2003/12/31 by Francisco Santos
Mathematics · #math.CO #msc:52B20 #msc:52B11
published as In "Integer Points in Polyhedra - Geometry, Number Theory, Algebra, Optimization", A. Barvinok, M. Beck, C. Haase, B. Reznick, and V. Welker (eds), Contemporary Mathematics 374, Amer. Math. Soc., Providence, 2005. ISBN 0-8218-3459-2. · This version has been accepted in "Proceedings of the Joint Summer Research Conference on Integer Points in Polyhedra" (Barvinok et al., eds.) Contemporary Mathematics, American Mathematical Society. Changes from v2: corrected a LaTeX problem with figures. Changes from v1: (1) some rephrasing, especially in the introduction. (2) the former proof of Theorem 5.4 was incorrect. The bound in the statement has been changed to match the new proof
We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) l, we show that the numbers of regular and non-regular triangulations of Δl×Δk grow, respectively, as kΘ(k) and 2Ω(k2). For the special case of l=2, we relate triangulations to certain class of lozenge tilings. This allows us to compute the exact number of triangulations up to k=15, show that the number grows as eβk2/2 + o(k2) where β≃ 0.32309594 and prove that the set of all triangulations is connected under geometric bistellar flips. The latter has as a corollary that the toric Hilbert scheme of the determinantal ideal of 2× 2 minors of a 3× k matrix is connected, for every k. We include ``Cayley Trick pictures'' of all the triangulations of Δ2× Δ2 and Δ2× Δ3, as well as one non-regular triangulation of Δ2× Δ5 and one of Δ3× Δ3.