Document Type
Discussion Paper
Publication Date
4-1-2004
CFDP Number
1458
CFDP Pages
24
Abstract
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.
Recommended Citation
Scarf, Herbert E. and Woods, Kevin M., "Neighborhood Complexes and Generating Functions" (2004). Cowles Foundation Discussion Papers. 1734.
https://elischolar.library.yale.edu/cowles-discussion-paper-series/1734