A Closer Look at Chapoton's q-Ehrhart Polynomials

TL;DR

This paper explores Chapoton's q-Ehrhart polynomials for lattice polytopes, providing new formulas, analyzing their properties, and extending results to rational polytopes using Brion's Theorem.

Contribution

It offers explicit formulas, asymptotic behavior, and reciprocity results for Chapoton's polynomials, extending their applicability to rational polytopes.

Findings

Derived explicit formulas for Chapoton's q-Ehrhart polynomials

Analyzed the leading coefficient and asymptotic behavior as t approaches infinity

Extended structural and reciprocity theorems to rational polytopes

Abstract

If $\mathcal{P}$ is a lattice polytope (i.e., $\mathcal{P}$ is the convex hull of finitely many integer points in $\mathbb{R}^d$), Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|t \mathcal{P} \cap \mathbb{Z}^d|$ is a polynomial in the integer variable $t$. Chapoton (2016) proved that, given a fixed integral form $\lambda: \mathbb{Z}^d \to \mathbb{Z}$, there exists a polynomial $\text{cha}_\mathcal{P}^\lambda(q,x) \in \mathbb{Q}(q)[x]$ such that the refined enumeration function $\sum_{ \mathbf{m} \in t \mathcal{P} } q^{ \lambda(\mathbf{m}) }$ equals the evaluation $\text{cha}_\mathcal{P}^\lambda (q, [t]_q)$ where, as usual, $[t]_q := \frac{ q^t - 1 }{ q-1 }$; naturally, for $q=1$ we recover the Ehrhart polynomial. Our motivating goal is to view Chapoton's work through the lens of Brion's Theorem (1988), which expresses the integer-point structure of a given polytope via that of its vertex cones. It turns out that this viewpoint naturally yields various refinements and extensions of Chapoton's results, including explicit formulas for $\text{cha}_\mathcal{P}^\lambda(q,x)$, its leading coefficient, and its behavior as $t \to \infty$. We also prove an analogue of Chapoton's structural and reciprocity theorems for rational polytopes (i.e., with vertices in $\mathbb{Q}^d$).