Gröbner Bases Tutorial I.pdf

8/10/2019 Grbner Bases Tutorial I.pdf

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GrbnerBases Tutorial

David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove

Extension andClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Grbner Bases TutorialPart I: Grbner Bases and the Geometry of Elimination

David A. Cox

Department of Mathematics and Computer Science

Amherst College

ISSAC 2007 Tutorial
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2/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Outline

1 Grbner Basics

Notation and Definitions

Grbner BasesThe Consistency and Finiteness Theorems

2 Elimination Theory

The Elimination TheoremThe Extension and Closure Theorems

3 Prove Extension and Closure Theorems

The Extension Theorem

The Closure Theorem

An Example

Constructible Sets

4 References
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3/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Begin Grbner Basics

kfield(often algebraically closed)

x =x11 xnn monomialin x1, . . . ,xncx,c kterminx1, . . . ,xnk[x] =k[x1, . . . ,xn]polynomial ringinnvariables

An =An(k)n-dimensionalaffine spaceoverk

V(I) =V(f1, . . . , fs) AnvarietyofI= f1, . . . , fsI(V)

k[x]idealof the varietyV

An

I= {f k[x] | m fm I} theradicalofI

Recall thatI is aradical idealifI=

I.
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4/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Monomial Orders

Definition

Amonomial orderis a total order>on the set of monomialsx satisfying:

x>x impliesxx>xx for allx

x>1 for allx =1We often think of a monomial order as a total order on the

set of exponent vectors Nn.Lemma

A monomial order is a well-ordering on the set of all

monomials.


5/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Examples of Monomial Orders

Examples

Lex orderwithx1> >xn: x

>lexx

iff

1>1, or1=1 and2> 2, or . . .

Weighted orderusingwRn

+ and lex to break ties:x>w,lexx iff

w>w, orw=w andx>lexx

Graded lex orderwithx1> >xn: This is>w,lex forw= (1, . . . ,1)

Weighted orders will appear in Part II when we discuss theGrbner walk.


6/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Leading Terms

Definition

Fix a monomial order>and letf k[x]be nonzero. Writef=cx+ terms with exponent vectors=

such thatc=0 andx

>x

wherever=andx

appears in a nonzero term off. Then:

LT(f) =cx is theleading termof f

LM

(f

) =x is theleading monomialof f

LC(f) =cis theleading coefficientof f

The leading term LT(f)is sometimes called theinitial term,

denoted in(f).


7/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

The Division Algorithm

Division Algorithm

Given nonzero polynomialsf, f1, . . . , fs

k[x]and a

monomial order>, there existr,q1, . . . ,qs k[x]with thefollowing properties:

f= q1 f1+ + qsfs+ r.

No term ofris divisible by any of LT(f1), . . . ,LT(fs).LT(f) =max>{LT(qi) LT(fi) | qi=0}.

Definition

Any representation

f=q1 f1+ + qsfssatisfying the third bullet is astandard representationoff.


8/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

The Ideal of Leading Terms

Definition

Given an idealI k[x]and a monomial order>, theideal ofleading termsis the monomial ideal

LT(I)

:=

LT(f)

|f

I.

IfI= f1, . . . , fs, then

LT(f1), . . . ,LT(fs)

LT(I)

,

though equalityneed notoccur.

This is where Grbner bases enter the picture!


9/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Grbner Bases

Fix a monomial order>onk[x].

Definition

Given an idealI k[x]a finite setG I\{0}is aGrbnerbasisforIunder>if

LT(g)

|g

G

=

LT(I)

.

Definition

A Grbner basisGisreducedif for everyg G,LT(g)divides no term of any element ofG\{g}.LC(g) =1.

TheoremEvery ideal has a unique reduced Grbner basis under>.


10/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Criteria to be a Grbner Basis

Given>andg,h k[x] \{0}, we get theS-polynomial

S(g,h) :=

x

LT(g)g x

LT(h)h, x

=lcm(LM(g),LM(h)).

Three Criteria

(SR)G I is a Grbner basis of I every f I hasa standard representation using G.

(Buchberger)G is a Grbner basis ofG forevery g,h G, S(g,h)has a standard representationusing G.

(LCM)G is a Grbner basis ofG for everyg,h

G, S(g,h) =

GA , where A

=0 implies

LT(A )


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David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

The Consistency Theorem

Fix an idealI k[x], wherek is algebraically closed.Nullstellensatz

(Strong)I(V(I)) = I.(Weak)V(I) = /0 1 I I=k[x].

The Consistency TheoremThe following are equivalent:

I=k[x].

1 / I.V(I) = /0.I has a Grbner basis consisting of nonconstant

polynomials.

I has a reduced Grbner basis= {1}.


12/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

The Finiteness Theorem

Fix an idealI k[x], wherekis algebraically closed. Alsofix a monomial order>.

The Finiteness TheoremThe following are equivalent:

V(I) An is finite.

k[x]/I is a finite-dimensional vector space over k.I has a Grbner basis G wherei, G has a elementwhose leading monomial is a power of xi.

Only finitely many monomials are not in

LT(I)

.

When these conditions are satisfied:

# solutions dimkk[x]/I.

Equality holds I is radical.dimkk[x]/I= # solutions counted with multiplicity.


13/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Begin Elimination Theory

Given

k[x,y] =k[x1, . . . ,xs,ys+1, . . . ,yn],

we write monomials asxy.

Definition

A monomial order>onk[x,y]eliminatesxwhenever

x>x xy> xy

for ally

,y

.

Example

Lex withx1>

>xneliminatesx=

{x1, . . . ,xs

} s.


14/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

The Elimination Theorem

Fix an idealI k[x,y].Definition

I k[y]is theelimination idealofI that eliminatesx.

Theorem

Let G be a Grbner basis of I for a monomial order>thateliminatesx. Then G k[y]is a Grbner basis of theelimination ideal I k[y]for the monomial order on k[y]induced by>.

Proof

f I k[y]has standard representationf= gGAgg. If

Ag=0, thenLT

(g) LT

(Agg) LT

(f) k[y], sog G k[y].SR CriterionG k[y]is a Grbner basis ofI k[y].


15/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Partial Solutions

GivenI k[x,y] =k[x1, . . . ,xs,ys+1, . . . ,yn], the eliminationidealI

k[y]will be denoted

Is:=I k[y] k[y].

Definition

Thevariety of partial solutionsis

V(Is) Ans.

Question

How do the partial solutionsV(Is) Ans relate to theoriginal varietyV :=V(I)

An?


16/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Partial Solutions

Given coordinatesx1, . . . ,xs,ys+1, . . . ,yn, let

s:An Ans

denote projection onto the lastn scoordinates.

An idealI k[x,y]gives:V=V(I) An ands(V) Ans.Is=I

k[y]

k[y]andV(Is)

Ans.

Lemma

s(V) V(Is).


17/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Partial Solutions Dont Always Extend

In A3, considerV=V(xy

1,y

z).

Using lex orderwithx>y>z,I = xy 1,y zhas Grbner basis

xy 1,y z. ThusI1= y z, so thepartial solutions are

the liney=z. TheThe partial solution(0,0)doesnotextend.

x

z

y

the plane y= z

the solutions the partial

solutions

the arrows ,indicate theprojection 1


18/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

The Extension Theorem

LetI k[x,y2, . . . ,yn] = k[x,y]with varietyV=V(I) An,and letI1:=I

k[y]be the first elimination ideal. We

assume thatkis algebraically closed.

Theorem

Letb= (a2, . . . ,an)

V(I1)be a partial solution. If the ideal I

contains a polynomial f such that

f=c(y)xN + terms of degree


19/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

Prove


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Zariski Closure

Definition

Given a subsetS An, theZariski closureofSis thesmallest varietyS An containingS.

Lemma

The Zariski closure of S An is S=V(I(S)).

Example

Over C, the set Zn Cn has Zariski closure Zn = Cn.


20/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

ProveExtension andClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

The Closure Theorem

LetV=V(I) An and letkbe algebraically closed.Theorem

V(Is) =s(V).

ThusV(Is)is the smallest variety inAns containings(V).Furthermore, there is an affine variety

W V(Is) Ans

with the following properties:

V(Is)\ W=V(Is).V(Is)\ W s(V).

Thus most partial solutions inV(Is)come from actualsolutions, i.e, the projection ofVfills up most ofV(Is).


21/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Prove Extension and Closure Theorems

Traditional proofs of the Extension and Closure Theorems

use resultants or more abstract methods from algebraic

geometry.

Recently, Peter Schauenberg wrote

A Grbner-based treatment of elimination theory for affinevarieties

(Journal of Symbolic Computation, to appear). This paper

uses Grbner bases to give new proofs of the Extension

and Closure Theorems.

Part I of the tutorial will conclude with these proofs. We

begin with the Extension Theorem.


22/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Prove the Extension Theorem

Fix a partial solutionb= (a2, . . . ,an) V(I1) An1.

Notation for the ProofLetf k[x,y2, . . . ,yn] =k[x,y].

We write

f=c(y)LCx(f)

xM + terms of degree


23/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Lemma

LetGbe a Grbner basis of Iusing>that eliminatesx.

Lemma

There isg Gwith LCx(g) =0.Proof

Letf= g

GAggbe a standard representation of f

Iwith

LCx(f) =0. Thus

LT(f) = max{LT(Agg) | Ag= 0}.

Since>eliminatesx, it follows that

degx(f) =max{degx(Agg) | Ag=0}LCx(f) = degx(Agg)=degx(f)

LCx(Ag) LCx(g)


24/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Main Claim

By the lemma, we can pickg Gwith LCx(g) = 0 andM:=degx(g)minimal. Note thatM=degx(g)> 0.

Main Claim

{f| f I} = g k[x].

Consequence:If g(a) =0 for somea k, then

f(a,b) =f(a) =0

for allf I, so(a,b) = (a,a2, . . . ,an) V=V(I).This proves the Extension Theorem!

Strategy to prove Main Claim: Showh

g

for allh

G.


25/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Claim

Considerh Gwith degx(h)


26/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Prove the Claim

LCx(g) xMmh=S= GA

max{degx(A) + degx()} = degx(S)


27/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Finish the Claim

The inequality degx(h) degx(h)2 applies toallh Gwith degx(h)


28/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Prove the Main Claim

Proof. Forh G, we showh gby induction on degx(h).Base Case: degx(h)


29/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

The Closure Theorem

We next give Schauenbergs proof of the Closure Theorem.

Fix a partial solutionb= (as+1, . . . ,an)

V(Is)

Ans and a

Grbner basisGofI for>that eliminatesx= (x1, . . . ,xs).

Notation for the Proof

Letf

k[x1, . . . ,xs,ys+1, . . . ,yn] =k[x,y].

We write

f= c(y)

LCs(f)x(f) + terms


30/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems


The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

A Special Case

Special Case

Ifb V(Is)satisfiesLCs(g) =0 for allg G\ k[y],

thenb s(V).Proof. If we can finda= (a1, . . . ,as)such that g(a) =0 forallg

G, theng(a,b) =0 for allg

G. This implies

(a,b) V,

andb

s(V)follows.


31/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Prove the Special Case

LetG= { | G\ k[y]}. Takeg,h G\ k[y]and set

S:= LCs(g) x

x(h)h LCs(h) x

x(g)g

wherex =lcm(x(g),x(h)). A standard representation

S= GA gives

LCs(g) x

x(h)h LCs(h) x

x(g)g=S= GA.

Then LCs(g) =0 and LCs(h) =0 imply:S= GAis a lcm representation.

Sis theS-polynomial ofg, hup to a constant.


32/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

EliminationTheory

The Elimination

Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Finish the Special Case

S= GAis a lcm representation.

Sis theS-polynomial ofg, hup to a nonzero constant.These tell us that for every

g, h

G={|

G\

k[y]},

theS-polynomial ofg, hhas a lcm representation withrespect toG.LCM Criterion Gis a Grbner basis ofG.

Since LTx(g) = 0 forg G\ k[y], gis nonconstant for everyg G. Consistency Theorem V(G) = /0.

Hence there exitsasuch that g(a

) =0 for allg

G.


33/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

Elimination

TheoryThe Elimination

Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Saturation

Fix idealsI,J k[x].Definition

ThesaturationofIwith respect toJ is

I: J := {f k[x] | JMf IforM 0}.

To see what this means geometrically, let

V :=V(I)

W :=V(J).

Then:

Lemma

V\ W=V(I:J).

P f f h Cl Th


34/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

Elimination


Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Proof of the Closure Theorem

Proof. The goal is to find a varietyW V(Is)such thatV(Is)\ W s(V) and V(Is) \ W=V(Is).

LetGbe a reduced Grbner basis of I that eliminatesx. Set

J:=Is+gG\k[y] LCs(g)

.

ThenV(J) =

gG\k[y]V(Is) V(LCs(g)).

Notice that

b V(Is)\ V(J) LCs(g) =0g G\ k[y].BySpecial Case,b s(V). Hence

V(Is) \ V(J) s(V).

T C


35/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

Elimination


Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Two Cases

Case 1.V(Is) \ V(LCs(g)) =V(Is)forallg G\ k[y].

Intersecting finitely many open dense sets is dense, so

V(Is) \ V(J) =gG\k[y] V(Is)\ V(LCs(g)) =V(Is).

Thus the theorem holds withW=V(J).

Case 2.V(Is) \ V(LCs(g)) V(Is)forsomeg G\ k[y].

The strategy will be to enlargeI. First suppose that

V(Is) V(LCs(g)). Then:LCs(g)vanishes onV(Is)and hence onV. HenceNullstellensatz LCs(g)

I.

Greduced LCs(g) / I(since LT(LCs(g))divides LT(g)).

Fi i h th P f


36/40


David A. Cox

GrbnerBasics

Notation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

Elimination


Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Finish the Proof

Together, these bullets imply that

I I+ LCs(g)

I,

so in particular,V(I) =V(I+ LCs(g)). So it suffices to provethe theorem forI+ LCs(g)whenV(Is) V(LCs(g)).

On the other hand, whenV(Is

)

V(LCs

(g)), we have a unionof proper subsets

V(Is) =V(Is)\ V(LCs(g)) (V(Is) V(LCs(g)))

=V(Is: LCs(g)

) V(Is+ LCs(g))Hence it suffices to prove the theorem for

I I+Is: LCs(g)

and I I+

LCs(g)

.

ACCthese enlargements occur only finitely often.

E l


37/40


David A. Cox

Grbner

BasicsNotation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

Elimination


Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Example

Consider A4 with variablesw,x,y,z. Let1: A4 A3 beprojection onto the last three coordinates. Set

f=x

5

+ x

4

+ 2 yzandI= (y z)(fw 1),(yw 1)(fw 1). This definesV:= V(I) = V(y

z,yw

1)

V((x5 + x4 + 2

yz)w

1)

A4.

Then:

{g1,g2} = {(y z)(fw 1),(yw 1)(fw 1)}is aGrbner basis ofIfor lex withw>x>y>z.

I1=0.Furthermore,

g1=

(y

z)(x5 + x4 + 2

yz)

w+

g2= y(x5 + x4 + 2 yz)w2 +

Continue Example


38/40


David A. Cox

Grbner

BasicsNotation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

Elimination


Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Continue Example

As in the proof of the Closure Theorem, let

J=I1+gG

\k[x,y,z] LC1(g)

= (y z)(x5 + x4 + 2 yz) y(x5 + x4 + 2 yz)= y(y z)(x5 + x4 + 2 yz)2.

This satisfies Case 1 of the proof, so that

V(I1) \ V(J) = A3 \ V(J) 1(V).

However,

1(V) =A3 \ V(x5 + x4 + 2 yz)

V(yz,x5 + x4 + 2

yz)

\V(x5 + x4 + 2,y,z).

Constructible Sets


39/40


David A. Cox

Grbner

BasicsNotation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

Elimination


Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

Constructible Sets

Definition

A setS

An isconstructibleif there are affine varieties

Wi ViAn,i=1, . . . ,N, such that

S=N

i=1

Vi\ Wi).

Theorem

Let k be algebraically closed ands: An

Ans be

projection onto the last n s coordinates. If V An is anaffine variety, thens(V) Ans is constructible.

The proof of the Closure Theorem can be adapted to give

analgorithmfor writings(V)as a constructible set.

References


40/40


David A. Cox

Grbner

BasicsNotation and

Definitions

Grbner Bases

The Consistency and

Finiteness Theorems

Elimination


Theorem

The Extension and

Closure Theorems

ProveExtension and

ClosureTheorems

The Extension

Theorem

The Closure

Theorem

An Example

Constructible Sets

References

References

T. Becker, V. Weispfenning,Grbner Bases, GraduateTexts in Mathematics141, Springer, New York, 1993.

D. Cox, J. Little, D. OShea,Ideals, Varieties and

Algorithms, Third Edition, Undergraduate Texts inMathematics, Springer, New York, 2007.

P. Schauenberg,A Grbner-based treatment of

elimination theory for affine varieties, Journal ofSymbolic Computation, to appear.

Documents

Gröbner Bases Tutorial I.pdf