Ehrhart quasi-polynomials and parallel translations

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 \(\operatorname{ehr}_P\) 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 \(P+ \boldsymbol v\) for all \(\boldsymbol v \in \mathbb Q^d\)? 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 \(\operatorname{ehr}_{P+\boldsymbol v}\) change in the torus \(\mathbb R^d/\mathbb Z^d\). We also prove that information of \(\operatorname{ehr}_{P+\boldsymbol v}\) for all \(\boldsymbol v \in \mathbb Q^d\) 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 \(\operatorname{ehr}_{P+\boldsymbol v}\) is symmetric for all \(\boldsymbol v \in \mathbb Q^d\).Mathematics Subject Classifications: 52C07, 52C35Keywords: Ehrhart quasi-polynomials, rational polytopes, toric arrangements, conic divisorial ideals