Ehrhart Series in Discrete and Graph Geometry

• Ehrhart series is a rational generating function that encodes the lattice-point counts in dilated rational polytopes via the Ehrhart h*-polynomial.
• It extends classical lattice-point enumeration to edge polytopes of graphs, integrating methods from discrete geometry, combinatorial optimization, and commutative algebra.
• Factorization approaches and Gröbner basis techniques enable explicit analysis of normality, factorization properties, and polynomial root distributions in Ehrhart series.

The Ehrhart series is a generating function that encodes the lattice-point enumeration in the positive integer dilations of a rational polytope. It constitutes a central object in discrete and convex geometry, commutative algebra, and combinatorial optimization, providing a unified framework for the paper of counting lattice points, Hilbert series, and the interplay between polyhedral geometry and algebraic structures such as semigroup rings and edge rings. The classical theory, developed by Ehrhart and subsequently refined by Stanley and others, has been generalized in various directions, most notably to combinatorially defined polytopes arising from graphs, as well as to polytopes with additional structures or symmetries.

1. Fundamentals of the Ehrhart Series

Given an integral polytope $P \subset \mathbb{R}^n$ (i.e., a convex hull of finitely many integer points), the Ehrhart function $L_P(m)$ enumerates the number of lattice points contained in the $m$-th dilate $mP$. Ehrhart's theorem asserts that $L_P(m)$ is a polynomial in $m$ of degree $\dim(P)$ for $m \in \mathbb{Z}_{\ge 0}$, known as the Ehrhart polynomial.

The Ehrhart series, or generating function, is defined as:

$E_P(t) = \sum_{m=0}^\infty L_P(m) t^m\,.$

It is rational and can always be expressed as:

$E_P(t) = \frac{h^*_P(t)}{(1-t)^{d+1}}$

where $d = \dim(P)$ and $h^*_P(t)$ is the Ehrhart $h^*$-polynomial (also called the $\delta$-polynomial), whose coefficients encode deep arithmetic and geometric information. Fundamental results, such as Stanley's nonnegativity and monotonicity theorems, establish that the coefficients of $h^*_P(t)$ are nonnegative integers and are monotone under inclusion for integral polytopes.

For rational polytopes, $L_P(m)$ is a quasipolynomial and the denominator in $E_P(t)$ reflects the denominator of $P$.

2. Ehrhart Series of Edge Polytopes of Graphs

A major extension of the classical Ehrhart theory concerns edge polytopes associated to finite graphs, leading to new algebraic and combinatorial phenomena.

Given a connected simple graph $G=(V,E)$, the edge polytope $\mathcal{P}_G \subset \mathbb{R}^{|V|}$ is defined as the convex hull of the incidence vectors of the edges:

$$\mathcal{P}_G = \operatorname{conv}\{ \epsilon(i) + \epsilon(j) : \{i,j\} \in E \}$$

where $\epsilon(i)$ is the $i$th unit vector.

The counting function $i_G(m) = | m \mathcal{P}_G \cap \mathbb{Z}^n |$ leads to the graph-Ehrhart series:

$$H_G(t) = \sum_{m=0}^\infty i_G(m) t^m = \frac{i_G^*(t)}{(1-t)^{D+1}}$$

where $D = \dim \mathcal{P}_G$ and $i_G^*(t)$ is the numerator polynomial.

The interpretation of $H_G(t)$ in algebraic terms leverages the edge ring $K[G]$ associated to $G$, a semigroup algebra generated by monomials representing the edges. The Hilbert series of $K[G]$ coincides with $H_G(t)$ if and only if $K[G]$ is normal, which is characterized combinatorially by the "odd cycle condition": every pair of odd cycles in $G$ either shares a vertex or is connected by a single edge. If this condition fails, the Ehrhart ring is strictly larger and requires the introduction of additional indeterminates reflecting exceptional pairs of odd cycles (Matsui, 2011).

3. Combinatorial Generators and Non-Normal Graphs

For graphs failing the odd cycle condition, the defining ideal of the Ehrhart ring of $\mathcal{P}_G$ acquires additional structure informed by the combinatorial configuration of $G$. The minimal generators are constructed from:

• Edge walks/trivial binomials: Derived from even closed walks, generating the toric ideal associated to the edge ring.
• "Exceptional cycle" binomials: For each exceptional pair of odd cycles $(C_i, C_j)$, a new variable $\theta_{ij}$ is introduced, and binomials of the form $\theta_{ij}^2 - C_iC_j$ are enforced, where $C_i, C_j$ denote monomials corresponding to their respective cycles.
• Path-connected cycle binomials: These have the form $$\theta_{ij} N_{ij}^{(p)\pm} - C_i^{\mp} C_j^{\mp} N_{ij}^{(p)\mp}$$, with $N_{ij}^{(p)\pm}$ being minimal alternating products of edges along paths connecting $C_i$ and $C_j$.
• Relations among the $\theta$-variables: More complex binomials capturing cycle interactions beyond pairs.

The structure of this ideal, described in Theorem 2.8 in (Matsui, 2011), allows for explicit Gröbner basis computations and reflects the precise way in which the cyclomatic and path structure of $G$ influences the Ehrhart series.

4. Factoring Properties of the Graph Ehrhart Series

The analysis in (Matsui, 2011) yields two principal factorization properties useful both theoretically and computationally:

(a) Biconnected and Oddment Factorization:

When $G$ is decomposed into biconnected components, especially distinguishing those supporting odd cycles ("oddments"), the Ehrhart series factors as:

$H_G(t) = H_{G_0}(t) \prod_{i=1}^{r'} H_{B_i}(t)$

where $G_0$ is the core containing all oddments and $B_i$ range over the remaining biconnected components.

(b) Single-edge Gluing with a Bipartite Component:

If $G$ is obtained by identifying an edge $e$ shared between subgraphs $G^{(1)}$ and a bipartite $G^{(2)}$ (with $e$ in a cycle of $G^{(2)}$), then:

$$H_G(t) = \frac{H_{G^{(1)}}(t) \cdot H_{G^{(2)}}(t)}{1-t}$$

This correction by $1-t$ removes the overcount due to the shared edge.

These structural decompositions parallel the behavior of the lcm-lattice associated with the ideal generators and facilitate both inductive proofs and practical computations.

5. Applications: Root Location for Ehrhart Polynomials of Polygon Trees

An application of the above factorization properties is the precise control of the roots of Ehrhart polynomials for classes of graphs such as bipartite polygon trees—graphs formed by gluing even cycles along edges iteratively.

For a bipartite polygon tree $G$ with $e$ edges and $f_{2n}$ cycles of length $2n$, the Ehrhart series assumes the form:

$$H_G(t) = \frac{\prod\, (1 + t + \ldots + t^{n-1})^{f_{2n}}}{(1-t)^{e-f}}$$

This explicit rational form provides immediate access to the numerator polynomial, and by applying a result of Rodriguez–Villegas, it is shown that all roots of the Ehrhart polynomial are either negative integers or lie on the vertical line

$\Re(x) = -\frac{e - \sum n f_{2n}}{2}$

Establishing such explicit root location aligns with conjectures about the distribution of Ehrhart polynomial roots and connects to ongoing work on the algebraic and geometric structure of their coefficients (Matsui, 2011).

6. Broader Implications and Combinatorial–Algebraic Synthesis

The theoretical framework developed for the Ehrhart series in the context of edge polytopes advances the understanding of the intricate relationship between graph combinatorics, commutative algebra, and discrete geometry. The explicit construction of the Ehrhart ring and its generators provides tools for studying normality, factorization, and root location phenomena. The methodology generalizes to other combinatorially defined polytopes, offering templates for analyzing non-normal rings and their Hilbert/Ehrhart series.

Furthermore, the factorization patterns echo broader themes in polyhedral geometry, such as decomposition along irreducible or structural building blocks, and their algebraic shadows in the structure of semigroup rings and their defining relations. The precise interplay between cycles, paths, and algebraic generators elucidates the influence of underlying combinatorial structures on enumerative, geometric, and algebraic properties.

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References:

• "Ehrhart series for Connected Simple Graphs" (Matsui, 2011)