Face Numbers Of Scarf Complexes

Let A be a (d + 1) \Theta d real matrix whose row vectors positively span R d and which is generic in the sense of B'ar'any and Scarf [BS]. Such a matrix determines a certain infinite d-dimensional simplicial complex \Sigma, as described by B'ar'any, Howe and Scarf [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 = (f0 ; f1 ; : : : ; fd\Gamma1 ) is the f-vector of certain triangulated (d \Gamma 1)-ball T embedded in \Sigma. When A has integer entries it is also, as shown by work of Peeva and Sturmfels [PS], the sequence of Betti numbers of the minimal free resolution of k[x1 ; : : : ; xd+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 f0 ; f1 ; : : : ; f b d\Gamma3 2 c determine the other numbers via linear relations, and that there are additional non-linear relations. In more precise (and more technical)..