Unimodular triangulations and Ehrhart theory for two families of Hermite normal form simplices

Source abstract

We study regular unimodular triangulations, the integer decomposition property, and Ehrhart-theoretic properties of two families of Hermite normal form simplices. We first consider the one-row case associated with the vector $(N - 1, \dots ,N - 1 , N)\in \mathbb{N}^d$, and completely characterize when the corresponding simplices admit a regular unimodular triangulation. Our constructions are explicit and also yield closed formulas for the $h^\ast$-polynomial and the local $h^\ast$-polynomial. Moreover, we prove Ehrhart positivity and derive explicit dimension-dependent conditions under which the Ehrhart polynomial is not unimodal. Finally, we extend our approach to the two-row cases associated with $(1, \dots ,1 , N)\in\mathbb{N}^d$ and $(M-1, \dots ,M-1, M, 0)\in\mathbb{N}^d$. In these cases, we construct regular unimodular triangulations, derive closed formulas for the $h^\ast$-polynomial and the local $h^\ast$-polynomial, and prove Ehrhart positivity.