On Counting Integral Points in a Convex Rational Polytope

Given a convex rational polytope b = x ∈ n+ Ax= b , we consider the function b → f b , which counts the nonnegative integral points of b . A closed form expression of its -transform z → z is easily obtained so that f b can be computed as the inverse -transform of . 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 z and whose main computational effort is to solve a linear system. This latter approach is particularly attractive for relatively small values of m, where m is the number of nontrivial constraints (or rows of A).