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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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    References listed on IDEAS

    as
    1. I. Bárány & H. E. Scarf & D. Shallcross, 2008. "The topological structure of maximal lattice free convex bodies: The general case," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 11, pages 191-205, Palgrave Macmillan.
    2. Herbert E. Scarf & David F. Shallcross, 2008. "The Frobenius Problem and Maximal Lattice Free Bodies," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 7, pages 149-153, Palgrave Macmillan.
    3. Herbert E. Scarf, 2008. "Production Sets with Indivisibilities Part II. The Case of Two Activities," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 3, pages 39-67, Palgrave Macmillan.
    4. David Shallcross, 1992. "Neighbors of the Origin for Four by Three Matrices," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 608-614, August.
    5. Herbert E. Scarf & David F. Shallcross, 2008. "The Frobenius Problem and Maximal Lattice Free Bodies," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 7, pages 149-153, Palgrave Macmillan.
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    More about this item

    Keywords

    Large Firm; Simplicial Complex; Hilbert Series; Neighborhood Complex; Frobenius Number;
    All these keywords.

    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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