{
  "abstract": "For each dimension $d$, we determine the minimum number of facets of a smooth $d$-dimensional lattice polytope whose Ehrhart polynomial has at least one negative coefficient. This minimum is six in dimension $d=3$, seven in dimension $d=4$, and $d+2$ in every dimension $d\\ge 5$. The lower bounds follow from the classification of smooth lattice polytopes with at most $d+2$ facets, together with explicit formulas for Ehrhart polynomials in dimensions at most $4$. We also prove that a lattice polytope of dimension at least $3$ with a smooth vertex cannot be $2$-integral with respect to any ordered lattice basis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27819",
  "author_total": 4,
  "authors": [
    {
      "name": "Feihu Liu"
    },
    {
      "name": "Jinlong Tang"
    },
    {
      "name": "Sihao Tao"
    },
    {
      "name": "Zihao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a4tv2dht329ej.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27819",
  "fulltext_url": null,
  "id": "a4tv2dht329ej",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27819",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223145850",
  "pdf_url": "https://arxiv.org/pdf/2610.27819",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a4tv2dht329ej.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27819",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Ehrhart positivity of smooth polytopes with few facets",
  "title_text": "Ehrhart positivity of smooth polytopes with few facets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/a4tv2dht329ej",
  "venue": "arXiv",
  "versions": []
}