Toric Graph Ideals - Mathboocher/writings/torictalk.pdf · Toric Graph Ideals Bryan Brown, Laura...
Transcript of Toric Graph Ideals - Mathboocher/writings/torictalk.pdf · Toric Graph Ideals Bryan Brown, Laura...
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 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].
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
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
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
φG (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
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?
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
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
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
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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 .
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 10 / 21
Fact!
Fact: MG ⊂ GG
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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)
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MG and GG Can Be Vastly Different Sizes
MG = 70, but GG = 3360
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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
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!
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