{
  "abstract": "Let $p_n$ be a random polynomial of degree $n$ whose roots are independent and identically distributed according to a rotationally invariant probability measure $μ_0$ on the complex plane with finite logarithmic moment. If $k_n/n\\to t\\in(0,1)$, we prove that, as $n \\to \\infty$, the empirical zero measure of the $k_n$-th derivative of $p_n$ converges weakly in probability to a deterministic rotationally invariant probability measure. We describe the limiting measure explicitly in terms of the radial quantile function of $μ_0$. This proves a conjecture of Hoskins and Kabluchko [Exp. Math. 32 (2023), no. 4].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23909",
  "author_total": 1,
  "authors": [
    {
      "name": "Sean O'Rourke"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/85r6jy5xa6q38.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23909",
  "fulltext_url": null,
  "id": "85r6jy5xa6q38",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23909",
  "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.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7214042735",
  "pdf_url": "https://arxiv.org/pdf/2609.23909",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/85r6jy5xa6q38.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.23909",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23909"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Repeated differentiation of random polynomials with i.i.d. rotationally invariant roots",
  "title_text": "Repeated differentiation of random polynomials with i.i.d. rotationally invariant roots",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/85r6jy5xa6q38",
  "venue": "arXiv",
  "versions": []
}