Face numbers of Scarf complexes
2000 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 24, no 3-Feb, 185-196 p.Article in journal (Refereed) Published
Let A be a (d + 1) x d real matrix whose row vectors positively span R-d and which is generic in the sense of Barany and Scarf [BS1]. Such a matrix determines a certain infinite d-dimensional simplicial complex Sigma, as described by Barany et al. [BHS]. The group Z(d) acts on Sigma with finitely many orbits. Let f(i) be the number of orbits of (i + 1)-simplices of Sigma. The sequence f = (f(0), f(1),..., f(d-1)) is the f-vector of a certain triangulated (d - 1)-ball T embedded in Sigma. When A has integer entries it is also, as shown by the work of Peeva and Sturmfels [PS], the sequence of Betti numbers of the minimal free resolution of k[x(1),...,x(d+1)]/I, where I is the lattice ideal determined by A. In this paper we study relations among the numbers f(i). It is shown that f(0), f(1),..., f([(d-3)/2]) determine the other numbers via linear relations, and that there are additional nonlinear relations. In more precise (and more technical) terms, our analysis shows that f is linearly determined by a certain M-sequence (g(0), g(1),..., g([(d-1)/2])). namely, the g-vector of the (d - 2)-sphere bounding T. Although T is in general not a cone over its boundary, it turns out that its f-vector behaves as if it were.
Place, publisher, year, edition, pages
2000. Vol. 24, no 3-Feb, 185-196 p.
resolutions, polytopes, sets
IdentifiersURN: urn:nbn:se:kth:diva-19849ISI: 000087814300005OAI: oai:DiVA.org:kth-19849DiVA: diva2:338541
QC 201005252010-08-102010-08-10Bibliographically approved