Examples of IDP lattice polytopes with non-log-concave $h^*$-vector
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| Format: | Preprint |
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2025
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| _version_ | 1866916756992294912 |
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| author | Hofscheier, Johannes Kurylenko, Vadym Nill, Benjamin |
| author_facet | Hofscheier, Johannes Kurylenko, Vadym Nill, Benjamin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_18896 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Examples of IDP lattice polytopes with non-log-concave $h^*$-vector Hofscheier, Johannes Kurylenko, Vadym Nill, Benjamin Combinatorics 52B20, 05A20, 68T05 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. |
| title | Examples of IDP lattice polytopes with non-log-concave $h^*$-vector |
| topic | Combinatorics 52B20, 05A20, 68T05 |
| url | https://arxiv.org/abs/2505.18896 |