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Neighborhood Complexes and Generating Functions for Affine Semigroups

In: Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research

Author

Listed:
  • Herbert E. Scarf

    (Yale University)

  • Kevin M. Woods

    (University of California)

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.

Suggested Citation

  • Herbert E. Scarf & Kevin M. Woods, 2008. "Neighborhood Complexes and Generating Functions for Affine Semigroups," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 12, pages 207-225, Palgrave Macmillan.
    • Handle: RePEc:pal:palchp:978-1-137-02441-1_12
    DOI: 10.1057/9781137024411_12
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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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