{
  "abstract": "The full shadow of a complex polynomial $P$ of degree $d\\ge2$ is the closure of all zeros of $(P^n)^{(m)}$, $n\\ge1$, $0\\le m<dn$. We prove that it consists of a compact connected limiting core containing all roots and critical points of $P$, together with isolated transient zeros, each occurring at only finitely many powers. The core is the Hausdorff limit of the zero sets pooled over all derivative orders as $n\\to\\infty$ and the support of their limiting normalized root-counting measure. For $m/n\\toα\\in(0,d)$, we establish weak convergence of normalized root-counting measures and Hausdorff convergence of zero sets to the limiting supports, both stable under coefficient perturbations. We exhibit quartics with infinitely many transients. A bounded operator constructed from polynomial differentiators realizes the full shadow, core, and transients as its spectrum, Fredholm essential spectrum, and discrete spectrum, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09602",
  "author_total": 2,
  "authors": [
    {
      "name": "Christian Hägg"
    },
    {
      "name": "Boris Shapiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/85p8bcsp6tyqq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09602",
  "fulltext_url": null,
  "id": "85p8bcsp6tyqq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09602",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7220950798",
  "pdf_url": "https://arxiv.org/pdf/2610.09602",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/85p8bcsp6tyqq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09602",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09602"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "The full shadow of a polynomial I. The limiting core and the transient set",
  "title_text": "The full shadow of a polynomial I. The limiting core and the transient set",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/85p8bcsp6tyqq",
  "venue": "arXiv",
  "versions": []
}