{
  "abstract": "The optimal Sobolev regularity of the roots of a smooth curve of monic complex polynomials is $W^{1,q}$, for $q \\in [1,\\frac{d}{d-1})$, where $d$ is the degree. This result is stable in the sense that the solution map from $C^d$ coefficients to $W^{1,q}$ roots is continuous. We prove that, after a natural Lipschitz reparameterization, the roots are Lipschitz and the map from $C^d$ coefficients to reparameterization and reparameterized roots is continuous with respect to the $W^{1,q}$ topology on the target spaces, for all $q \\in [1,\\infty)$. The result is based on a convergence and reparameterization theorem for Almgren's $Q$-valued Sobolev functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39389",
  "author_total": 2,
  "authors": [
    {
      "name": "Adam Parusiński"
    },
    {
      "name": "Armin Rainer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6j2xzvyfx19k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39389",
  "fulltext_url": null,
  "id": "c6j2xzvyfx19k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39389",
  "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.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7215042034",
  "pdf_url": "https://arxiv.org/pdf/2609.39389",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6j2xzvyfx19k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39389",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39389"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Natural reparameterization and continuity of the solution map for polynomials",
  "title_text": "Natural reparameterization and continuity of the solution map for polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/c6j2xzvyfx19k",
  "venue": "arXiv",
  "versions": []
}