{
  "abstract": "Scott and Sokal asked whether every homogeneous polynomial with the half-plane property has a completely monotone negative power. We answer this question affirmatively and prove the Riesz-positivity conjecture of Michałek, Sturmfels, Uhler and Zwiernik, restated by Kozhasov, Michałek and Sturmfels: in $n$ variables, every exponent $α\\ge4096n^2$ is admissible, independently of the degree and coefficients. For complete hyperbolic polynomials the Riesz density is strictly log-concave, with relative Gaussian error at most $512n^2/α$ and explicit curvature bounds. We characterize admissible exponents by a common spectral Dirichlet law, prove $n\\le m+αm(m-1)$ with its equality case, and obtain the sharp degree-dependent gap $0<α<1/(2(m-1))$ whenever the degree-$m$ polynomial has a nonlinear irreducible factor. A nonnegative fourth-order defect of $-\\log p$ vanishes at one point precisely for products of positive integer powers of Euclidean Jordan determinants. We classify the corresponding logarithmic Monge-Amp{è}re equation and answer the question of Etingof, Kazhdan and Polishchuk about polynomial multiplicative Legendre transforms within irreducible complete hyperbolic polynomials; the general question has counterexamples, the Clifford quartics of Kogiso and Sato. The classification extends to reducible polynomials under a boundary-visibility hypothesis, and asymptotic common-power formulas for the Riesz densities force exact Jordan formulas.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31592",
  "author_total": 1,
  "authors": [
    {
      "name": "Dongsheng Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0j2zzppz2begb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31592",
  "fulltext_url": null,
  "id": "0j2zzppz2begb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31592",
  "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.AG"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7214643126",
  "pdf_url": "https://arxiv.org/pdf/2609.31592",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0j2zzppz2begb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.31592",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31592"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Riesz kernels of hyperbolic polynomials: positivity, admissible exponents and Jordan rigidity",
  "title_text": "Riesz kernels of hyperbolic polynomials: positivity, admissible exponents and Jordan rigidity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/0j2zzppz2begb",
  "venue": "arXiv",
  "versions": []
}