Rational Ehrhart quasi-polynomials

TL;DR

This paper extends Ehrhart's theorem to rational dilation factors, showing that the count of integral points in rational polytopes forms a rational quasi-polynomial with coefficients that are piecewise polynomial functions related by derivation.

Contribution

It introduces the concept of rational Ehrhart quasi-polynomials and characterizes their coefficients as piecewise polynomial functions connected through derivation.

Findings

Number of integral points in rational polytopes is a rational quasi-polynomial.

Coefficients are piecewise polynomial functions.

Coefficients are related by derivation.

Abstract

Ehrhart's famous theorem states that the number of integral points in a rational polytope is a quasi-polynomial in the integral dilation factor. We study the case of rational dilation factors and it turns out that the number of integral points can still be written as a rational quasi-polynomial. Furthermore the coefficients of this rational quasi-polynomial are piecewise polynomial functions and related to each other by derivation.