{"paper":{"title":"Unimodality on $\\delta$-vectors of lattice polytopes and two related properties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Akihiro Higashitani","submitted_at":"2014-11-19T15:21:13Z","abstract_excerpt":"In this paper, we investigate two properties concerning the unimodality of the $\\delta$-vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes $\\mathcal{P}$ of dimension $d$, we prove that the dilated lattice polytopes $n\\mathcal{P}$ have strictly log-concave and strictly alternatingly increasing $\\delta$-vectors if $n > \\max\\{s,d+1-s\\}$, where $s$ is the degree of the $\\delta$-polynomial of $\\mathcal{P}$. The bound $\\max\\{s,d+1-s\\}$ for $n$ is reasonable. We also provide several kinds of unimodal (or non-unimodal) $\\delta$-vectors. Concre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5250","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}