Ehrhart quasi-polynomials of rational polytopes by real dilations

TL;DR

This paper investigates the Ehrhart function of rational polytopes, revealing it as a real-variable quasi-polynomial with explicit coefficient formulas for simplices and extending reciprocity laws.

Contribution

It introduces a novel analysis of Ehrhart functions with real dilations, providing explicit formulas for coefficients and extending reciprocity to real parameters.

Findings

Ehrhart function is a quasi-polynomial in real variable t.

Coefficient functions are explicitly given for rational simplices.

Reciprocity law extends to real dilations.

Abstract

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 $\sigma$, the coefficient functions $c_k(\sigma,t)$ are given explicitly in terms of vertex information of the simplex $\sigma$. Moreover, the reciprocity law still holds.