A Closer Look at Chapoton's q-Ehrhart Polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914526686871552 |
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| author | Beck, Matthias Kunze, Thomas |
| author_facet | Beck, Matthias Kunze, Thomas |
| contents | 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 $\mathbb{Q}^d$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22900 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Closer Look at Chapoton's q-Ehrhart Polynomials Beck, Matthias Kunze, Thomas Combinatorics 52B20 (Primary) 05A15 (Secondary) 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 $\mathbb{Q}^d$). |
| title | A Closer Look at Chapoton's q-Ehrhart Polynomials |
| topic | Combinatorics 52B20 (Primary) 05A15 (Secondary) |
| url | https://arxiv.org/abs/2505.22900 |