Neighborhood complexes and generating functions for affine semigroups

Given a_1,a_2,...,a_n in 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)=\sum_{b\in 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 Z^n. 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.

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