Neighborhood Complexes and Generating Functions for Affine Semigroups By

Given a1, a2, . . . , an ∈ Zd , we examine the set, G, of all non-negative integer combinations of these ai . In particular, we examine the generating function f (z) = ∑ b∈G z b. 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 Zn . In the generic case, this follows from algebraic results of Bayer and Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice.