{
  "abstract": "Let $R_n$ be the infimum of the inradii of $\\{z:|p(z)|<1\\}$ over monic degree-$n$ polynomials whose zeros lie in the closed unit disk. We prove $nR_n\\toπ/2$, matching the asymptotic obstruction supplied by $z^n-1$. We first establish the exact universal radius $2^{1/n}-1$ for disks centered at zeros, which recovers the $(\\log2)/n$ bound. Small inradius then forces radial concentration of the zeros and decay of their low reciprocal moments. These estimates give an entire limit with a modulus reflection identity; a second rescaling produces an exponential tangent and strict sublevel disks of every radius below $π/2$. The proof is accompanied by a Lean~4 formalization and a step-by-step source index.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06461",
  "author_total": 2,
  "authors": [
    {
      "name": "YuJian Geng"
    },
    {
      "name": "Dong Qiu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bhf0qt47rvraf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06461",
  "fulltext_url": null,
  "id": "bhf0qt47rvraf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06461",
  "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",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7212020369",
  "pdf_url": "https://arxiv.org/pdf/2609.06461",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bhf0qt47rvraf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06461",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06461"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "From $\\log 2$ to $π/2$: the sharp asymptotic inradius of polynomial lemniscates",
  "title_text": "From log 2 to π/2: the sharp asymptotic inradius of polynomial lemniscates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/bhf0qt47rvraf",
  "venue": "arXiv",
  "versions": []
}