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### Transcript of Toric Graph Ideals - boocher/writings/torictalk.pdf · PDF fileToric Graph Ideals Bryan...

• 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

• Introduction

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

v2 v3

v1 v4

A

B

C

D

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

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

Im(G ) = k[v1v2, v2v3, v3v4, v4v1].

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 2 / 21

• Introduction (cont.)

v2 v3

v1 v4

A

B

C

D

G (AC BD) = 0

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

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 3 / 21

• Graph Ideals

Definition

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

vi vjeij

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 4 / 21

• An Even Cycle

A

B

C

D

E

F

H

I

v1

v2

v3 v4

v5

v6

v7v8

G (ACEH BDFI ) = v1v2v3v4v5v6v7v8 v2v3v4v5v6v7v8v1 = 0

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 5 / 21

• An Odd Cycle

A B

C

v1

v2 v3

Lets 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?

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 6 / 21

• 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

e2j1 k

j=1

e2j

Closed even walk:

w : w1e1 w2

e2 ... e2k2 w2k1e2k1 w2k

e2k w1wi V , ei E

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 7 / 21

• 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

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 8 / 21

• An Example: The Complete Graph

A

B

C

D

E F

v1

v2 v3

v4

AC BD(AC EF )

EF BD

| MG |= 2

ACBD,ACEF ,BDEF seems like a more completedescription of the toric ideal.

The set{ACBD,ACEF ,BDEF}forms what is called a GraverBasis for the ideal IG .

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 9 / 21

• Graver Bases

Notation: GG - Graver Basis

Definition

A binomial xv+ xv IG is called primitive if @ another binomial

xu+ 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 .

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 10 / 21

• Fact!

Fact: MG GG

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 11 / 21

• BIG QUESTION:

For what graphs G is MG = GG?

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 12 / 21

• 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)

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 13 / 21

• MG and GG Can Be Vastly Different Sizes

MG = 70, but GG = 3360

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 14 / 21

• Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 15 / 21

• An Example Where MG = GG

A

E F

B

C D

(Minimal) closed even walks are allcycles, corresponding to

p2 = CF DE

p3 = AF BE

but pi / pj , pk

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

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 16 / 21

• A Few More Examples Where MG = GG

|GG | = 3

|GG | = 6

|GG | = 3

|GG | = 3

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 17 / 21

• Contructing Larger Graphs Where MG = GG From SmallerOnes

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 18 / 21

• 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

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 19 / 21

• 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!

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 20 / 21

• 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!

Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 20 / 21

• References

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

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