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arXiv · 2505.18896

Examples of IDP lattice polytopes with non-log-concave $h^*$-vector

Johannes Hofscheier·Vadym Kurylenko·Benjamin Nill
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Abstract

Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a $k$th dilation is a sum of $k$ lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, called the $h^*$-polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that $h^*$-vectors of IDP polytopes do not have to be log-concave. This answers a question of Luis Ferroni and Akihiro Higashitani. As this is an ongoing project, this paper will be updated with more details and examples in the near future.

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Johannes Hofscheier, Vadym Kurylenko, Benjamin Nill. 2025-05-24. Examples of IDP lattice polytopes with non-log-concave $h^*$-vector. https://arxiv.org/abs/2505.18896

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