Neighborhood Complexes and Generating Functions for Affine Semigroups

Given a1, a2, . . . , an ∈ Z , we examine the set, G, of all nonnegative integer combinations of these ai. In particular, we examine the generating function f(z) = ∑ b∈G z . We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the Scarf complex) on a particular lattice in Z. In the generic case, this follows from algebraic results of D. Bayer and B. Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice.