Ehrhart quasi-polynomials and parallel translations

TL;DR

This paper develops a method to compute Ehrhart quasi-polynomials for translated rational polytopes using toric arrangements, revealing how these polynomials vary and characterizing polytopes based on their translation-invariant properties.

Contribution

It introduces a novel approach to analyze Ehrhart quasi-polynomials of translated polytopes via toric arrangements and characterizes polytopes with symmetric Ehrhart quasi-polynomials under translation.

Findings

Method to compute Ehrhart quasi-polynomials for all translations

Visualization of polynomial changes in the torus

Characterization of polytopes with translation-invariant Ehrhart symmetry

Abstract

Given a rational polytope $P \subset \mathbb R^d$, the numerical function counting lattice points in the integral dilations of $P$ is known to become a quasi-polynomial, called the Ehrhart quasi-polynomial $\mathrm{ehr}_P$ of $P$. In this paper we study the following problem: Given a rational $d$-polytope $P \subset \mathbb R^d$, is there a nice way to know Ehrhart quasi-polynomials of translated polytopes $P+ \mathbf v$ for all $\mathbf v \in \mathbb Q^d$? We provide a way to compute such Ehrhart quasi-polynomials using a certain toric arrangement and lattice point counting functions of translated cones of $P$. This method allows us to visualize how constituent polynomials of $\mathrm{ehr}_{P+\mathbf v}$ change in the torus $\mathbb R^d/\mathbb Z^d$. We also prove that information of $\mathrm{ehr}_{P+\mathbf v}$ for all $\mathbf v \in \mathbb Q^d$ determines the rational $d$-polytope $P \subset \mathbb R^d$ up to translations by integer vectors, and characterize all rational $d$-polytopes $P \subset \mathbb R^d$ such that $\mathrm{ehr}_{P+\mathbf v}$ is symmetric for all $\mathbf v \in \mathbb Q^d$.