arXiv subjects

Jhon B. Caicedo

Publications and source records attributed to Jhon B. Caicedo.

3 recordsLinked to original sources

arXiv2026

Sylvester simplices: Triangulations and Ehrhart-theoretic aspects

The Sylvester simplex $\mathsf{Sylv}_d^k$ is a $d$-dimensional lattice simplex with exactly $k$ interior lattice points. Sylvester simplices are conjectured to be the volume maximizers among all $d$-dimensional lattice polytopes with exactly $k$ interior lattice points for any $k\geq 1$. Even stronger, it is conjectured that they maximize (entry-wise) the $h^\ast$-vector among all $d$-dimensional lattice polytopes with exactly $k$ interior lattice points. Yet, Sylvester simplices seem to be rarely studied in their own right. In particular, their Ehrhart-theoretic properties are far from being well understood. In the present article, we tackle this problem. We describe flag, regular and unimodular triangulations for the Sylvester simplices, and prove that their $h^\ast$-vectors are unimodal. Moreover, we explicitly determine the values of some entries of their $f^\ast$-vectors, and prove that they are Ehrhart magic positive up to dimension $6$ but not in dimension $7$. We conclude by detailing tables of Ehrhart-theoretic quantities (numbers of lattice points, Ehrhart polynomials, local and boundary $h^\ast$-vectors, $f^\ast$-vectors) for Sylvester simplices of dimensions 7 and lower.

math.CO↗

arXiv2026

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

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.

math.CO↗

arXiv2025

Ehrhart non-positivity and unimodular triangulations for classes of s-lecture hall simplices

Counting lattice points and triangulating polytopes is a prominent subject in discrete geometry, yet proving Ehrhart positivity or existence of unimodular triangulations remain of utmost difficulty in general, even for ``easy'' simplices. We study these questions for classes of s-lecture hall simplices. Inspired by a question of Olsen, we present a new natural class of sequences s for which the s-lecture hall simplices are not Ehrhart positive, by explicitly estimating a negative coefficient. Meanwhile, motivated by a conjecture of Hibi, Olsen and Tsuchiya, we extend the previously known classes of sequences s for which the s-lecture hall simplex admits a flag, regular and unimodular triangulation. The triangulations we construct are explicit.

math.CO↗