{
  "abstract": "Sendov's conjecture, first formulated in 1958, asserts that if a complex polynomial $f$ of degree $n \\geq 2$ has all its zeros in the closed unit disk ${z \\in \\mathbb{C} : |z| \\leq 1}$, then, for every zero $λ_0$ of $f$, there exists a critical point $ζ$ of $f$ such that $|ζ-λ_0| \\leq 1$. The conjecture was previously known to hold for polynomials of degree $n \\leq 8$, as well as in several special higher-degree cases. In 2022, using compactness methods, balayage, and the argument principle, Tao~\\cite{Tao22} proved that there exists an absolute constant $n_0$ such that Sendov's conjecture holds for all $n \\geq n_0$. However, Tao's argument does not provide an explicit admissible value of $n_0$. In the present paper, we make Tao's result effective and prove that one may take $$ n_0 = 10^{200000}. $$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20256",
  "author_total": 1,
  "authors": [
    {
      "name": "Teng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8jj8s9p7y764c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20256",
  "fulltext_url": null,
  "id": "8jj8s9p7y764c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20256",
  "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"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7213585460",
  "pdf_url": "https://arxiv.org/pdf/2609.20256",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8jj8s9p7y764c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20256",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20256"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sendov's conjecture holds for every degree $n\\ge 10^{200000}$",
  "title_text": "Sendov's conjecture holds for every degree n≥ 10^200000",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/8jj8s9p7y764c",
  "venue": "arXiv",
  "versions": []
}