On Counting Integral Points in a Convex Rational Polytope

Given a convex rational polytope ?( b) := { x ? R n+ |Ax = b}, we consider the functionb ?f( b), which counts the nonnegative integral points of ?( b). A closed form expression of its Z-transformz?F( z) is easily obtained so thatf( b) can be computed as the inverse Z-transform of F. We then provide two variants of an inversion algorithm. As a by-product, one of the algorithms provides the Ehrhart polynomial of a convex integer polytope ?. We also provide an alternative that avoids the complex integration of F( z) and whose main computational effort is to solve a linear system. This latter approach is particularly attractive for relatively small values ofm, wherem is the number of nontrivial constraints (or rows ofA).