Motivation

Theorem. (Steingrímsson 2001)

Question

Do other counting polynomials in graph theory have

the same property?

The chromatic polynomial of a graph

is the Hilbert function of a relative Stanley‐Reisner ideal.

fix an orientation and a spanning tree

a flow is determinedby its values on

the non‐tree edgesview ‐flow as a point

relative polytopal complex

A pulling triangulation of

cube diagonals facet diagonals

trianglesin

in

‐minimaltetrahedra

Theorem

the modular and integral flow polynomials and

the modular and integral tension polynomials of

are Hilbert functions of relative Stanley‐Reisner ideals.

Let be a graph. Then

Relative Polytopal Complexes

For any set the Ehrhart function

A ‐dimensional polytope is integral if all vertices of have integer coordinates.

is called compressed if every pulling triangulation of is unimodular.

A relative polytopal complex is a pair of polytopal complexes.

is given by

.

is the set of points contained in but not in , that is,

.

Bounds on the Coefficients

Theorem.

A polynomial is the Hilbert function

if and only if

for all

of some relative Stanley‐Reisner ideal

Better Bounds on the Coefficients

[See separate article arXiv:1004.3470.]...exploiting the geometry of inside‐out polytopes.

Let denote the modular flow or tension polynomial of a graph.

Let and define the ‐vector of

the polynomial by

Then 1.

2.3. is an ‐vector.

for

Theorem.

Hilbert equals Ehrhart

If all faces of are compressed lattice polytopes,

then

for any subcomplex

there exists a relative Stanley‐Reisner ideal

such that for all

Let be a polytopal complex.

Ehrhart

polynomial

Hilbert

function

Theorem.

.

start with a graph

a ‐flow on

network matrix

nowhere‐zero ‐flow lies

off hyperplanes inflow on non‐tree edges

flow on tree edges

(in black)

first two determinedby network matrix

determined by cube

network matrix hyperplanes & boundary of the cube

is integral and compressed

#tetrahedra with 4 facets removed

#tetrahedra with 3 facets removed #open tetrahedra

#open triangles

label vertices ofwith

exploded view of

striped triangles introduced by triangulation,not contained in

two viewsof

subdivision of the cube

Fact. If is an integral relative polytopal complex,

then is a polynomial in called the

Ehrhart polynomial of .

Stanley‐Reisner ideal

Stanley‐Reisner ring

Relative Stanley‐Reisner ideal

The Hilbert functioncounts monomials ofdegree in

Example:

Relative Stanley‐Reisner Ideals

The Hilbert function of is

.

Flows and Tensions

modular flow polynomial

integral flow polynomial

#nowhere‐zero

#nowhere‐zero ‐flows

‐flows

such that at every vertex flow is conserved

‐flow

‐flow

modular tension polynomial

integral tension polynomial

#nowhere‐zero

#nowhere‐zero

‐tensions

‐tensions

such that along every cycle tension is conserved

‐tension

‐tension

Example: ‐flows

Counting Polynomials as Hilbert FunctionsFelix BreuerFreie Universität Berlin

Aaron Dall