{
  "abstract": "Very recently, Lech Mazur proved the celebrated Sendov conjecture, and Terence Tao subsequently distilled the main ideas of the proof in a blog post. In this paper, we establish a quantitative strengthening of Sendov's conjecture, namely the quadratic Tang--Zhang inequality. Let $p$ be a polynomial of degree $n\\ge2$ whose zeros lie in the closed unit disk, and let $ζ_1,\\ldots,ζ_{n-1}$ denote its critical points, counted with multiplicity. We prove that, for every zero $a$ of $p$, $$ \\sum_{j=1}^{n-1}\\frac{1}{|a-ζ_j|^2}\\ge n-1. $$ Moreover, equality holds if and only if $p(z)=c(z^n-ω)$ for some $c\\in\\mathbb C\\setminus\\{0\\}$ and $|ω|=1$. We also provide a Lean 4 formalization of the main results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19126",
  "author_total": 1,
  "authors": [
    {
      "name": "Teng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8vd74yta3wrs7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19126",
  "fulltext_url": null,
  "id": "8vd74yta3wrs7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19126",
  "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": "W7213499011",
  "pdf_url": "https://arxiv.org/pdf/2609.19126",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8vd74yta3wrs7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.19126",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19126"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Beyond Sendov's conjecture: the quadratic Tang--Zhang inequality",
  "title_text": "Beyond Sendov's conjecture: the quadratic Tang--Zhang inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/8vd74yta3wrs7",
  "venue": "arXiv",
  "versions": []
}