English

Ehrhart quasi-polynomials of rational polytopes by real dilations

Combinatorics 2026-05-12 v1

Abstract

This paper is to study the Ehrhart function L(P,t)L(P,t) of a rational nn-polytope PP, defined as the number of lattice points of dilated polytopes tPtP with real numbers t0t\geq 0. It turns out that L(P,t)L(P,t) is a quasi-polynomial of real variable tt in the sense that L(P,t)=k=0nck(P,t)tk,t0, L(P,t)=\sum_{k=0}^{n} c_k(P,t)t^k, \quad t\geq 0, where ck(P,t)c_k(P,t) are periodic piecewise polynomials of degree nkn-k if affP{\rm aff}\,P contains the origin, and are periodic functions vanishing almost everywhere otherwise. When PP is a rational simplex σ\sigma, the coefficient functions ck(σ,t)c_k(\sigma,t) are given explicitly in terms of vertex information of the simplex σ\sigma. Moreover, the reciprocity law still holds.

Keywords

Cite

@article{arxiv.2605.10514,
  title  = {Ehrhart quasi-polynomials of rational polytopes by real dilations},
  author = {Ying Cao and Beifang Chen},
  journal= {arXiv preprint arXiv:2605.10514},
  year   = {2026}
}