Neighborhood Complexes and Generating Functions for Affine Semigroups

Given a1,a2,...,an E Z^d$, we examine the set, G, of all non-negative integer combinations of these ai. In particular, we examine the generating function f(z) = [Sigma]b E Gzb. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes calle...

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Bibliographic Details
Published in:Discrete & computational geometry 2006-03, Vol.35 (3), p.385-403
Main Authors: Scarf, Herbert E., Woods, Kevin M.
Format: Article
Language:English
Subjects:
Mathematical functions
Mathematical models
Online Access:Get full text
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Summary:Given a1,a2,...,an E Z^d$, we examine the set, G, of all non-negative integer combinations of these ai. In particular, we examine the generating function f(z) = [Sigma]b E Gzb. 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 Zn. 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. [PUBLICATION ABSTRACT]
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-005-1222-y