{
  "abstract": "Let $p$ be a complex polynomial of degree $n\\ge2$ whose zeros lie in the closed unit disk. Sendov's conjecture asserts that every zero of $p$ lies within distance one of a zero of $p'$. Mazur's recent proof, and Tao's streamlined exposition of it, reduce a hypothetical counterexample to a scalar lower bound \\[ 1\\le F(η,α,n) \\] together with two upper bounds for the endpoint parameter $η$ and the uniform restriction $0<α\\le 17$. We complete this scalar reduction by a two-stage optimization argument. First, $F$ is nondecreasing in $η$, so $η$ may be replaced by a piecewise polar envelope $η^{\\ast}(α)$. Writing $s=(n-1)/2$ converts the degree to a half-integer variable and gives \\[ F(η^{\\ast}(α),α,n) = E(α,s) + κ(α,s) \\int_0^1 t^3 \\widehatβ(t;α,s)^{\\,s-\\frac{3}{2}} \\,dt. \\] On each half-unit $α$-slab, $E$ and $κ$ decrease with $α$, whereas $\\widehatβ$ increases. Convexity in $t$ reduces all but the first mesh interval to eight explicit nodal terms. The first interval satisfies a uniform bound $3/200$. Each nodal upper term is a strictly log-concave function of the half-integer $s$, so its global discrete maximum is certified by two adjacent ratio evaluations. A fixed finite certificate over the $34$ half-unit slabs gives \\[ F(η^{\\ast}(α),α,n)<\\frac{97}{100}<1, \\] contradicting the scalar lower bound. The main contribution is a substantial reduction of the computer assistance required after the Mazur--Tao scalar reduction: all continuous optimization and the unbounded degree parameter are handled analytically, leaving only a small fixed collection of explicit one-variable inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04246",
  "author_total": 1,
  "authors": [
    {
      "name": "Senjian An"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0m4k3qfeg48jn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04246",
  "fulltext_url": null,
  "id": "0m4k3qfeg48jn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04246",
  "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.CV"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7211894198",
  "pdf_url": "https://arxiv.org/pdf/2609.04246",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0m4k3qfeg48jn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04246",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04246"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Scalar Optimization Proof of Sendov's Conjecture with Reduced Computer Assistance",
  "title_text": "A Scalar Optimization Proof of Sendov's Conjecture with Reduced Computer Assistance",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/0m4k3qfeg48jn",
  "venue": "arXiv",
  "versions": []
}