{
  "abstract": "Let \\(λ\\succγ\\succμ\\) be equal-degree exponent partitions with at most three parts, and let \\begin{equation*} P_{λ,γ,μ} = J_λ+J_μ-2J_γ, \\end{equation*} where \\(J_ν\\) denotes the symmetric monomial orbit sum associated with \\(ν\\). We prove a necessary and sufficient collision criterion for the positivity of this three-point majorization difference in three variables. For nonnegative integer exponent partitions, \\begin{equation*} P_{λ,γ,μ}(x,y,z)\\ge0 \\qquad(x,y,z>0) \\end{equation*} if and only if \\begin{equation*} P_{λ,γ,μ}(t,1,1)\\ge0 \\qquad(t>0). \\end{equation*} Thus the positivity of a genuinely three-variable symmetric polynomial of this form is completely determined by its one-variable restriction to the locus where two variables coincide. The theorem gives a uniform and effectively checkable criterion for an infinite class of symmetric polynomial inequalities. For a fixed integer chain, the global three-variable problem is reduced to a single univariate polynomial inequality, which can often be verified exactly by factorization, Sturm's theorem, or other one-variable methods. The criterion extends well beyond the classical Schur family, produces explicit inequalities outside the Schur pattern, and yields genuine strengthenings and refinements of Schur's inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22465",
  "author_total": 1,
  "authors": [
    {
      "name": "Jian Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/098x3fc3by75f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22465",
  "fulltext_url": null,
  "id": "098x3fc3by75f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22465",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7214038008",
  "pdf_url": "https://arxiv.org/pdf/2609.22465",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/098x3fc3by75f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22465",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22465"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Collision Positivity for Three-Variable Symmetric Monomial Inequalities",
  "title_text": "Collision Positivity for Three-Variable Symmetric Monomial Inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/098x3fc3by75f",
  "venue": "arXiv",
  "versions": []
}