The minimum period of the Ehrhart quasi-polynomial of a rational polytope

If P subset of R-d is a rational polytope, then i p(n) := #(nP boolean AND Z(d)) is a quasi-polynomial in n, called the Ehrhart quasi-polynomial of P. The minimum period of i p(n) must divide D(P) = min{n is an element of Z(> 0): nP is an integral polytope}. Few examples are known where the minimum period is not exactly D(P). We show that for any D, there is a 2-dimensional triangle P such that D(P) = D but such that the minimum period of i p(n) is 1, that is, i p(n) is a polynomial in n. We also characterize all polygons P such that i p (n) is a polynomial. In addition, we provide a counterexample to a conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart quasi-polynomial. (c) 2004 Elsevier Inc. All rights reserved.