Neighborhood Complexes and Generating Functions for Affine Semigroups

Herbert Scarf and Kevin M. Woods ()
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Kevin M. Woods: University of California

Chapter 12 in Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, 2008, pp 207-225 from Palgrave Macmillan

Abstract: Abstract 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.

Keywords: Large Firm; Simplicial Complex; Hilbert Series; Neighborhood Complex; Frobenius Number (search for similar items in EconPapers)
Date: 2008
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Working Paper: Neighborhood Complexes and Generating Functions for Affine Semigroups (2004)
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Persistent link: https://EconPapers.repec.org/RePEc:pal:palchp:978-1-137-02441-1_12

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DOI: 10.1057/9781137024411_12

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