Toric Graph Ideals - Mathboocher/writings/torictalk.pdf · Toric Graph Ideals Bryan Brown, Laura...

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Toric Graph Ideals Bryan Brown, Laura Lyman, Amy Nesky Berkeley RTG July 31st, 2013 Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 1 / 21

Transcript of Toric Graph Ideals - Mathboocher/writings/torictalk.pdf · Toric Graph Ideals Bryan Brown, Laura...

Page 1: Toric Graph Ideals - Mathboocher/writings/torictalk.pdf · Toric Graph Ideals Bryan Brown, Laura Lyman, Amy Nesky ... E F v 1 v 2 v 3 v 4 AC BD ... Amy Nesky (Berkeley RTG) Toric

Toric Graph Ideals

Bryan Brown, Laura Lyman, Amy Nesky

Berkeley RTG

July 31st, 2013

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 1 / 21

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Introduction

k is a field. Let G denote the following graph.

v2 v3

v1 v4

A

B

C

D

Define the homomorphism φG : k[A,B,C ,D]→ k[v1, v2, v3, v4] by

A 7→ v1v2 B 7→ v2v3 C 7→ v3v4 D 7→ v4v1.

Im(φG ) = k[v1v2, v2v3, v3v4, v4v1].

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Introduction (cont.)

v2 v3

v1 v4

A

B

C

D

φG (AC − BD) = 0

In fact, it is easy to show that ker(φG ) = 〈AC − BD〉.We call this ideal, ker(φG ), the Toric Ideal of the graph G .

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Graph Ideals

Definition

Let G = (V ,E ) be an undirected graph. We can define the mapφG : k[E ]→ k[V ] by eij 7→ vivj as shown below. The ideal ker(φG ) iscalled the Toric Ideal associated to G and is denoted IG .

vi vjeij

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An Even Cycle

A

B

C

D

E

F

H

I

v1

v2

v3 v4

v5

v6

v7v8

φG (ACEH − BDFI ) = v1v2v3v4v5v6v7v8 − v2v3v4v5v6v7v8v1 = 0

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An Odd Cycle

A B

C

v1

v2 v3

Let’s try to find a non-zero elementin the toric ideal

AB · · · − C · · ·

For this graph, the toric ideal is...is..hm..trivial? What happened here?

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Things We Want To Be True That Are!

Proposition

IG is a homogeneous ideal generated by the following binomials whichcorrespond to closed even walks of the graph:

Bw =k∏

j=1

e2j−1 −k∏

j=1

e2j

Closed even walk:

w : w1e1−−−−→ w2

e2−−−−→ ...e2k−2−−−−→ w2k−1

e2k−1−−−−→ w2ke2k−−−−→ w1

wi ∈ V , ei ∈ E

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Minimal Generating Set

Definition

If 〈s1, ..., sn〉 = I , then we say it is a minimal generating set if no subsetof it generates I .

Notation:MG - any minimal generating set

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An Example: The Complete Graph

A

B

C

D

E F

v1

v2 v3

v4

AC − BD−(AC − EF )

EF − BD

| MG |= 2

〈AC−BD,AC−EF ,BD−EF 〉seems like a more completedescription of the toric ideal.

The set{AC−BD,AC−EF ,BD−EF}forms what is called a GraverBasis for the ideal IG .

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Graver Bases

Notation: GG - Graver Basis

Definition

A binomial xv+ − xv− ∈ IG is called primitive if @ another binomialxu+ − xu− ∈ IG s.t. xu+ |xv+

and xu− |xv−.

The set of all primitive binomials in IG is called its Graver basis andis denoted GG .

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Fact!

Fact: MG ⊂ GG

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BIG QUESTION:

For what graphs G is MG = GG?

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An Example of When MG 6= GG

D

CA B

H J

F E

MG = {AJF − BEH,BD − CE}GG = {AJF − BEH,BD − CE ,CAJF − DB2H,DAJF − CE 2H}

Note: CAJF − DB2H = BH(CE − BD) + C (AJF − BEH)

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MG and GG Can Be Vastly Different Sizes

MG = 70, but GG = 3360

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An Example Where MG = GG

A

E F

B

C D

(Minimal) closed even walks are allcycles, corresponding to

p1 = AD − BC

p2 = CF − DE

p3 = AF − BE

but pi /∈ 〈pj , pk〉

for {i , j , k} = {1, 2, 3}

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A Few More Examples Where MG = GG

|GG | = 3

|GG | = 6

|GG | = 3

|GG | = 3

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Contructing Larger Graphs Where MG = GG From SmallerOnes

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Universal Grobner Basis

Another kind of basis with some algorithmically nice properities iscalled a Universal Grobner Basis, denoted UG , and this set laysnestled in between the minimal generating set of a toric ideal and theGraver Basis

Proposition

MG ⊂ UG ⊂ GG for any minimal generating set MG

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Some Results

Theorem

MG = UG ⇔MG = GG . If IG sastisfies either of these conditions we callit robust.

Theorem

IG is robust ⇔ (Graph theoretic conditions).

Thank You!

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Some Results

Theorem

MG = UG ⇔MG = GG . If IG sastisfies either of these conditions we callit robust.

Theorem

IG is robust ⇔ (Graph theoretic conditions).

Thank You!

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References

Sturmfels, Bernd. Grobner Bases and Convex Polytopes. AmericanMathematical Society: Providence, RI, 1991.

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