Encyclopaedia of Mathematical Sciences Invariant Theory and Algebraic Transformation Groups VIIIRevaz V. Gamkrelidze · Vladimir l. Popov Subseries Editors

130

Harm DerksenGregor Kemper

Computational Invariant Theory Second Edition

Encyclopaedia of Mathematical Sciences

Encyclopaedia of Mathematical SciencesVolume 130

Invariant Theory and Algebraic Transformation Groups VIII

Subseries Editors:Revaz V. Gamkrelidze Vladimir l. Popov

More information about this series at http://www.springer.com/series/855

http://www.springer.com/series/855

Harm Derksen • Gregor Kemper

ComputationalInvariant Theory

Second Enlarged Edition with two Appendices byVladimir L. Popov, and an Addendum byNorbert A. Campo and Vladimir L. Popov

123

Harm DerksenDepartment of MathematicsUniversity of MichiganAnn Arbor, MIUSA

Gregor KemperZentrum Mathematik - M11Technische UniversitRat MRunchenGarching, Germany

ISSN 0938-0396Encyclopaedia of Mathematical SciencesISBN 978-3-662-48420-3 ISBN 978-3-662-48422-7 (eBook)DOI 10.1007/978-3-662-48422-7

Library of Congress Control Number: 2015955260

Mathematics Subject Classification (2010): 13H10, 13P10

Springer Heidelberg New York Dordrecht London© Springer-Verlag Berlin Heidelberg 2002, 2015This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made.

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Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media(www.springer.com)

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To Maureen, William, Claire

To Elisabeth, Martin, Stefan

Preface to the Second Edition

It is more than 10 years ago that the first edition of this book has appeared. Sincethen, the field of computational invariant theory has been enjoying a lot of attentionand activity, resulting in some important and, to us, exciting developments. This iswhy we think that it is time for a second enlarged and revised edition. Apart fromcorrecting some mistakes and reorganizing the presentation here and there, we haveadded the following material: further results about separating invariants and theircomputation (Sects. 2.4 and 4.9.1), Symonds’ degree bound (Sect. 3.3.2), Hughes’and Kemper’s extension of Molien’s formula (Sect. 3.4.2), King’s algorithm forcomputing fundamental invariants (Sect. 3.8.2), Broer’s criterion for the quasi-Gorenstein property (Sect. 3.9.11), Dufresne’s generalization of Serre’s result onpolynomial invariant rings and her result with Jeffries (Sect. 3.12.2), Kamke’salgorithm for computing invariants of finite groups acting on algebras (Sect. 3.13),Kemper’s and Derksen’s algorithm for computing invariants of reductive groups inpositive characteristic (Sect. 3.13), algorithms by Müller-Quade and Beth, Hubertand Kogan, and Kamke and Kemper for computing invariant fields and localizationsof invariant rings (Sect. 4.10.1), and work by van den Essen, Freudenburg, Greueland Pfister, Kemper, Sancho de Salas, and Tanimoto on invariants of the additivegroup and of connected solvable groups (Sect. 4.10.5).

Last but not least, this edition contains two new appendices, written by VladimirPopov, on algorithms for deciding the containment of orbit closures and on astratification of Hilbert’s nullcone. The second appendix has an addendum, authoredby Norbert A’Campo and Vladimir Popov, containing the source code of a programfor computing this stratification.

We would like to thank Bram Broer, Emilie Dufresne, Vladimir Popov, JimShank, and Peter Symonds for valuable comments on a pre-circulated version ofthis edition, Vladimir Popov and Norbert A’Campo for their contributions to the

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book, and Ruth Allewelt at Springer-Verlag for managing the production processand for gently pushing us to finally finish our work and hand over the files.

Ann Arbor, MI, USA Harm DerksenMunich, Germany Gregor KemperJuly 2015

Preface to the First Edition

Invariant theory is a subject with a long tradition and an astounding ability torejuvenate itself whenever it reappears on the mathematical stage. Throughoutthe history of invariant theory, two features of it have always been at the centerof attention: computation and applications. This book is about the computationalaspects of invariant theory. We present algorithms for calculating the invariant ringof a group that is linearly reductive or finite, including the modular case. Thesealgorithms form the central pillars around which the book is built. To prepare theground for the algorithms, we present Gröbner basis methods and some generaltheory of invariants. Moreover, the algorithms and their behavior depend heavily onstructural properties of the invariant ring to be computed. Large parts of the bookare devoted to studying such properties. Finally, most of the applications of invarianttheory depend on the ability to calculate invariant rings. The last chapter of this bookprovides a sample of applications inside and outside of mathematics.

Acknowledgments. Vladimir Popov and Bernd Sturmfels brought us together asa team of authors. In early 1999 Vladimir Popov asked us to write a contribution onalgorithmic invariant theory for Springer’s Encyclopaedia series. After we agreedto do that, it was an invitation by Bernd Sturmfels to spend two weeks togetherin Berkeley that really got us started on this book project. We thank Bernd forhis strong encouragement and very helpful advice. During the stay at Berkeley, westarted outlining the book, making decisions about notation, and, of course, contents.After that, we worked separately and communicated by e-mail. Most of the workwas done at MIT; Queen’s University at Kingston, Ontario, Canada; the Universityof Heidelberg; and the University of Michigan at Ann Arbor. In early 2001 we spentanother week together at Queen’s University, where we finalized most of the book.Our thanks go to Eddy Campbell, Ian Hughes, and David Wehlau for inviting us toQueen’s.

The book benefited greatly from numerous comments, suggestions, and cor-rections we received from a number of people who read a pre-circulated version.Among these people are Karin Gatermann, Steven Gilbert, Julia Hartmann, GerhardHiß, Jürgen Klüners, Hanspeter Kraft, Martin Lorenz, Kay Magaard, Gunter Malle,

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B. Heinrich Matzat, Vladimir Popov, Jim Shank, Bernd Sturmfels, Nicolas Thiéry,David Wehlau, and Jerzy Weyman. We owe them many thanks for working throughthe manuscript and offering their expertise. The first author likes to thank theNational Science Foundation for partial support under the grant 0102193. Last butnot least, we are grateful to the anonymous referees for further valuable commentsand to Ms. Ruth Allewelt and Dr. Martin Peters at Springer-Verlag for the swift andefficient handling of the manuscript.

Ann Arbor, MI, USA Harm DerksenHeidelberg, Germany Gregor KemperMarch 2002

Contents

1 Constructive Ideal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Ideals and Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Monomial Orderings .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.4 The Buchberger Algorithm .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Elimination Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Image Closure of Morphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 Relations Between Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.3 The Intersection of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.4 The Colon Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.5 The Krull Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Syzygy Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Computing Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Free Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 Hilbert Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.1 Computation of Hilbert Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 The Radical Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.1 Reduction to Dimension Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.2 Positive Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.6 Normalization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Invariant Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1 Invariant Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Reductive Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.1 Linearly Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.2 Other Notions of Reductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3 Categorical Quotients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4 Separating Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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2.5 Homogeneous Systems of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.5.1 Hilbert’s Nullcone.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.5.2 Existence of Homogeneous Systems of Parameters . . . . . . . . 56

2.6 The Cohen-Macaulay Property of Invariant Rings . . . . . . . . . . . . . . . . . . . 572.6.1 The Cohen-Macaulay Property .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.6.2 The Hochster-Roberts Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.7 Hilbert Series of Invariant Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3 Invariant Theory of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.1 Homogeneous Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.1.1 The Linear Algebra Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.1.2 The Reynolds Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 Noether’s Degree Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3 Degree Bounds in the Modular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3.1 Richman’s Lower Degree Bound .. . . . . . . . . . . . . . . . . . . . . . . . . . . 793.3.2 Symonds’ Degree Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4 Molien’s Formula .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.4.1 Characters and Molien’s Formula .. . . . . . . . . . . . . . . . . . . . . . . . . . 843.4.2 Extensions to the Modular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.4.3 Extended Hilbert Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5 Primary Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.5.1 Dade’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.5.2 An Algorithm for Optimal Homogeneous

Systems Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.5.3 Constraints on the Degrees of Primary Invariants .. . . . . . . . . 94

3.6 Cohen-Macaulayness .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.7 Secondary Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.7.1 The Nonmodular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.7.2 The Modular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.8 Minimal Algebra Generators and Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.8.1 Algebra Generators from Primary and

Secondary Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.8.2 Direct Computation of Algebra Generators:

King’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.8.3 Computing Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.9 Properties of Invariant Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.9.1 The Cohen-Macaulay Property .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.9.2 Free Resolutions and Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.9.3 The Hilbert Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.9.4 Polynomial Invariant Rings and Reflection Groups . . . . . . . . 1153.9.5 The Gorenstein Property .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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3.10 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.10.1 Direct Products of Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . 1233.10.2 Göbel’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.10.3 SAGBI Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.11 Ad Hoc Methods .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.11.1 Finding Primary Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.11.2 Finding Secondary Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343.11.3 The Other Exceptional Reflection Groups .. . . . . . . . . . . . . . . . . 138

3.12 Separating Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.12.1 Degree Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.12.2 Polynomial Separating Subalgebras and

Reflection Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.13 Actions on Finitely Generated Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4 Invariant Theory of Infinite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534.1 Computing Invariants of Linearly Reductive Groups . . . . . . . . . . . . . . . . 153

4.1.1 The Heart of the Algorithm.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534.1.2 The Input: The Group and the Representation.. . . . . . . . . . . . . 1564.1.3 The Algorithm.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.2 Improvements and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.2.1 Localization of the Invariant Ring. . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.2.2 Generalization to Arbitrary Graded Rings . . . . . . . . . . . . . . . . . . 1694.2.3 Covariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.3 Invariants of Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1744.4 Invariants of SLn and GLn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.4.1 Binary Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.5 The Reynolds Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.5.1 The Dual Space KŒG�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.5.2 The Reynolds Operator for Semi-simple Groups.. . . . . . . . . . 1864.5.3 Cayley’s Omega Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

4.6 Computing Hilbert Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.6.1 A Generalization of Molien’s Formula . . . . . . . . . . . . . . . . . . . . . 1984.6.2 Hilbert Series of Invariant Rings of Tori . . . . . . . . . . . . . . . . . . . . 2024.6.3 Hilbert Series of Invariant Rings of Connected

Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.6.4 Hilbert Series and the Residue Theorem.. . . . . . . . . . . . . . . . . . . 206

4.7 Degree Bounds for Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.7.1 Degree Bounds for Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2194.7.2 Degree Bounds for Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

4.8 Properties of Invariant Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2264.9 Computing Invariants of Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 227

4.9.1 Computing Separating Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 2284.9.2 Computing the Purely Inseparable Closure . . . . . . . . . . . . . . . . . 2324.9.3 Actions on Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

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4.10 Invariant Fields and Localizations of Invariant Rings . . . . . . . . . . . . . . . 2404.10.1 Extendend Derksen Ideals and CAGEs . . . . . . . . . . . . . . . . . . . . . 2414.10.2 The Italian Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2454.10.3 Geometric Aspects of Extended Derksen Ideals. . . . . . . . . . . . 2464.10.4 Computational Aspects of Extended Derksen Ideals . . . . . . . 2484.10.5 The Additive Group .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2544.10.6 Invariant Rings and Quasi-affine Varieties . . . . . . . . . . . . . . . . . . 258

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

5 Applications of Invariant Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2655.1 Cohomology of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2655.2 Galois Group Computation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

5.2.1 Approximating Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2695.2.2 The Symbolic Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

5.3 Noether’s Problem and Generic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 2725.4 Systems of Algebraic Equations with Symmetries. . . . . . . . . . . . . . . . . . . 2755.5 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2765.6 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2785.7 Coding Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2815.8 Equivariant Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2835.9 Material Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2855.10 Computer Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

5.10.1 View Invariants of 3D Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2885.10.2 Invariants of n Points on a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2895.10.3 Moment Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

A Linear Algebraic Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297A.1 Linear Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297A.2 The Lie Algebra of a Linear Algebraic Group . . . . . . . . . . . . . . . . . . . . . . . 299A.3 Reductive and Semi-simple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303A.4 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304A.5 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

B Is One of the Two Orbits in the Closure of the Other? . . . . . . . . . . . . . . . . . . 309B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309B.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310B.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312B.4 Defining the Set G � L by Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

C Stratification of the Nullcone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323C.2 The Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Contents xv

C.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330C.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Addendum to Appendix C: The Source Code of HNC . . . . . . . . . . . . . . . . . . . . . . . 345References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Introduction

Like the Arabian phoenix rising out of the ashes, the theory of invariants, pronounced deadat the turn of the century, is once again at the forefront of mathematics. During its longeclipse, the language of modern algebra was developed, a sharp tool now at last beingapplied to the very purpose for which it was invented. (Kung and Rota [1])

A brief history Invariant theory is a mathematical discipline with a long tradition,going back at least one hundred and fifty years. Sometimes it has blossomed;sometimes it has lain dormant. But through all phases of its existence, invarianttheory has had a significant computational component. Indeed, the period of“classical invariant theory,” in the late 1800s, was championed by true mastersof computation like Aronhold, Clebsch, Gordan, Cayley, Sylvester, and Cremona.This classical period culminated with two landmark papers by Hilbert. In thefirst [2], he showed that invariant rings of the classical groups are finitely generated.His nonconstructive proof was harshly criticized by Gordan (see p. 42 in thisbook). Hilbert replied in the second paper [3] by giving constructive methods forfinding all invariants under the special and general linear group. Hilbert’s papersclosed the chapter of classical invariant theory and sent this line of research into anearly dormant state for some decades, but they also sparked the development ofcommutative algebra and algebraic geometry. Indeed, Hilbert’s papers on invarianttheory [2, 3] contain such fundamental results as the Nullstellensatz, the basistheorem, the rationality of what is now called the Hilbert series, and the syzygytheorem. The rise of algebraic geometry and commutative algebra had a stronginfluence on invariant theory—which never really went to sleep—as might be bestdocumented by the books by Mumford et al. [4] (whose first edition was publishedin 1965) and Kraft [5].

The advent in the 1960s and 1970s of computational methods based on Gröbnerbases1 brought a decisive turn. These methods initiated the development of

1It may be surprising that Gröbner bases themselves came much earlier. They appeared in an 1899paper of Gordan [6], where he re-proved Hilbert’s finiteness theorem for invariant rings.

xvii

xviii Introduction

computational commutative algebra as a new field of research, and consequentlythey revived invariant theory. In fact, new algorithms and fast computers make manycalculations now feasible that in the classical period were either simply impossibleor carried a prohibitive cost. Furthermore, a heightened interest in modulo pquestions led to a strong activity in modular invariant theory. An important role inboosting interest in computational invariant theory was also played by Sturmfels’sbook Algorithms in Invariant Theory [7]. Various other monographs (such asBenson [8], Smith [9], Neusel and Smith [10], Dolgachev [11], Lorenz [12],Neusel [13], and Campbell and Wehlau [14]) and numerous research articles oninvariant theory have appeared quite recently, all evidence of a field in ferment.

Aims of this book This book focuses on algorithmic methods in invariant theory.A central topic is the question how to find a generating set for the invariant ring.We deal with this question in the case of finite groups and reductive groups. In thecase of finite groups, we emphasize the modular case, in which the characteristic ofthe ground field divides the group order. In this case, many interesting theoreticalquestions in invariant theory of finite groups are still open, and new phenomenatend to occur. The scope of this book is not limited to the discussion of algorithms.A recurrent theme in invariant theory is the investigation of structural properties ofinvariant rings and their links with properties of the corresponding linear groups. Inthis book, we consider primarily the properties of invariant rings that are susceptibleto algorithmic computation (such as the depth) or are of high relevance to thebehavior and feasibility of algorithms (such as degree bounds). We often considerthe geometric “incarnation” of invariants and examine, for example, the question ofseparating orbits by invariants. In addition, this book has a chapter on applicationsof invariant theory to several mathematical and nonmathematical fields. Althoughwe are nonexperts in most of the fields of application, we feel that it is important andhope it is worthwhile to include as much as we can from the applications side, sinceinvariant theory, as much as it is a discipline of its own, has always been drivenby what it was used for. Moreover, it is specifically the computational aspect ofinvariant theory that lends itself to applications particularly well.

Other books The list of monographs on invariant theory is quite long andincludes the books of Dieudonné and Carrell [15], Springer [16], Kraft andProcesi [17], Kraft [5], Popov [18], Sturmfels [7], Benson [8], Popov and Vin-berg [19], Dolgachev [20], Smith [9], Goodman and Wallach [21], Olver [22],Neusel and Smith [10], Dolgachev [11], Lorenz [12], Neusel [13], and Campbelland Wehlau [14]. We hope that our book will serve as a useful addition to these. Ourchoice of material differs in several ways from that of previous books. In particular,of the books mentioned, Sturmfels’ [7] is the only one that strongly emphasizesalgorithms and computation. Several points distinguish our book from [7]. First ofall, the first edition of this book appeared 9 years later than [7], enabling it to includemany new developments such as the first author’s algorithm for computing invariantrings of linearly reductive groups, new results on degree bounds, and the concept ofseparating invariants. Moreover, the modular case of invariant theory receives a fair

Introduction xix

amount of our attention in this book. On the other hand, [7] covers many aspectsof classical invariant theory and brings them together with modern algorithms. Incontrast, our book touches only occasionally on classical invariant theory. In itsemphasis on modular invariant theory, our book has something in common withBenson [8], Smith [9], Neusel and Smith [10], and Campbell and Wehlau [14]. Itis probably fair to say that most of the material covered in Chaps. 3 and 4 (the corechapters of this book) has never appeared in a book before.

Readership The intended readership of this book includes postgraduate studentsas well as researchers in geometry, computer algebra, and, of course, invarianttheory. The methods used in this book come from different areas of algebra, such asalgebraic geometry, (computational) commutative algebra, group and representationtheory, Lie theory, and homological algebra. This diversity entails some unevennessin the knowledge that we assume on the readers’ part. We have nevertheless triedto smooth out the bumps, so a good general knowledge of algebra should sufficeto understand almost all of the text. The book contains many examples and explicitcalculations that we hope are instructive. Generally, we aim to maximize the benefitsof this book to readers. We hope that it, or at least parts of it, can also be used as abasis for seminars.

Proofs When writing this book, we had to decide which proofs of particularstatements to include or omit. Our primary consideration was whether a proof is,in our view, instructive. Of course, other factors also had some weight, such asthe length of a proof, its novelty, its availability elsewhere in the literature, theimportance of the result, and its relevance to computational matters. Some degreeof arbitrariness is probably unavoidable in such decisions, but we do hope that ourchoices contribute to the readability of the book. When proofs are omitted, we givereferences.

Organization of the book Most of the algorithms presented in this book rely inone way or another on Gröbner basis methods. Therefore, we decided to devotethe first chapter to an introduction of Gröbner bases and methods in computationalcommutative algebra that are built on them. Since most of the material is alsocovered in several other books (see the references at the beginning of Chap. 1),we considered it justifiable and appropriate to give a concise presentation almostcompletely “unburdened” by proofs. The aim is to give the reader a quick overviewof the relevant techniques. We cover most of the standard applications of Gröbnerbases to ideal theory, such as the computation of elimination ideals, intersections,ideal quotients, dimension, syzygy modules and resolutions, radical ideals, andHilbert series. Our treatment in Sect. 1.6 of de Jong’s normalization algorithm goesbeyond the material found in the standard texts. For this reason, we have decided togive full proofs in Sect. 1.6.

The second chapter gives a general introduction to invariant theory. The goalis to acquaint the reader with the basic objects and problems and, perhaps mostimportant, to specify the notation. The presentation is enriched with many examples.

xx Introduction

In this chapter we aim to set the stage for later developments. In particular, Sects. 2.4through 2.7 are written with applications to Chaps. 3 and 4 in mind. In Sect. 2.6.2,we present a proof of the Hochster-Roberts theorem that is based on the concept oftight closure. Section 2.4 is devoted to separating invariants, a subject that attracteda lot of activity recently. Here we go back to one of the original purposes for whichinvariant theory was invented and ask whether a subset of the invariant ring mighthave the same properties of separating group orbits as the full invariant ring, even ifthe subset may not generate the invariant ring. As it turns out, it is always possibleto find a finite set with this property, even though the invariant ring itself may not befinitely generated (see Theorem 2.4.8).

Chapters 3 and 4 form the core of the book. In Chap. 3 we look at invariantsof finite groups. Here the modular case, in which the characteristic of the groundfield divides the group order, is included and indeed emphasized. The main goalof the chapter is to present algorithms for finding a finite set of generators ofthe invariant ring. As the reader will discover, these algorithms are much morecumbersome in the modular case. The importance of having algorithms for this caselies mainly in the fact that modular invariant theory is a field with many interestingproblems that remain unsolved. Therefore, it is useful to be able explore the terrainby computation. The main algorithms for computing generators and determiningproperties of invariant rings are presented in Sects. 3.1 through 3.9. Many of thealgorithms were developed by the second author. In Sects. 3.10 and 3.11, we discussmethods applicable to special situations and ad hoc methods. A number of notstrictly computational issues are addressed in Chap. 3, notably degree bounds. Wepresent a proof found by Benson, Fleischmann, and Fogarty for the Noether boundthat extends to the case of positive characteristic not dividing the group order,which was left open by Noether’s original argument. In Sect. 3.3.2, we presenta new degree bound, proved by Symonds, which holds in all characteristics andonly depends on the group order and the dimension of the representation. Such abound has not appeared in the literature before. In Sect. 3.12, we revisit the topic ofseparating subalgebras and show that the Noether bound always holds for separatinginvariants even when it fails for generating invariants. Section 3.13 deals with thecomputation of the invariant ring of a finite group acting on a finitely generatedalgebra which need not be a polynomial ring over a field.

The fourth chapter is devoted to invariants of linearly reductive groups. Wepresent a general algorithm for computing a finite set of generating invariants,which was found by the first author. This algorithm makes use of the Reynoldsoperator, which is studied systematically in Sect. 4.5. In Sect. 4.6, we discuss howthe Hilbert series of the invariant ring can be calculated by using an integral similarto Molien’s formula. As for finite groups, degree bounds are also an importantissue in the case of reductive groups. In Sect. 4.7, we discuss an improvement ofa degree bound given by Popov. An important special case of reductive groups istori. In Sect. 4.3, we present a new algorithm for computing generating invariantsof tori. Section 4.9 deals with computing invariant rings of reductive groups thatneed not be linearly reductive. Since invariant rings of such groups are guaranteedto be finitely generated by a celebrated result of Nagata, this constitutes (up to

References xxi

the question of actions on nonreduced algebras) the final step in making Nagata’sresult constructive. In Sect. 4.10, we go beyond reductive groups and discuss recenttechniques for computing invariant fields and localizations of invariant rings, fromwhich the invariant ring itself can be extracted if it happens to be finitely generated.

In Chap. 5, we embark on a tour of several applications of invariant theory. Westart with applications to different areas in algebra. Here we discuss the computationof cohomology rings of finite groups, solving systems of algebraic equations withsymmetries, the determination of Galois groups, and the construction of genericpolynomials via a positive solution of Noether’s problem. Then we move on to othermathematical disciplines. We address applications to graph theory, combinatorics,coding theory, and dynamical systems. Finally, we look at examples from computervision and materials science in which invariant theory can be a useful tool. Thischapter is incomplete in (at least) three ways. First, the scope of fields whereinvariant theory is applied is much bigger than the selection that we present here.We aim to present applications that we consider to be typical and that represent acertain bandwidth. Second, we are nonexperts in most of the fields addressed inthis chapter. Therefore, certain inaccuracies are unavoidable in our presentation,and many experts will probably find that we missed their favorite article on thesubject. We apologize in advance and ask readers to bring such shortcomings to ourattention. Third, we very intentionally limit ourselves to giving a short presentationof a few selected topics and examples for each field of application. We want toconvey to the reader more a taste of the subject matter than a comprehensivetreatment. So Chap. 5 is meant to operate a bit like a space probe originating fromour home planet (algebra) and traveling outward through the solar system, visitingsome planets but skipping others, and taking snapshots along the way.

Finally, the book has an appendix where we have compiled some standard factsabout algebraic groups. The material of the appendix is not a prerequisite for everypart of the book. In fact, it is required primarily for Sects. 4.5 through 4.8.

References

1. Joseph P. S. Kung, Gian-Carlo Rota, The invariant theory of binary forms, Bull. Am. Math.Soc., New Ser. 10 (1984), 27–85.

2. David Hilbert, Über die Theorie der algebraischen Formen, Math. Ann. 36 (1890), 473–534.3. David Hilbert, Über die vollen Invariantensysteme, Math. Ann. 42 (1893), 313–370.4. David Mumford, John Fogarty, Frances Kirwan, Geometric Invariant Theory, Ergebnisse der

Math. und ihrer Grenzgebiete 34, third edn., Springer-Verlag, Berlin, Heidelberg, New York1994.

5. Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathemat-ics D1, Vieweg, Braunschweig/Wiesbaden 1985.

6. Paul Gordan, Neuer Beweis des Hilbertschen Satzes über homogene Funktionen, NachrichtenKönig. Ges. der Wiss. zu Gött. (1899), 240–242.

7. Bernd Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, Wien, New York 1993.8. David J. Benson, Polynomial Invariants of Finite Groups, Lond. Math. Soc. Lecture Note

Ser. 190, Cambridge Univ. Press, Cambridge 1993.

xxii Introduction

9. Larry Smith, Polynomial Invariants of Finite Groups, A. K. Peters, Wellesley, Mass. 1995.10. Mara D. Neusel, Larry Smith, Invariant Theory of Finite Groups, Mathematical Surveys and

Monographs 94, American Mathematical Society, Providence, RI 2002.11. Igor Dolgachev, Lectures on Invariant Theory, London Mathematical Society Lecture Note

Series 296, Cambridge University Press, Cambridge 2003.12. Martin Lorenz, Multiplicative Invariant Theory, Encyclopaedia of Mathematical Sci-

ences 135, Springer-Verlag, Berlin 2005.13. Mara D. Neusel, Invariant Theory, Student Mathematical Library 36, American Mathematical

Society, Providence, RI 2007.14. H. E. A. Eddy Campbell, David L. Wehlau, Modular Invariant Theory, Encyclopaedia of

Mathematical Sciences 139, Springer-Verlag, Berlin 2011.15. Jean A. Dieudonné, James B. Carrell, Invariant Theory, Old and New, Academic Press,

New York 1971.16. Tonny A. Springer, Invariant Theory, Lecture Notes in Math. 585, Springer-Verlag, Berlin,

Heidelberg, New York 1977.17. Hanspeter Kraft, Claudio Procesi, A primer of invariant theory, Notes by G. Boffi, Brandeis

Lecture Notes 1. Updated version (2000) available at http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf, 1982.

18. Vladimir L. Popov, Groups, Generators, Syzygies and Orbits in Invariant Theory, Translationsof Mathematical Monographs 100, American Mathematical Society, Providence, RI 1992.

19. Vladimir L. Popov, Ernest B. Vinberg, Invariant theory, in: N. N. Parshin, I. R. Shafarevich,eds., Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences 55, Springer-Verlag,Berlin, Heidelberg 1994.

20. Igor V. Dolgachev, Introduction to Geometric Invariant Theory, Lecture Notes Series 25,Seoul National University Research Institute of Mathematics Global Analysis ResearchCenter, Seoul 1994.

21. Roe Goodman, Nolan R. Wallach, Representations and Invariants of the Classical Groups,Encyclopedia of Mathematics and its Applications 68, Cambridge University Press,Cambridge 1998.

22. Peter J. Olver, Classical Invariant Theory, London Mathematical Society Student Texts 44,Cambridge University Press, Cambridge 1999.

http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdfhttp://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf

Chapter 1Constructive Ideal Theory

In this chapter we will provide the basic algorithmic tools which will be used inlater chapters. More precisely, we introduce some algorithms of constructive idealtheory (or, almost synonymously, of computational commutative algebra), almost allof which are based on Gröbner bases. As the reader will find out, these algorithmsand thus Gröbner bases literally permeate this book. When Sturmfels’ book [1] waspublished, not much introductory literature on Gröbner bases and their applicationswas available. In contrast, we now have the books by Becker and Weispfenning [2],Adams and Loustaunau [3], Cox et al. [4], Vasconcelos [5], Cox et al. [23], Kreuzerand Robbiano [6, 7], Greuel and Pfister [8], Ene and Herzog [9], and chapters fromEisenbud [10] and Kemper [11]. This list of references could be continued further,particularly by adding books dealing with variants or special purposes of Gröbnerbases. We will draw heavily on these sources and restrict ourselves to giving arather short overview of the part of the theory that we require. The algorithmsintroduced in Sects. 1.1, 1.2 and 1.3 of this chapter have efficient implementationsin various computer algebra systems, such as CoCoA [12], MACAULAY2 [13],MAGMA [14], or SINGULAR [15], to name just a few, rather specialized ones. Thenormalization algorithm explained in Sect. 1.6 is implemented in MACAULAY2and SINGULAR.

We will be looking at ideals I � KŒx1; : : : ; xn� in a polynomial ring over a fieldK. For polynomials f1; : : : ; fk 2 KŒx1; : : : ; xn�, the ideal generated by the fi will bedenoted by . f1; : : : ; fk/KŒx1; : : : ; xn� or by . f1; : : : ; fk/ if no misunderstanding canarise. The algorithms in this chapter will be mostly about questions in algebraicgeometry, so let us introduce some basic notation. An affine variety is a subsetX of the n-dimensional affine space An D An.K/ WD Kn defined by a set S �KŒx1; : : : ; xn� of polynomials as

X D V.S/ WD f.�1; : : : ; �n/ 2 Kn j f .�1; : : : ; �n/ D 0 for all f 2 Sg:

© Springer-Verlag Berlin Heidelberg 2015H. Derksen, G. Kemper, Computational Invariant Theory, Encyclopaediaof Mathematical Sciences 130, DOI 10.1007/978-3-662-48422-7_1

1

2 1 Constructive Ideal Theory

When we talk about varieties, we usually assume that K is algebraically closed.(Otherwise, we could work in the language of schemes.) The Zariski topology onA

n is defined by taking the affine varieties as closed sets. An affine variety (or anyother subset of An) inherits the Zariski topology from An. A nonempty affine varietyX is called irreducible if it is not the union of two nonempty, closed proper subsets.(In the literature varieties are often defined to be irreducible, but we do not make thisassumption here.) The (Krull) dimension of X is the maximal length k of a strictlyincreasing chain

X0 ¤ X1 ¤ � � � ¤ Xk � X

of irreducible closed subsets.For an affine variety X D V.S/, let I be the radical ideal of the ideal in

KŒx1; : : : ; xn� generated by S. Then X D V.I/, and the quotient ring KŒX� WDKŒx1; : : : ; xn�=I is called the coordinate ring. X is irreducible if and only if KŒX�is an integral domain, and the dimension of X equals the Krull dimension of KŒX�,i.e., the maximal length of a strictly increasing chain of prime ideals in KŒX�. ByHilbert’s Nullstellensatz, we can identify KŒX� with a subset of the ring KX offunctions from X into K. Elements from KŒX� are called regular functions on X. IfX and Y are affine varieties, a morphism ': X ! Y is a mapping from X into Ysuch that the image of the induced mapping

'�: KŒY� ! KX; f 7! f ı ';

lies in KŒX�.

1.1 Ideals and Gröbner Bases

In this section we introduce the basic machinery of monomial orderings and Gröbnerbases.

1.1.1 Monomial Orderings

By a monomial in KŒx1; : : : ; xn� we understand an element of the form xe11 � � � xenn

with ei nonnegative integers. Let M be the set of all monomials. A term is anexpression c � t with 0 ¤ c 2 K and t 2 M. Thus every polynomial is a sum ofterms.

1.1 Ideals and Gröbner Bases 3

Definition 1.1.1 A monomial ordering is a total order “>” on M satisfying thefollowing conditions:

(i) t > 1 for all t 2 M n f1g,(ii) t1 > t2 implies st1 > st2 for all s; t1; t2 2 M.We also use a monomial ordering to compare terms. A nonzero polynomial f 2KŒx1; : : : ; xn� can be written uniquely as f D ct C g such that t 2 M, c 2 K n f0g,and every term of g is smaller (with respect to the order “>”) than t. Then we write

LT. f / D ct; LM. f / D t; and LC. f / D c

for the leading term, leading monomial, and leading coefficient of f . For f D 0,all three values are defined to be zero.

A monomial ordering is always a well-ordering. This follows from the fact thatideals in KŒx1; : : : ; xn� are finitely generated. We note that the usage of terminologyis not uniform in the literature. Some authors (e.g. Becker and Weispfenning [2])have monomials and terms interchanged, and some speak of initial or head terms,monomials and coefficients. Monomial orderings are often called term orders. Whenbrowsing through the literature one can find almost any combination of these piecesof terminology.

Example 1.1.2 We give a few examples of monomial orderings. Let t D xe11 � � � xennand t0 D xe011 � � � xe

0

nn be two distinct monomials.

(a) The lexicographic monomial ordering (with x1 > x2 > � � � > xn): t is consideredgreater than t0 if ei > e0i for the smallest i with ei ¤ e0i. We sometimes writet >lex t0 in this case. As an example, we have

LMlex.x1 C x2x4 C x23/ D x1:

The lexicographic monomial ordering is useful for solving systems of algebraicequations.

(b) The graded lexicographic monomial ordering: t >glex t0 if deg.t/ > deg.t0/, orif deg.t/ D deg.t0/ and t >lex t0. Here deg.t/ is the total degree e1 C � � � C en.For example,

LMglex.x1 C x2x4 C x23/ D x2x4:

The graded lexicographic monomial ordering can be generalized by using aweighted degree deg.t/ WD w1e1C� � �Cwnen with wi fixed positive real numbers.

(c) The graded reverse lexicographic monomial ordering (grevlex-ordering forshort): t >grevlex t0 if deg.t/ > deg.t0/, or if deg.t/ D deg.t0/ and ei < e0ifor the largest i with ei ¤ e0i. For example,

LMgrevlex.x1 C x2x4 C x23/ D x23:

4 1 Constructive Ideal Theory

The grevlex ordering is often very efficient for computations. It can also begeneralized by using a weighted degree.

(d) Block orderings: Let>1 be a monomial ordering on the monomials in x1; : : : ; xr,and >2 a monomial ordering on the monomials in xrC1; : : : ; xn. Then theblock ordering formed from >1 and >2 is defined as follows: t > t0 ifxe11 � � � xerr >1 xe

0

1

1 � � � xe0

rr , or if x

e11 � � � xerr D xe

0

1

1 � � � xe0

rr and x

erC1rC1 � � � xenn >2

xe0rC1rC1 � � � xe

0

nn . For example, the lexicographic monomial ordering is a block order-

ing. Block orderings are useful for the computation of elimination ideals (seeSect. 1.2). G

We say that a monomial ordering is graded if deg.t/ > deg.t0/ implies t > t0. Sothe orderings in (b) and (c) of the previous example are graded.

Given a monomial ordering, we write xi � xj if xi > xej for all nonnegativeintegers e. For example, in the lexicographic monomial ordering we have x1 �x2 � � � � � xn. Moreover, if “>” is a block ordering with blocks x1; : : : ; xr andxrC1; : : : ; xn, then xi � xj for i � r and j > r. If xi � xj for all j 2 J for someJ � f1; : : : ; ng, then xi is greater than any monomial in the indeterminates xj, j 2 J.This follows directly from Definition 1.1.1.

1.1.2 Gröbner Bases

We fix a monomial ordering on KŒx1; : : : ; xn�.

Definition 1.1.3 Let S � KŒx1; : : : ; xn� be a set of polynomials. We write

L.S/ D .LM.g/ j g 2 S/

for the ideal generated by the leading monomials from S. L.S/ is called the leadingideal of S (by some authors also called the initial ideal).

Let I � KŒx1; : : : ; xn� be an ideal. Then a finite subset G � I is called a Gröbnerbasis of I (with respect to the chosen monomial ordering) if

L.I/ D L.G/:

It is clear that a Gröbner basis of I generates I as an ideal. Indeed, a (hypothetical)element f 2 I n .G/ with minimal leading monomial could be transformed intog 2 I n .G/ with smaller leading monomial by subtracting a multiple of an elementfrom G, which yields a contradiction. It is also clear that Gröbner bases always exist.Indeed, fLM. f / j f 2 Ig generates L.I/ by definition, hence by the Noether propertya finite subset fLM. f1/; : : : ;LM. fm/g also generates L.I/, and so f f1; : : : ; fmg isa Gröbner basis. This argument, however, is nonconstructive. But we will see inSect. 1.1.4 that there is in fact an algorithm for computing Gröbner bases.

1.1 Ideals and Gröbner Bases 5

The most obvious question about an ideal I � KŒx1; : : : ; xn� that can be decidedwith Gröbner bases is whether I D KŒx1; : : : ; xn�. Indeed, this is the case if and onlyif G contains a (nonzero) constant polynomial.

1.1.3 Normal Forms

A central element in the construction and usage of Gröbner bases is the computationof so-called normal forms.

Definition 1.1.4 Let S D fg1; : : : ; gsg � KŒx1; : : : ; xn� be a finite set ofpolynomials.

(a) A polynomial f 2 KŒx1; : : : ; xn� is said to be in normal form with respect to Sif no term of f is divisible by the leading monomial of any g 2 S.

(b) If f and Qf are polynomials in KŒx1; : : : ; xn�, then Qf is said to be a normal formof f with respect to S if the following conditions hold:

(1) Qf is in normal form with respect to S.(2) There exist h1; : : : ; hs 2 KŒx1; : : : ; xn� with

f � Qf DsX

iD1higi and LM.higi/ � LM. f / for all i (1.1.1)

(in particular, f � Qf lies in the ideal generated by S);Example 1.1.5 Let S D fx1; x1 C 1g. Then 1 is congruent to 0 modulo .S/, but 0 isnot a normal form of 1. Moreover, f D x1 has two normal forms: 0 and �1. So ingeneral, normal forms are not uniquely determined.

Notice that S is not a Gröbner basis. GThe following algorithm, which mimics division with remainder in the univariate

case, calculates a normal form with respect to a finite set S of polynomials.

Algorithm 1.1.6 (Normal form) Given a polynomial f 2 KŒx1; : : : ; xn� and a finitesubset S D fg1; : : : ; gsg � KŒx1; : : : ; xn�, perform the following steps to obtaina normal form Qf of f with respect to S, together with polynomials h1; : : : ; hs 2KŒx1; : : : ; xn� satisfying (1.1.1).

(1) Set Qf WD f and hi WD 0 for all i, and repeat the steps (2)–(4).(2) If no term of Qf is divisible by the leading monomial of any gi 2 S, return Qf as a

normal form of f , and return the hi.(3) Let ct be the maximal term of Qf such that there exists gi 2 S with LM.gi/

dividing t.

6 1 Constructive Ideal Theory

(4) Set

Qf WD Qf � ctLT.gi/

gi and hi WD hi C ctLT.gi/

:

Of course the computation of the hi can be omitted if only a normal form isdesired. The termination of Algorithm 1.1.6 is guaranteed by the fact that themaximal monomials t of Qf divisible by some LM.gi/ form a strictly decreasingsequence, but such a sequence is finite by the well-ordering property. The resultof Algorithm 1.1.6 is in general not unique, since it depends on the choice of the giin step (3). However, if G is a Gröbner basis of an ideal I, then normal forms withrespect to G are unique. In fact, if Qf and Of are two normal forms of f with respectto G, then Qf � Of 2 I, so LM. Qf � Of / is divisible by some LM.g/ with g 2 G. Butif Qf ¤ Of , then LM. Qf � Of / must appear as a monomial in Qf or Of , contradicting thefact that Qf and Of are in normal form. In the case of a Gröbner basis G we writeQf DW NF. f / D NFG. f / for the normal form. An important property of the mapNFG : KŒx1; : : : ; xn� ! KŒx1; : : : ; xn� is that it is linear.

It should be mentioned that there is a variant of the normal form algorithm whichstops when the leading term of Qf is zero or not divisible by any LM.g/, g 2 S(“top-reduction”).

Using Algorithm 1.1.6, we obtain a membership test for ideals.

Algorithm 1.1.7 (Membership test in ideals) Let I � KŒx1; : : : ; xn� be an ideal,G a Gröbner basis of I, and f 2 KŒx1; : : : ; xn� a polynomial. Then

f 2 I ” NFG. f / D 0:

One can also substitute NFG. f / by the result of top-reducing f .

It is easy to see that the map NFG : KŒx1; : : : ; xn� ! KŒx1; : : : ; xn� is K-linear, andby Algorithm 1.1.7, its kernel is I. So the normal form provides a way to performexplicit calculations in the quotient ring KŒx1; : : : ; xn�=I. In fact, this was the mainobjective for which Gröbner bases were invented.

A Gröbner basis G of an ideal I can be transformed into a reduced Gröbnerbasis by iteratively substituting an element from G by a normal form with respect tothe other elements, until every element is in normal form. After deleting zero fromthe resulting set and making all leading coefficients equal to 1, the resulting monicreduced Gröbner basis is unique (i.e., it only depends on I and the chosen monomialordering, see Becker and Weispfenning [2, Theorem 5.43]).

1.1.4 The Buchberger Algorithm

In order to present Buchberger’s algorithm for the construction of Gröbner bases,we need to introduce s-polynomials. Let f ; g 2 KŒx1; : : : ; xn� be two nonzero

1.1 Ideals and Gröbner Bases 7

polynomials, and set t WD lcm.LM. f /;LM.g// (the least common multiple). Thenthe s-polynomial of f and g is defined as

spol. f ; g/ WD LC.g/ � tLM. f /

f � LC. f / � tLM.g/

g:

Note that the coefficients of t cancel in spol. f ; g/, and that spol. f ; g/ 2 . f ; g/. Thefollowing lemma is the key step toward finding an algorithm for the construction ofa Gröbner basis.

Lemma 1.1.8 (Buchberger [16]) Let G be a basis (=generating set) of an idealI � KŒx1; : : : ; xn�. Then the following statements are equivalent.(a) G is a Gröbner basis of I.(b) If f ; g 2 G, then spol. f ; g/ has 0 as a normal form with respect to G.

See Becker and Weispfenning [2, Theorem 5.48] or Kemper [11, Theorem 9.12]for a proof. We can give Buchberger’s algorithm in a rather coarse form now.

Algorithm 1.1.9 (Buchberger’s algorithm) Given a finite basis S for an ideal I �KŒx1; : : : ; xn�, construct a Gröbner basis (with respect to a given monomial ordering)of I by performing the following steps:

(1) Set G WD S and repeat steps (2)–(4).(2) For f ; g 2 G compute a normal form h of spol. f ; g/ with respect to G.(3) If h ¤ 0, include h into G.(4) If h was found to be zero for all f ; g 2 G, then G is the desired Gröbner basis.

This algorithm terminates after a finite number of steps since L.S/ strictlyincreases with every performance of steps (2)–(4).

Remark 1.1.10 The theoretical cost of Buchberger’s algorithm is extremely high.In fact, no general upper bound for the running time is known. But Möller andMora [17] proved an upper bound for the maximal degree of the Gröbner basiselements which depends doubly exponentially on the number of variables. Theyalso proved that this doubly exponential behavior cannot be improved. What makesthings even worse is the phenomenon of “intermediate expression swell,” meaningthat during the computation the number and size of polynomials can become muchbigger than in the final result. It is known that the memory space required for thecomputation of Gröbner bases increases at most exponentially with the size of theinput, and all problems with this behavior can be reduced to the problem of testingideal membership; so the problem of computing Gröbner bases is “EXPSPACE-complete.” We refer to von zur Gathen and Gerhard [18, Section 21.7] for a moredetailed account of what is known about the complexity of Gröbner bases.

In spite of all this bad news, practical experience shows that the algorithmoften terminates after a reasonable time (although this is usually not predictablein advance). Much depends on improvements of the algorithm given above, suchas omitting some pairs f ; g (by Buchberger’s first and second criterion, see Becker

8 1 Constructive Ideal Theory

and Weispfenning [2, Section 5.5]), by having a good strategy which pairs to treatfirst, and by choosing a suitable monomial ordering (if there is any freedom ofchoice). There are also algorithms which transform a Gröbner basis with respectto one monomial ordering into one with respect to another ordering (see Faugèreet al. [19], Collart et al. [20]). G

There is a variant of Buchberger’s algorithm which keeps track of how thepolynomials in the Gröbner basis G arise as linear combinations of the polynomialsin the original ideal basis S. This variant is called the extended Buchbergeralgorithm, and its output is an (ordered) Gröbner basis G D fg1; : : : ; grg and anr � s-matrix A with coefficients in KŒx1; : : : ; xn� such that

0

B@g1:::

gr

1

CA D A �

0

B@f1:::

fs

1

CA ;

where S D f f1; : : : ; fsg. On the other hand, it is straightforward to obtain an s � r-matrix B such that . f1; : : : ; fs/tr D B.g1; : : : ; gr/tr by applying the Normal FormAlgorithm 1.1.6 to the fi.

1.2 Elimination Ideals

Given an ideal I � KŒx1; : : : ; xn� and an integer k 2 f1; : : : ; ng, the eliminationideal of I with respect to xk; : : : ; xn is defined as the intersection I \ KŒxk; : : : ; xn�.It has the following geometric interpretation: If

�: An ! An�kC1; .�1; : : : ; �n/ 7! .�k; : : : ; �n/

is the canonical projection, then for K algebraically closed we have

� .V.I// D V.I \ KŒxk; : : : ; xn�/; (1.2.1)

where the left hand side is the Zariski-closure. (In scheme-theoretic language, �maps a prime ideal in KŒx1; : : : ; xn� to its intersection with KŒxk; : : : ; xn�, and wedo not need the hypothesis that K is algebraically closed.) An important feature ofGröbner bases is that they can be used to compute elimination ideals.

Algorithm 1.2.1 (Computing elimination ideals) Given an ideal I � KŒx1; : : : ;xn� and an integer k 2 f1; : : : ; ng, compute the elimination ideal I \ KŒxk; : : : ; xn� asfollows:

(1) Choose a monomial ordering such that xi � xj for i < k and j k (e.g., thelexicographic monomial ordering or a block ordering).

1.2 Elimination Ideals 9

(2) Compute a Gröbner basis G of I with respect to this monomial ordering.(3) G \ KŒxk; : : : ; xn� is a Gröbner basis of I \ KŒx1; : : : ; xn�.

It is elementary to see that this algorithm is correct (see Becker and Weispfen-ning [2, Proposition 6.15]). Equation (1.2.1) shows how elimination ideals can beused to solve a system of algebraic equations with a finite set of solutions.

We continue by presenting some applications of elimination ideals (and thus ofGröbner bases) which will be needed in the following chapters of this book.

1.2.1 Image Closure of Morphisms

Let X and Y be affine varieties and f : X ! Y a morphism. (Again we assume thatK is algebraically closed or use the language of schemes.) We want to compute theZariski-closure of the image f .X/. Assume that X is embedded into An and Y intoA

m for some n and m. Without loss of generality we can assume that Y D Am. If f isgiven by polynomials . f1; : : : ; fm/ with fi 2 KŒx1; : : : ; xn�, and X is given by an idealI � KŒx1; : : : ; xn�, then the graph of f is given by the ideal

J WD I � KŒx1; : : : ; xn; y1; : : : ; ym�C . f1 � y1; : : : ; fm � ym/ (1.2.2)

in KŒx1; : : : ; xn; y1; : : : ; ym�. Thus by Eq. (1.2.1), the closure of the image is

f .X/ D V .J \ KŒy1; : : : ; ym�/

(see Vasconcelos [5, Proposition 2.1.3]), and can therefore be calculated byAlgorithm 1.2.1.

1.2.2 Relations Between Polynomials

A further application of elimination ideals is the computation of relations betweenpolynomials. More precisely, let f1; : : : ; fm 2 KŒx1; : : : ; xn� be polynomials. We areinterested in the kernel of the homomorphism

˚ : KŒt1; : : : ; tm� ! KŒx1; : : : ; xn�; ti 7! fi;

of K-algebras (where t1; : : : ; tm are further indeterminates). The answer is asfollows: Define the ideal

J WD . f1 � t1; : : : ; fm � tm/

10 1 Constructive Ideal Theory

in KŒx1; : : : ; xn; t1; : : : ; tm�. Then it is easy to show that

ker.˚/ D J \ KŒt1; : : : ; tm�; (1.2.3)

so the desired kernel is again an elimination ideal (see Eisenbud [10, Proposi-tion 15.30]). More generally, taking J as in (1.2.2) and forming the elimination idealas above yields the preimage ˚�1.I/. Notice that generators for ker.˚/ togetherwith the fi provide a presentation of the algebra generated by the fi.

1.2.3 The Intersection of Ideals

The intersection of two ideals I; J � KŒx1; : : : ; xn� (which geometrically cor-responds to the union of varieties) can be computed as follows: With a newindeterminate t, form the ideal L in KŒx1; : : : ; xn; t� generated by

I � t C J � .1 � t/;

where the products are formed by multiplying each generator of I and J by t and1 � t, respectively. Then

I \ J D L \ KŒx1; : : : ; xn� (1.2.4)

(see Vasconcelos [5, Corollary 2.1.1]). A different method for computing the inter-section of I and J involves the calculation of a syzygy module (see Vasconcelos [5,p. 29]). We can apply any of these methods iteratively to obtain the intersection of afinite number of ideals, but there is also a direct method (involving further auxiliaryindeterminates) given by Becker and Weispfenning [2, Corollary 6.20].

1.2.4 The Colon Ideal

Given two ideals I; J � KŒx1; : : : ; xn�, it is often important to be able to calculatethe colon ideal

I W J WD fg 2 KŒx1; : : : ; xn� j gf 2 I 8f 2 Jg:Sometimes I W J is also referred to as the quotient ideal. The colon ideal has thefollowing geometric interpretation: If I is a radical ideal and K is algebraicallyclosed, then I W J is precisely the ideal of all polynomials vanishing on V.I/ n V.J/.The colon ideal is also of crucial importance for the computation of radical idealsand primary decomposition.

1.2 Elimination Ideals 11

If J D . f / is a principal ideal, we sometimes write I W f for the colon ideal I W . f /.If J D . f1; : : : ; fk/, then clearly

I W J Dk\

iD1I W fi;

which reduces the computation to the case that J is a principal ideal. But clearly

I W f D .I \ . f // � f �1 (1.2.5)

(see Vasconcelos [5, Proposition 2.1.4(a)]), which can be computed by Eq. (1.2.4).Thus colon ideals can be obtained by using any algorithm for the intersection ofideals.

For an ideal I � KŒx1; : : : ; xn� and a polynomial f 2 KŒx1; : : : ; xn� we can alsoconsider the ideal

I W f 1 WD[

i2NI W f i;

which is sometimes referred to as the saturation ideal of I with respect to f . Thesaturation ideal can be calculated by successively computing the colon ideals Ji WDI W f i D Ji�1 W f . This gives an ascending chain of ideals, thus eventually we getJkC1 D Jk, so I W f 1 D Jk. But there is a more efficient algorithm, based on thefollowing proposition.

Proposition 1.2.2 Let I � KŒx1; : : : ; xn� be an ideal and f 2 KŒx1; : : : ; xn�a polynomial. Introduce an additional indeterminate t and form the ideal J inKŒx1; : : : ; xn; t� generated by I and tf � 1. Then

I W f 1 D J \ KŒx1; : : : ; xn�:

A proof can be found in Becker and Weispfenning [2, Proposition 6.37].

1.2.5 The Krull Dimension

We define the dimension of an ideal I � KŒx1; : : : ; xn� to be the Krull dimension ofthe quotient KŒx1; : : : ; xn�=I. There is a method which computes the dimension byusing elimination ideals (Becker and Weispfenning [2, Section 6.3]). However, thistechnique involves a large number of Gröbner basis computations and is thereforenot very efficient. A better algorithm is based on the following lemma.

Lemma 1.2.3 If “>” is any monomial ordering, then the dimensions of I and ofthe leading ideal L.I/ with respect to “>” coincide.

12 1 Constructive Ideal Theory

A proof can be found in Kemper [11, Exercise 9.7] (solution given). The first stepis to establish the special case that “>” is graded (or, more generally, weight-graded)by using the (vector space) isomorphism KŒx1; : : : ; xn�=I ! KŒx1; : : : ; xn�=L.I/given by the normal form mapping, and the second step reduces the general caseto the graded one by using the so-called convex cone of the given monomialordering. Lemma 1.2.3 reduces our problem to the computation of the dimensionof L.I/, which is a monomial ideal. But the variety defined by a monomial ideal isa finite union of so-called coordinate subspaces, i.e., varieties of the form V.M/with M � fx1; : : : ; xng. Clearly such a variety is contained in the zero set of themonomial ideal J if and only if every generator of J involves at least one variable xilying in M. We obtain the following algorithm (see Cox et al. [4, Proposition 3 ofChap. 9, §1]).

Algorithm 1.2.4 (Dimension of an ideal) Given an ideal I � KŒx1; : : : ; xn�,calculate the dimension of I by performing the following steps:

(1) Compute a Gröbner basis G of I with respect to a any monomial ordering.(2) If G contains a nonzero constant, then I D KŒx1; : : : ; xn�, and the dimension is

(by convention) �1.(3) Otherwise, find a subset M � fx1; : : : ; xng of minimal cardinality such that for

every nonzero g 2 G the leading monomial LM.g/ involves at least one variablefrom M.

(4) The dimension of I is n � jMj.Step (3) of the above algorithm is purely combinatorial and therefore usually

much faster than the Gröbner basis computation. An optimized version of this stepcan be found in Becker and Weispfenning [2, Algorithm 9.6].

The set M � fx1; : : : ; xng occurring in Algorithm 1.2.4 has an interestinginterpretation. In fact, let M0 WD fx1; : : : ; xng n M be the complement of M. Thenfor every nonzero g 2 G the leading monomial LM.g/ involves at least one variablenot in M0. This implies that every nonzero polynomial in L.I/ involves a variablenot in M0, so L.I/ \ KŒM0� D f0g. From this it follows that

I \ KŒM0� D f0g: (1.2.6)

Indeed, if f 2 I \ KŒM0� were nonzero, then LM. f / would lie in L.I/ \ KŒM0�.Subsets M0 � fx1; : : : ; xng which satisfy (1.2.6) are called independent modulo I(see Becker and Weispfenning [2, Definition 6.46]). Consider the rational functionfield L WD K.M0/ in the variables lying in M0, and let LŒM� be the polynomial ringover L in the variables from M. Then (1.2.6) is equivalent to the condition that theideal ILŒM� generated by I in LŒM� is not equal to LŒM�. Since we have jM0j Ddim.I/, it follows that M0 is maximally independent modulo I. (Indeed, if thereexisted a strict superset of M0 which is independent modulo I, then dim.I/ > jMj,see Kemper [11, Theorem 5.9].) The maximality of M0 means that no nonemptysubset of M is independent modulo ILŒM�. By Algorithm 1.2.4, the dimension ofILŒM� must therefore be zero. Thus we have shown:

1.3 Syzygy Modules 13

Proposition 1.2.5 Let I ¤ KŒx1; : : : ; xn� be an ideal and M � fx1; : : : ; xng as inAlgorithm 1.2.4. Set M0 WD fx1; : : : ; xng n M, and take the rational function fieldL WD K.M0/ in the variables lying in M0, and the polynomial ring LŒM�. Then theideal J WD ILŒM� generated by I in LŒM� is not equal to LŒM�, and dim.J/ D 0.

1.3 Syzygy Modules

In this section we write R WD KŒx1; : : : ; xn� for the polynomial ring and Rk for afree R-module of rank k. The standard basis vectors of Rk are denoted by e1; : : : ; ek.Given polynomials f1; : : : ; fk 2 R, we ask for the set of all .h1; : : : ; hk/ 2 Rk suchthat h1f1 C � � � C hkfk D 0. This set is a submodule of Rk, called the syzygy moduleof f1; : : : ; fk and denoted by Syz. f1; : : : ; fk/. More generally, we ask for the kernel ofan R-homomorphism ': Rk ! Rl between two free R-modules. If fi WD '.ei/ 2 Rl,then the kernel of ' consists of all .h1; : : : ; hk/ 2 Rk with h1f1 C � � � C hkfk D 0.Again Syz. f1; : : : ; fk/ WD ker.'/ is called the syzygy module of the fi.

1.3.1 Computing Syzygies

In order to explain an algorithm which computes syzygy modules, we have to givea brief introduction into Gröbner bases of submodules of Rk. A monomial in Rk isan expression of the form tei with t a monomial in R. The notion of a monomialordering is given as in Definition 1.1.1, with condition (i) replaced by tei > ei forall i and 1 ¤ t a monomial in R, and demanding (ii) for monomials t1; t2 2 Rkand s 2 R. Given a monomial ordering, we can now define the leading submoduleL.M/ of a submodule M � Rk and the concept of a Gröbner basis of M as inDefinition 1.1.3. Normal forms are calculated by Algorithm 1.1.6, with the extraspecification that tei is said to be divisible by t0ej if i D j and t divides t0, so thequotients are always elements in R. Moreover, the s-polynomial of f and g 2 Rkwith LM. f / D tei and LM.g/ D t0ej is defined to be zero if i ¤ j. With theseprovisions, Buchberger’s algorithm can be formulated as in Algorithm 1.1.9.

Suppose that G D fg1; : : : ; gkg is a Gröbner basis of a submodule M � Rl. Thenfor gi; gj 2 G we have that NFG.spol.gi; gj// D 0, so there exist h1; : : : ; hk 2 R with

spol.gi; gj/ D h1g1 C � � � C hkgk; (1.3.1)

and the hi can be computed by the Normal Form Algorithm 1.1.6. Sincespol.gi; gj/ is an R-linear combination of gi and gj, Eq. (1.3.1) yields a syzygyri;j 2 Syz.g1; : : : ; gk/. Of course ri;j D 0 if the leading monomials of gi and of gj liein different components of Rl.

The following monomial ordering “>G” on Rk, which depends on G, wasintroduced by Schreyer [21]: tei is considered bigger than t0ej if t LM.gi/ >

14 1 Constructive Ideal Theory

t0 LM.gj/ (with “>” the given ordering on Rl), or if t LM.gi/ D t0 LM.gj/ and i < j.It is easy to see that “>G” satisfies the properties of a monomial ordering.

Theorem 1.3.1 (Schreyer [21]) Let G D fg1; : : : ; gkg be a Gröbner basis withrespect to an arbitrary monomial ordering “>” of a submodule M � Rl. Then, withthe above notation, the ri;j (1 � i < j � k) form a Gröbner basis of Syz.g1; : : : ; gk/with respect to the monomial ordering “>G”.

This settles the case of syzygies for Gröbner bases. Now assume thatf1; : : : ; fk 2 Rl are arbitrary. Using the extended Buchberger algorithm (see at theend of Sect. 1.1), we can calculate a Gröbner basis fg1; : : : ; gk0g of the submodulegenerated by f1; : : : ; fk, along with representations of the gi as R-linear combinationsof the fj. Using the Normal Form Algorithm 1.1.6, we can also express the fj in termsof the gi. The choice of the fj and gi is equivalent to giving homomorphisms Rk ! Rland Rk

0 ! Rl, and expressing the fj in terms of the gi and vice versa is equivalent togiving homomorphisms ' and such that the diagram

0 � N �Rk ′ � Rl

�

�

Rk

commutes (both along ' and ), where N WD Syz.g1; : : : ; gk0/ can be computedby Theorem 1.3.1. The following lemma tells us how to compute Syz. f1; : : : ; fk/ Dker.Rk ! Rl/.Lemma 1.3.2 Let A be a commutative ring and

0 � N �M1 � M

�

�θ

M2

a commutative diagram (both along ' and ) of A-modules, with the upper rowexact. Then we have an exact sequence

0 �! .id � ı '/.M1/ �! N ˚ .id �' ı /.M2/ �! M2 ��! M

with maps

.id � ı '/.M1/ ! N ˚ .id �' ı /.M2/; m 7! .m;�'.m//; andN ˚ .id �' ı /.M2/ ! M2; .n;m/ 7! '.n/C m:

1.3 Syzygy Modules 15

In particular,

ker.�/ D '.N/C .id �' ı /.M2/:

Proof It follows by a simple diagram chase that .id � ı '/.M1/ � N, so the firstmap is indeed into N ˚ .id �' ı /.M2/. We show the exactness at M2. Again by adiagram chase �.'.n/C m/ D 0 for n 2 N and m 2 .id �' ı /.M2/. Conversely,for m 2 ker.�/ we have

m D '. .m//C .id �' ı /.m/

with .m/ 2 N. To show the exactness at N ˚ .id �' ı /.M2/, take.n;m2 � '. .m2/// 2 N ˚ .id �' ı /.M2/ with '.n/Cm2�'. .m2// D 0. Then

n D .id � ı '/.n � .m2// 2 .id � ı '/.M1/;

and .n;m2 � '. .m2/// D .n;�'.n//. This completes the proof. utIn summary, we obtain the following algorithm.

Algorithm 1.3.3 (Calculation of a syzygy module) Given elements f1; : : : ;fk 2 Rl, perform the following steps to find the syzygy module Syz. f1; : : : ; fk/:(1) Using the extended Buchberger algorithm, calculate a Gröbner basis

fg1; : : : ; gk0g of the submodule of Rl generated by the fi together with a matrixA 2 Rk0�k such that

0

B@g1:::

gk0

1

CA D A �

0

B@f1:::

fk

1

CA :

(2) Using the Normal Form Algorithm 1.1.6, compute a matrix B 2 Rk�k0 with0

B@f1:::

fk

1

CA D B �

0

B@g1:::

gk0

1

CA :

(3) For 1 � i < j � k0, compute the syzygies ri;j 2 Syz.g1; : : : ; gk0/ given byEq. (1.3.1).

(4) Syz. f1; : : : ; fk/ is generated by the ri;j � A and the rows of Ik � BA.

16 1 Constructive Ideal Theory

1.3.2 Free Resolutions

For a submodule M � Rl (with R D KŒx1; : : : ; xn� as before) with generating setf1; : : : ; fk, we can compute generators for N WD Syz. f1; : : : ; fk/ � Rk by usingAlgorithm 1.3.3. Continuing by computing the syzygies of these generators andso on, we obtain a free resolution of M, i.e., an exact sequence

0 �! Fr �! Fr�1 �! � � � �! F2 �! F1 �! F0 �! M �! 0 (1.3.2)

with the Fi free R-modules. Hilbert’s syzygy theorem (see Eisenbud [10, Corol-lary 19.8] or the “original” reference Hilbert [22]) guarantees that there exists a freeresolution of finite length (bounded by n, in fact), as given above. Free resolutionsare of great interest because they contain a lot of information about the structure ofM. Theorem 1.3.1 provides the following method for calculating a free resolutionwith only a single Gröbner basis computation.

Algorithm 1.3.4 (Schreyer’s algorithm) Let M � Rl be a submodule given by agenerating set. Obtain a free resolution of M as follows:

(1) Compute a Gröbner basis G D fg1; : : : ; gkg of M with respect to an arbitrarymonomial ordering “>”. Set i WD 0 and repeat steps (2)–(4).

(2) Set Fi WD Rk and obtain the map Fi ! Fi�1 (with F�1 WD M) from (1.3.2) by.h1; : : : ; hk/ 7! h1g1 C � � � C hkgk.

(3) Compute the relations ri;j from Eq. (1.3.1). By Theorem 1.3.1, the ri;j form aGröbner basis with respect to “>G” of the kernel of the map defined in (2).

(4) If all ri;j are zero, the resolution is complete. Otherwise, let G � Rk be the setof the nonzero ri;j and set i WD i C 1.

The termination of Algorithm 1.3.4 after at most n iterations is guaranteed by(the proof of) Theorem 2.1 in Chap. 6 of Cox et al. [23] (which provides a new,constructive proof of Hilbert’s syzygy theorem).

Now suppose that the polynomial ring R is made into a graded algebra bydefining the degrees deg.xi/ of the indeterminates to be positive integers. Thenthe free module Rl can be made into a graded R-module by defining the deg.ei/to be integers. Moreover, suppose that M is a graded submodule, i.e., generatedby homogeneous elements. Then we want to find a graded free resolution, i.e.,one that consists of graded free modules Fi with all mappings degree-preserving.Applying Buchberger’s algorithm to a homogeneous generating set of M yieldsa homogeneous Gröbner basis, too, and by inspection of the way in which thesyzygies ri;j are formed from Eq. (1.3.1), we see that the resolution obtained byAlgorithm 1.3.4 is indeed graded (with the proper choice of the degrees of the freegenerators, i.e., each generator gets the same degree as the relation to which it ismapped).

In the case that Rl is graded and M is a graded submodule, we are also interestedin obtaining a minimal free resolution of M, i.e., a free resolution such that thefree generators of each Fi are mapped to a minimal generating set of the image of

1.4 Hilbert Series 17

Fi. Such a resolution is unique up to isomorphism of complexes (see Eisenbud [10,Theorem 20.2]), and in particular its length is unique. This length is called thehomological dimension of M, written as hdim.M/, and is an important structuralinvariant of M. A graded resolution (1.3.2) calculated by Algorithm 1.3.4 is usuallynot minimal, so how can it be transformed into a minimal resolution, preferablywithout computing any further Gröbner bases? As a first step, we can use linearalgebra to select a minimal subset of the free generators of F0 whose image in Mgenerates M. Thus we obtain a free submodule F00 � F0 and a commutative diagram

· · · � F1 ρ � F0 � M

�

�

F ′0 �M,

where ' ı D id. Lemma 1.3.2 yields an exact sequence

0 �! .id � ı '/.F0/ �! im.�/ '�! F00 �! M: (1.3.3)

Observe that .id � ı'/maps a free generator ei from F0 either to zero (if it is alsoa generator of F00) or to a nonzero element of .id � ı '/.F0/ corresponding to therepresentation of the image of ei in M in terms of the images of those ej containedin F00. These nonzero elements are linearly independent, hence .id � ı '/.F0/ isa free module. We can use linear algebra to compute preimages under � of the freegenerators of .id � ı'/.F0/ in F1. This yields a free submodule bF1 � F1 such that�.bF1/ D .id � ı '/.F0/ and the restriction of � to bF1 is injective. Now it is easy tosee that (1.3.3) and (1.3.2) lead to the exact sequence

0 �! Fr �! Fr�1 �! � � � �! F3 �! F2 ˚ bF1 �! F1 'ı��! F00 �! M �! 0:Thus we have managed to replace (1.3.2) by a free resolution with the firstfree module minimal. Iterating this process, we obtain the desired minimal freeresolution of M. Notice that the only computationally significant steps are theselection of minimal generators for M and the computation of preimages of ei � .'.ei// for some free generators ei of F0. Both of these are accomplished by linearalgebra. Thus a minimal resolution of M can be computed by just one Gröbner basiscomputation and linear algebra.

1.4 Hilbert Series

In this section, we prove some results about Hilbert series of rings, and how we canuse Gröbner bases to compute them.

18 1 Constructive Ideal Theory

Definition 1.4.1 For a graded vector space V D L1dDk Vd with k an integer and Vdfinite dimensional for all d we define the Hilbert series of V as the formal Laurentseries

H.V; t/ WD1X

dDkdim.Vd/td:

In the literature, Hilbert series are sometimes called Poincaré series. In ourapplications, V will always be a graded algebra or a graded module.

Example 1.4.2 Let us compute the Hilbert series of KŒx1; : : : ; xn�. There are�nCd�1

n�1�

monomials of degree d, therefore the Hilbert series is

H.KŒx1; : : : ; xn�; t/ D1X

dD0

n C d � 1

n � 1

!td:

This is exactly the power series expansion of .1 � t/�n. GRemark 1.4.3 If V and W are two graded vector spaces, then the tensor productV ˝ W also has a natural grading, namely

.V ˝ W/d DM

d1Cd2DdVd1 ˝ Wd2 :

It is obvious from this formula that H.V ˝ W; t/ D H.V; t/H.W; t/. Suppose thatR D KŒx1; : : : ; xn� and xi has degree di > 0. Then we have R D KŒx1� ˝ KŒx2� ˝� � � ˝ KŒxn� as graded algebras and H.KŒxi�; t/ D .1 � tdi /�1. It follows that

H.R; t/ D 1.1 � td1 / � � � .1 � tdn/ (1.4.1)

GRemark 1.4.4 If

0 ! V.1/ ! V.2/ ! � � � ! V.r/ ! 0 (1.4.2)

is an exact sequence of graded vector spaces (all maps respect degree) with V.i/dfinite dimensional for all i and d, then

rX

iD1.�1/iH.V.i/; t/ D 0:

This is clear because the degree d part of (1.4.2) is exact for all d. G

1.4 Hilbert Series 19

Proposition 1.4.5 (Hilbert) If R D L1dD0 Rd is a finitely generated graded algebraover a field K D R0, then H.R; t/ is the power series of a rational function. Theradius of convergence of this power series is at least 1. Moreover, if M D L1dDk Mdis a finitely generated graded R-module, then H.M; t/ is the Laurent series of arational function (which may have a pole at 0).

Proof Let A D KŒx1; x2; : : : ; xn� be the polynomial ring, graded in such a way thatdeg.xi/ D di > 0. Then H.A; t/ is a rational function by (1.4.1), and the radius ofconvergence of the power series is 1 if n > 0, and 1 if n D 0. For any integer e,we define the A-module A.e/ by A.e/ D L1dD�e A.e/d with A.e/d WD AeCd. It isclear that H.A.e/; t/ D t�eH.A; t/ is again a rational function. A module is free ifit is isomorphic to a direct sum

Li A.ei/. The Hilbert series of a finitely generated

free module M is a rational function. If M is a finitely generated A-module, thenby Hilbert’s syzygy theorem (see Eisenbud [10, Theorem 1.13]), there exists aresolution

0 ! F.r/ ! F.r�1/ ! � � � ! F.1/ ! F.0/ ! M ! 0; (1.4.3)

where F.i/ is a finitely generated free A-module for all i, and the sequence is exact.It follows from Remark 1.4.4 that

H.M; t/ DrX

iD0.�1/iH.F.i/; t/; (1.4.4)

so H.M; t/ is a rational function. If M is nonnegatively graded, then the same is truefor all Fi, so the radius of convergence of H.M; t/ is at least 1.

Let R be an arbitrary finitely generated graded algebra over K D R0. Then forsome n and some d1; : : : ; dn > 0, there exists a homogeneous ideal I � A such thatA=I Š R. Hence R is a finitely generated, nonnegatively graded A-module, and theclaim follows. Moreover, any finitely generated graded R-module M is also a finitelygenerated graded A-module. ut

The above proof gives an easy way to compute the Hilbert series of a gradedmodule M over a graded polynomial ring R D KŒx1; : : : ; xn�, if we have a gradedfree resolution (1.4.3) of M. Indeed, we only have to combine (1.4.4) and (1.4.1).A graded free resolution can be calculated by Algorithm 1.3.4, which involves thecomputation of a Gröbner basis of M. Given a Gröbner basis of M, there is also amore direct way to find the Hilbert series, which will be discussed in Sect. 1.4.1.

The Hilbert series encodes geometric information as the following lemma shows.

Lemma 1.4.6 Let R D Ld�0 Rd be a graded algebra, finitely generated over thefield R0 D K. Then r WD dim.R/ is equal to the pole order of H.R; t/ at t D 1.Proof The proof requires the concept of homogeneous systems of parameters.For the definition and the proof of existence, we refer forward to Sect. 2.5.2. Letf1; : : : ; fr be a homogeneous system of parameters for R, and set A WD KŒ f1; : : : ; fr�.