{
  "abstract": "The Gaussian curvature of the graphs of cubic polynomials $f(z)=(z-a)(z-b)(z-c)$, with triangle of zeros $\\triangle=\\triangle(a,b,c)$, is analyzed. We introduce the composed Pythagorean mean $\\mathtt{B}_{\\triangle}$, a combination of the three classical Pythagorean means, which plays a central role in the description of the critical points of the Gaussian curvature. When $\\triangle$ is not equilateral, the curvature has exactly five critical points: a global maximum at the barycenter, two local minima, and two saddle points, forming a rhombus whose vertices lie on the axes of the Siebeck-Marden ellipse of the triangle. We also initiate a comparative study of $\\mathtt{B}_{\\triangle}$ with the skew of the triangle by describing a family of polynomials whose zeros determine a specific family of isosceles triangles. The cubic case provides a concrete model strongly suggesting that the critical geometry of the Gaussian curvature of the graph can reveal both analytic and algebraic properties of the underlying holomorphic function.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38613",
  "author_total": 3,
  "authors": [
    {
      "name": "Baruc S. Cabrera"
    },
    {
      "name": "Manuel Cruz-López"
    },
    {
      "name": "Otto Romero-Germán"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2d72kcs34zxsg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38613",
  "fulltext_url": null,
  "id": "2d72kcs34zxsg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38613",
  "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.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.38613",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2d72kcs34zxsg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38613",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38613"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On Gaussian Curvature of Graphs of Cubic Polynomials",
  "title_text": "On Gaussian Curvature of Graphs of Cubic Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/2d72kcs34zxsg",
  "venue": "arXiv",
  "versions": []
}