Neighborhood Complexes and Generating Functions for Affine Semigroups

Given a1, a2,…, a n ∈ ℤ d , we examine the set, G, of all non-negative integer combinations of these a i . In particular, we examine the generating function f(z)∑b∈Gz 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 ℤ n . 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.