{
  "abstract": "We prove that the proportion of $0/1$-equivalence classes of $d$-dimensional $(0,1)$-polytopes that are normal tends to zero at least at a double exponential rate as $d\\to\\infty$. As a consequence, the same holds for any of the following classes given by the type of triangulation possible: (a) quadratic, (b) flag unimodular, (c) regular unimodular, or (d) unimodular, among others. We classify the $0/1$-equivalence classes of $d$-dimensional $(0,1)$-polytopes for $d\\leq5$ according to whether they admit a unimodular, flag unimodular, or quadratic triangulation. In dimension five, exactly $175$ out of $1{,}226{,}525$ classes have a flag unimodular triangulation, but no quadratic triangulation. Among them, there are polytopes whose toric rings are not Koszul; thus, we find the first polytopes that have a flag unimodular triangulation, but whose toric ring is not Koszul. In contrast with the matroid case, we exhibit a delta-matroid polytope that is not normal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02778",
  "author_total": 1,
  "authors": [
    {
      "name": "Santiago Morales"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cgtb4vw8c5nhj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02778",
  "fulltext_url": null,
  "id": "cgtb4vw8c5nhj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02778",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7207669696",
  "pdf_url": "https://arxiv.org/pdf/2609.02778",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cgtb4vw8c5nhj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02778",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02778"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Most $(0,1)$-polytopes are not normal",
  "title_text": "Most (0,1)-polytopes are not normal",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/cgtb4vw8c5nhj",
  "venue": "arXiv",
  "versions": []
}