{
  "abstract": "Let $p$ be a complex polynomial of degree $n\\ge2$, and let $σ_2(p)$ be the root mean square distance of its zeros from their centroid, including multiplicities. We prove that every zero of $p$ lies within distance $σ_2(p)$ of a critical point. This proves Borcea's quadratic conjecture and, in particular, gives a new proof of Sendov's conjecture. Two polynomial integration identities prove the result analytically for $n\\ge2048$. For the remaining degrees, we derive explicit necessary inequalities, including a variable integration endpoint and an exact centered coefficient identity. Directed integer arithmetic certifies a finite covering of the resulting parameter domains. Degrees $4$--$26$ use $150\\,845$ certified three-dimensional regions; degrees $27$--$2047$ use a further $27\\,429$ regions, with degree intervals allowed. The cases of degree 2 and 3 were already covered by Khavinson--Pereira--Putinar--Saff--Shimorin.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12499",
  "author_total": 2,
  "authors": [
    {
      "name": "Ludovick Bouthat"
    },
    {
      "name": "Shaun Fallat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ahqbnpqcjzy3p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12499",
  "fulltext_url": null,
  "id": "ahqbnpqcjzy3p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12499",
  "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.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7221565344",
  "pdf_url": "https://arxiv.org/pdf/2610.12499",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ahqbnpqcjzy3p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12499",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12499"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A proof of Borcea's quadratic variance conjecture",
  "title_text": "A proof of Borcea's quadratic variance conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/ahqbnpqcjzy3p",
  "venue": "arXiv",
  "versions": []
}