symbolic powers of ideals - university of …jvanderwoude2/graph-ideals/dordt...summer seminar...

Summer Seminar

Symbolic Powers of Ideals

Thomas Kamp and Jason Vander Woude

Dordt College

June 28, 2017

Thomas Kamp and Jason Vander Woude Symbolic Powers of Ideals

Summer Seminar

Background of Our Research

I 1992 On the Ideal Theory of GraphsI They show that a graph G is bipartite (contains only even

cycles) if and only if its corresponding edge ideal satisfiesI(n) = In for all n ≥ 1.

I 2004 Symbolic Powers of Edge IdealsI I(n+r) ⊇ In+r + Ir−1 · 〈x1x2 · · ·x2n+1〉 for all r ≥ 1


Summer Seminar

An Introduction to Abstract Algebra and Graph Theory

Graphs:

I A collection of vertices and edges connecting those vertices

Cycles:

x2x3

x1

x1x2

x3

x4

x5x6

x7

x8

x9

3-Cycle 9-Cycle


Summer Seminar

Monomials

Polynomials

x2 + 5x + 6 2x4 + 7x2 + 3x + 1

Multivariable Polynomials

x21x2x33 + 5x1x4 + 6x2 2x41 + 7x21x4x

35x

76 + 3x51 + 1

Monomials

x21x2x33 x21x4x

35x

76


Summer Seminar

Ideals

An ideal is a set with special additive and multiplicativeproperties.

Example:

I The set of all even numbers (all numbers divisible by 2) isan ideal of the integers, and we’ll call this ideal E.

I The sum of any two even numbers is an even number.I Any integer times an even number is an even number.

I E2 is the set of all numbers divisible by 22 = 4.

I E3 is the set of all numbers divisible by 23 = 8.

Ideals can also be generated from graphs.


Summer Seminar

Research Goals

I One of the research goals was to prove a pair of conjecturesstated in the 2004 paper regarding odd cycles of length2n + 1.

I I(t) = It for 1 ≤ t ≤ nI I(n+1) = In+1 + (x1x2 · · ·x2n+1)

I We have been able to prove both of these conjectures,which has opened up the door for further research for therest of the summer.


Summer Seminar

So, what did we actually do?

I Explore background information

I Collect data and explore patterns using Macaulay2

I Formulate and test conjectures until one seems adequateI I(t) = It + IS〈t〉

I Prove the conjectureI When is something considered to be true in mathematics?I Often requires splitting the conjecture into sub-conjectures

and proving them individually or splitting them furtherI Also requires considerable amounts of backtracking

I Write a formal proof in a way that is easy to comprehendI Shockingly difficult


Summer Seminar

Graphing Monomials

We graphically represent the monomials by reducing each pairof consecutive vertices to an edge so that we get as manyedges as we can. Consider the monomial x21x

32x3x4x5.

x1

x2

x3x4

x5

(x1x2)2(x2x3)(x4x5)


Summer Seminar

Graphing Monomials

However, each monomial does not necessarily have to perfectlyreduce, which creates ancillary vertices. Consider themonomial x21x

32x3x4x

25.

x1

x2

x3x4

x5

(x1x2)2(x2x3)(x4x5)x5


Summer Seminar

Graphing Monomials

Another problem we encountered was that each monomial doesnot necessarily correspond to a unique graph.

x21x32x

23x4x5

x1

x2

x3x4

x5

x1

x2

x3x4

x5

(x1x2)2(x2x3)x3(x4x5) (x1x2)(x2x3)

2x4(x5x1)


Summer Seminar

Graphing Monomials

We had to be concerned with an infinite number of possiblegraphs of varying sizes, so we wanted to understand whatgraphs can and cannot look like.

x1x2

x3

x4

x5x6

x7

x8

x9x1

x2

x3

x4

x5x6

x7

x8

x9

Assuming one of each ancillary.


Summer Seminar

Graphing MonomialsBut even after turning those two ancillaries into an edge, it isstill not good enough!

Assuming one of each ancillary.


Summer Seminar

Graphing Monomials


Summer Seminar

Theorem (Kamp-Vander Woude)

Let G be an odd cycle of size 2n + 1, I be its edge ideal, and V ′

denote a minimal vertex cover. Then

I(t) = It + ({xα|deg(xα) < 2t and ∀V ′ ⊆ V (G), wV ′(xα) ≥ t})


symbolic powers of ideals - university of …jvanderwoude2/graph-ideals/dordt...summer seminar...

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