eu Teo Mora - [EMAvol0… · ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS FOUNDING EDITOR G.-C. ROTA Editorial Board P. Flajolet, M. Ismail, E. Lutwak 40 N. White (ed.)