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arXiv · 2508.18890

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

Jhon B. Caicedo·Martina Juhnke·Germain Poullot
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Abstract

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.

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Jhon B. Caicedo, Martina Juhnke, Germain Poullot. 2025-12-19. Ehrhart non-positivity and unimodular triangulations for classes of s-lecture hall simplices. https://arxiv.org/abs/2508.18890

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