Ehrhart quasi-polynomials of rational polytopes by real dilations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910209219231744 |
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| author | Cao, Ying Chen, Beifang |
| author_facet | Cao, Ying Chen, Beifang |
| contents | This paper is to study the Ehrhart function $L(P,t)$ of a rational $n$-polytope $P$, defined as the number of lattice points of dilated polytopes $tP$ with real numbers $t\geq 0$. It turns out that $L(P,t)$ is a quasi-polynomial of real variable $t$ in the sense that \[ L(P,t)=\sum_{k=0}^{n} c_k(P,t)t^k, \quad t\geq 0, \] where $c_k(P,t)$ are periodic piecewise polynomials of degree $n-k$ if ${\rm aff}\,P$ contains the origin, and are periodic functions vanishing almost everywhere otherwise. When $P$ is a rational simplex $σ$, the coefficient functions $c_k(σ,t)$ are given explicitly in terms of vertex information of the simplex $σ$. Moreover, the reciprocity law still holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10514 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ehrhart quasi-polynomials of rational polytopes by real dilations Cao, Ying Chen, Beifang Combinatorics This paper is to study the Ehrhart function $L(P,t)$ of a rational $n$-polytope $P$, defined as the number of lattice points of dilated polytopes $tP$ with real numbers $t\geq 0$. It turns out that $L(P,t)$ is a quasi-polynomial of real variable $t$ in the sense that \[ L(P,t)=\sum_{k=0}^{n} c_k(P,t)t^k, \quad t\geq 0, \] where $c_k(P,t)$ are periodic piecewise polynomials of degree $n-k$ if ${\rm aff}\,P$ contains the origin, and are periodic functions vanishing almost everywhere otherwise. When $P$ is a rational simplex $σ$, the coefficient functions $c_k(σ,t)$ are given explicitly in terms of vertex information of the simplex $σ$. Moreover, the reciprocity law still holds. |
| title | Ehrhart quasi-polynomials of rational polytopes by real dilations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.10514 |