{
  "abstract": "We study degree-three rational sphere maps in two complex variables. After a standard normalization, the denominator of such a map takes the form \\[ g_σ(z)=1+σ_1 z_1^2+σ_2 z_2^2, \\qquad σ_1,σ_2\\geq 0. \\] A basic question is: which pairs $(σ_1,σ_2)$ can actually occur as the denominator of a degree-three rational sphere map? The first main result of the paper gives a complete answer: such a denominator occurs if and only if \\[ 0\\leq σ_1,σ_2<1, \\qquad \\sqrt{1-σ_1^2}+\\sqrt{1-σ_2^2}>1. \\] Our approach converts the sphere-mapping condition into a finite-dimensional Gram-matrix positivity problem. Furthermore, for each admissible parameter $ σ=(σ_1,σ_2), $ we determine all possible minimal target dimensions in which the corresponding denominator $g_σ$ can be realized. We also give a Gram-matrix normal form for maps with a fixed denominator and compute, for each admissible $σ$, the dimension of the moduli space of equivalence classes of rational sphere maps realizing $g_σ$. Finally, we extend the Gram-matrix method to arbitrary source dimension and obtain a general sufficient condition for the existence of degree-three rational sphere maps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29451",
  "author_total": 7,
  "authors": [
    {
      "name": "Eden Danielsen-Jensen"
    },
    {
      "name": "Dusty Grundmeier"
    },
    {
      "name": "Abdullah Al Helal"
    },
    {
      "name": "Valentin D. Kunz"
    },
    {
      "name": "Jiri Lebl"
    },
    {
      "name": "Ming Xiao"
    },
    {
      "name": "Weixia Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/enhedgd2br4g9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29451",
  "fulltext_url": null,
  "id": "enhedgd2br4g9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29451",
  "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"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7204914555",
  "pdf_url": "https://arxiv.org/pdf/2608.29451",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/enhedgd2br4g9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2608.29451",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29451"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Degree-Three Rational Sphere Maps: Sharp Denominator Region and Gram Normal Forms",
  "title_text": "Degree-Three Rational Sphere Maps: Sharp Denominator Region and Gram Normal Forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/enhedgd2br4g9",
  "venue": "arXiv",
  "versions": []
}