Neighborhood complexes and generating functions for affine semigroups (Q2492644)

For given integral vectors \(a_1,\dots,a_m\in\mathbb Z^n\) let \(G\) be the set of all non-negative integer combinations of these vectors, i.e., the additive semigroup generated by \(a_1,\dots,a_m\). By the work of \textit{A. Barvinok} and \textit{K. Woods} [J. Am. Math. Soc. 16, No. 4, 957--979 (2003; Zbl 1017.05008)] it is known that the generating function of \(G\), i.e., \(f(z)=\sum_{b\in G} z^b\) can be written as a short rational function. In the paper under review the authors present a (geometric) representation of \(f\) as a rational function using the so called neighborhood or \textit{H. E. Scarf} complex [see e.g., Math. Program. 79, No. 1--3(B), 355--368 (1997; Zbl 0887.90124)] of a certain lattice. If that lattice is ``generic'', the result follows from more algebraic results of \textit{D. Bayer, I. Peeva} and \textit{B. Sturmfels } [Math. Res. Lett. 5, No. 1--2, 31--46 (1998; Zbl 0909.13010)]. Here the authors use a purely geometric approach which is based on topological properties of these complexes [cf. \textit{I. Bárány, H. E. Scarf} and \textit{D. Shallcross}, Math. Program. 80, No. 1(A), 1--15 (1998; Zbl 0894.90176)].