{
  "abstract": "For a complex polynomial $P$, its full shadow is the closure of the set of zeros of all derivatives of all powers $P^n$, and its limiting core is the Hausdorff limit, as $n\\to\\infty$, of the set of zeros of all derivatives of $P^n$. These sets retain geometric information that weak limits of root-counting measures discard. We determine complete zero-set limits for arrays of polynomials whose roots approach finitely many locations. A cluster containing $o(D)$ roots of a degree-$D$ polynomial can survive outside the limiting support of the derivative zeros, which depends only on the positive limiting masses: under repeated differentiation, the zeros it produces follow a straight path, selected by an explicit min--max condition, and preserve the cluster's exact multiplicity. When such a location instead carries a small positive mass $\\varepsilon$, the support of the limiting zero distribution at a fixed ratio of derivative order to degree develops an arc on the $\\sqrt{\\varepsilon}$ scale with a universal local semicircle profile. We use these mechanisms to prove that polynomial shadows can have robust and arbitrarily complicated topology. For every $k\\ge2$, there are polynomials with $k+2$ distinct root locations whose limiting cores and full shadows have at least $\\binom{k}{2}$ holes; this persists under coefficient perturbations and among square-free polynomials without a reflection axis. In a four-location family, the core and full-shadow areas tend to zero while one hole has area tending to $8/1221$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40842",
  "author_total": 2,
  "authors": [
    {
      "name": "Christian Hägg"
    },
    {
      "name": "Boris Shapiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5exx09k5e9p02.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40842",
  "fulltext_url": null,
  "id": "5exx09k5e9p02",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40842",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CV"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40842",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5exx09k5e9p02.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40842",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40842"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The full shadow of a polynomial II. Vanishing root masses and holes",
  "title_text": "The full shadow of a polynomial II. Vanishing root masses and holes",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/5exx09k5e9p02",
  "venue": "arXiv",
  "versions": []
}