ISSN:
1432-0444
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract. Let A be a (d+1) × d real matrix whose row vectors positively span R d and which is generic in the sense of Bárány and Scarf [BS1]. Such a matrix determines a certain infinite d -dimensional simplicial complex Σ , as described by Bárány et al.[BHS]. The group Z d acts on Σ with finitely many orbits. Let f i be the number of orbits of (i+1) -simplices of Σ . The sequence f=(f 0 ,f 1 , . . ., f d-1 ) is the f -vector of a certain triangulated (d-1) -ball T embedded in Σ . 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 $f_0,f_1, . . . , f_{\lfloor (d-3)/2 \rfloor}$ 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 $(g_0,g_1, . . . , g_{\lfloor (d-1)/2 \rfloor})$ , 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s004540010026
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