{
  "abstract": "Let $P = \\sum_ω a_ωx^ω$ be a degree $d$ homogeneous polynomial in $x_0, \\dots, x_n$ with real coefficients. Pólya showed that, if $P$ is strictly positive on the standard $n$-simplex, then $(x_0 + \\cdots + x_n)^M P$ has strictly positive coefficients for all $M \\gg 0$. The least such $M$ is known as the \\emph{Pólya exponent} of $P$. de Klerk-Laurent-Parrilo showed that the Pólya exponent is $O(d^2) P^{(0)}_{\\max}/λ(P)$, where $P^{(0)}_{\\max} := \\max_ωa_ω/\\binom{d}ω$ and $λ(P)$ is the infimum of $P$ on the standard simplex. Pólya's Positivstellensatz is a consequence of the classical approximation of the multivariate hypergeometric distribution by the multinomial distribution, as was already implicitly known to de Klerk-Laurent-Sun. By applying known estimates of this approximation from probability theory, we obtain a different $O_n(d) L(P)/λ(P)$ upper bound of the Pólya exponent, where $L(P) := \\max |a_ω|/\\binom{d}ω$. An example shows that $O_n(d)$ is tight.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14535",
  "author_total": 2,
  "authors": [
    {
      "name": "Yulong Li"
    },
    {
      "name": "Colin Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5gmzq6tssgwx7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14535",
  "fulltext_url": null,
  "id": "5gmzq6tssgwx7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14535",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7221753895",
  "pdf_url": "https://arxiv.org/pdf/2610.14535",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5gmzq6tssgwx7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14535",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14535"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A degree-linear upper bound for the Pólya exponent of forms with fixed arity",
  "title_text": "A degree-linear upper bound for the Pólya exponent of forms with fixed arity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/5gmzq6tssgwx7",
  "venue": "arXiv",
  "versions": []
}