A Closer Look at Chapoton's q-Ehrhart Polynomials
Matthias Beck, Thomas Kunze
TL;DR
This paper revisits Chapoton's $q$-Ehrhart polynomials through Brion's vertex-cone decomposition, proving explicit, vertex-sum formulas for the refined counts $\operatorname{ehr}_\mathcal{P}^\lambda(q,t)$ and the associated Chapoton polynomials $\operatorname{cha}_\mathcal{P}^\lambda(q,x)$ for lattice polytopes with generic, positive $\lambda$; the main representation is $\operatorname{cha}_\mathcal{P}^\lambda(q,x) = \sum_{\mathbf{v}} \rho_\mathbf{v}^\lambda(q) \bigl((q-1)x+1\bigr)^{\lambda(\mathbf{v})}$, where $\rho_\mathbf{v}^\lambda(q)$ encodes the vertex-cone geometry and its poles. It further obtains rational analogues for polytopes with rational vertices, giving families of polynomials $\operatorname{cha}_\mathcal{P}^{\lambda,r}(q,x)$ and reciprocity relations, and applies the framework to concrete families such as the unit cube, standard simplex, unimodular simplex, and lecture hall simplices, yielding explicit closed forms and recursions. The results deepen the understanding of refined lattice-point enumeration and connect Chapoton's polynomials to Euler–Mahonian statistics, Sylvester waves, and integer partitions. The work provides tools for analyzing the asymptotics ($t\to\infty$) and pole structure, as well as rational-polytope refinements, with potential applications in combinatorial geometry and related areas.
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 $λ: \mathbb{Z}^d \to \mathbb{Z}$, there exists a polynomial $\text{cha}_\mathcal{P}^λ(q,x) \in \mathbb{Q}(q)[x]$ such that the refined enumeration function $\sum_{ \mathbf{m} \in t \mathcal{P} } q^{ λ(\mathbf{m}) }$ equals the evaluation $\text{cha}_\mathcal{P}^λ(q, [t]_q)$ 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 $\text{cha}_\mathcal{P}^λ(q,x)$, 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 $\mathbb{Q}^d$).