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Is description of all regular triangluations of $\Delta^n\times \Delta^k$ known? (Regular triangulations are those which correspond to vertices of Gelfand--Kapranov--Zelevinsky secondary polytope, or, equivalently, those which correspond to domains of linearity of a convex function.) Estimates of the number of triangulations is also of interest.

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    $\begingroup$ Perhaps Francisco Santos' paper, "The Cayley trick and triangulations of products of simplices," is relevant. (arXiv abstract.) He computes the exact number of triangulations of $\Delta^2 \times \Delta^k$ up to $k \le 15$, and includes some discussion of $\Delta^3 \times \Delta^3$. $\endgroup$ – Joseph O'Rourke Mar 12 '16 at 18:25
  • $\begingroup$ They should be the same as combinatorial types of tropical hyperplane arrangements. See arxiv.org/pdf/math/0605598.pdf $\endgroup$ – Sam Hopkins Mar 12 '16 at 23:49
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    $\begingroup$ I would say no "combinatorial" description that distinguishes regular from non-regular triangulations of $\Delta^n\times \Delta^k$ is known. For example, my paper cited by Jo O'Rourke above contains a nice description and a counting algorithm for triangulations of $\Delta^2\times \Delta^k$, but none of them distinguish the regular from the non-regular. In terms of asymptotics, in the same paper I show that the number of regular triangulations of $\Delta^n\times \Delta^k$ is less than $\left(\frac{e}{2}kn\right)^{n(n-1)(k-1)}$. $\endgroup$ – Francisco Santos Jan 31 '17 at 20:05

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