Generalized Ehrhart polynomials

TL;DR

This paper extends Ehrhart's theorem to polytopes with vertices as rational functions of n, showing the lattice point count remains a quasi-polynomial for large n, thus broadening the scope of lattice point enumeration.

Contribution

It generalizes Ehrhart's theorem to polytopes with vertices as rational functions, connecting lattice point counts to solutions of parametrized Diophantine equations.

Findings

Number of lattice points is a quasi-polynomial for large n.

Established a link between Ehrhart's conjecture and Diophantine equations.

Extended classical lattice point enumeration to rational function vertices.

Abstract

Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of P(n) to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in P(n) is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related.