Ehrhart quasi-polynomials of almost integral polytopes

TL;DR

This paper explores the Ehrhart quasi-polynomials of almost integral polytopes, revealing how their geometric properties relate to algebraic features, and characterizing certain polytopes through these polynomials.

Contribution

It establishes a connection between the shape of almost integral polytopes and the algebraic properties of their Ehrhart quasi-polynomials, including characterizations of lattice zonotopes and centrally symmetric polytopes.

Findings

Lattice zonotopes are characterized by their Ehrhart quasi-polynomials.

Centrally symmetric lattice polytopes are characterized similarly.

The shape of polytopes influences the algebraic properties of their Ehrhart quasi-polynomials.

Abstract

A lattice polytope translated by a rational vector is called an almost integral polytope. In this paper we investigate Ehrhart quasi-polynomials of almost integral polytopes. We study the relationship between the shape of the polytopes and algebraic properties of the Ehrhart quasi-polynomials. In particular, we prove that lattice zonotopes and centrally symmetric lattice polytopes are characterized by Ehrhart quasi-polynomials of their rational translations.