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Discussion Paper

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Given a 1 , a 2 ,…, a n in Z d , we examine the set, G , of all nonnegative 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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