The Cayley trick and triangulations of products of simplices
Abstract
Description
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 $Δ^l\timesΔ^k$ grow, respectively, as $k^{Θ(k)}$ and $2^{Ω(k^2)}$.
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 $e^{βk^2/2 + o(k^2)}$ where $β\simeq 0.32309594$ 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 $2\times 2$ minors of a $3\times k$ matrix is connected, for every $k$.
We include ``Cayley Trick pictures'' of all the triangulations of $Δ^2\times Δ^2$ and $Δ^2\times Δ^3$, as well as one non-regular triangulation of $Δ^2\times Δ^5$ and one of $Δ^3\times Δ^3$.
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
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
Keywords
Citation
Consulte el texto completo en el siguiente enlace:
https://arxiv.org/abs/math/0312069
http://arxiv.org/abs/math/0312069
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.
http://arxiv.org/abs/math/0312069
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.