Ehrhart quasi-polynomials and parallel translations

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 $\mathbf{v}\in {\mathbb{Q}}^d$.

28 pages, to appear in Combinatorial Theory