September 29, 2025Open Access

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

Key Points

h*-vectors in IDP polytopes may lack log-concavity, challenging previous assumptions about their structure.
The numerator of the Ehrhart series for IDP polytopes does not always yield a unimodal coefficient vector, indicating a more complex relationship.
Preliminary research identifies specific cases where IDP polytopes demonstrate non-log-concave h*-vectors, answering queries from other mathematicians.
Ongoing investigations will provide additional examples, potentially reshaping understanding of IDP polytopes and their properties.

Abstract

Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i. e. , any lattice point in a kth 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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Cite This Study

Hofscheier et al. (Sat,) studied this question.

synapsesocial.com/papers/68da58d8c1728099cfd11296 https://doi.org/https://doi.org/10.48550/arxiv.2505.18896

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