Computing the Period of an Ehrhart Quasi-Polynomial

If P ⊂ Rd is a rational polytope, then iP (t) := #(tP ∩Zd) is a quasi-polynomial in t, called the Ehrhart quasi-polynomial of P . A period of iP (t) is D(P ), the smallest D ∈ Z+ such that D ·P has integral vertices. Often, D(P ) is the minimum period of iP (t), but, in several interesting examples, the minimum period is smaller. We prove that, for fixed d, there is a polynomial time algorithm which, given a rational polytope P ⊂ Rd and an integer n, decides whether n is a period of iP (t). In particular, there is a polynomial time algorithm to decide whether iP (t) is a polynomial. We conjecture that, for fixed d, there is a polynomial time algorithm to compute the minimum period of iP (t). The tools we use are rational generating functions.