{
  "abstract": "Motivated by Scott's inequality for lattice polygons, we study normalized volume bounds for lattice polytopes $\\mathcal{P}$ with $h^*$-polynomials $1+at^k+bt^\\ell$, where $a,b\\ge1$ and $1\\le k<\\ell$. For $\\ell>2k$, the coefficients satisfy $a+1\\le b$, and consequently ${\\rm vol}(\\mathcal{P})\\le2b$. In the complementary range $k<\\ell\\le2k$, we establish the dimension-independent bound \\[ {\\rm vol}(\\mathcal{P})\\le(2b+1)(k+\\ell). \\] On the boundary $\\ell=2k$ with $b\\ge2$, the sharper Scott-type inequality ${\\rm vol}(\\mathcal{P})\\le4k(b+1)$ holds, confirming a conjecture of Higashitani, Nill, and the author. Equality holds if and only if $k=1$ and $a=3b+3$. Moreover, when $\\ell=2k$ and $k,b\\ge2$, we obtain an upper bound depending only on $b$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35287",
  "author_total": 1,
  "authors": [
    {
      "name": "Akiyoshi Tsuchiya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/875jq4dcc419c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35287",
  "fulltext_url": null,
  "id": "875jq4dcc419c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35287",
  "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": "W7223868550",
  "pdf_url": "https://arxiv.org/pdf/2610.35287",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/875jq4dcc419c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35287",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35287"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Scott-type inequalities for trinomial $h^*$-polynomials",
  "title_text": "Scott-type inequalities for trinomial h^*-polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/875jq4dcc419c",
  "venue": "arXiv",
  "versions": []
}