{
  "abstract": "We classify graphs of complex polynomials in one variable up to isometry, both for the inner (length) distance and for the outer (Euclidean) distance of $\\mathbb{C}^2\\cong\\mathbb{R}^4$. For polynomials $p,q$ with $°p\\ge 2$ we show that the graphs of $p$ and $q$ are inner isometric if and only if they are outer isometric, if and only if $p(z)=α\\,q(βz+h)+k$ or $p(z)=α\\,\\bar q(βz+h)+k$ for some $α,β,h,k\\in\\mathbb{C}$ with $|α|=|β|=1$, where $\\bar q$ denotes the polynomial with conjugate coefficients. Moreover, every inner isometry between such graphs is the restriction of an isometry of $\\mathbb{C}^2$; in this sense graphs of complex polynomials are isometrically rigid. We contrast this with the real case, where the two distances behave very differently: the graphs of any two real polynomials are inner isometric, whereas, for degree at least $2$, outer isometries are generated by translations, reflections in horizontal and vertical lines and half-turns.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14291",
  "author_total": 3,
  "authors": [
    {
      "name": "Alexandre Fernandes"
    },
    {
      "name": "Claudia Fernandes"
    },
    {
      "name": "Edson Sampaio"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/70em49s58w0x3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14291",
  "fulltext_url": null,
  "id": "70em49s58w0x3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14291",
  "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.DG"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7221806834",
  "pdf_url": "https://arxiv.org/pdf/2610.14291",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/70em49s58w0x3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14291",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14291"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Inner and outer isometries of graphs of polynomials",
  "title_text": "Inner and outer isometries of graphs of polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/70em49s58w0x3",
  "venue": "arXiv",
  "versions": []
}