{
  "abstract": "For over two centuries, the esteemed mathematicians Gauss and Fourier conjectured that each real critical point of the rational function $p^{'}(s)/p(s)$ is associated with a pair of non-real zeros of the real polynomial $p(s)$. The conjecture remains unresolved. The real zeros of $(p^{'}(s)/p(s))^{'}$ that are real multiple zero of $p^{'}(s)$ are defined as the second kind of zeros of $(p^{'}(s)/p(s))^{'}$. In addition to the second kind of zeros, the other real zeros of $(p^{'}(s)/p(s))^{'}$ are referred to as the first kind of zeros of $(p^{'}(s)/p(s))^{'}$. We demonstrate that the first kind of zeros of $(p^{'}(s)/p(s))^{'}$ emerge from the intersection of root loci emitted from non-real zeros of $p(s)$, thereby providing a comprehensive geometric interpretation of their relationship. Moreover, each first kind of zero of $(p^{'}(s)/p(s))^{'}$ is connected via root loci to a pair of conjugate non-real zeros of $p(s)$. This establishes the precise geometric relationship conjectured by Gauss and Fourier, fully resolving their original problem for first-kind zeros and completing a foundational chapter in polynomial theory. We have proven Gauss-Fourier conjecture about the first kind of zeros.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30620",
  "author_total": 2,
  "authors": [
    {
      "name": "Lande Ma"
    },
    {
      "name": "Zhaokun Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dgkfwskj0nk6b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30620",
  "fulltext_url": null,
  "id": "dgkfwskj0nk6b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30620",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7223299985",
  "pdf_url": "https://arxiv.org/pdf/2610.30620",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dgkfwskj0nk6b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30620",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30620"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Comprehensive Resolution of the Gauss-Fourier Conjecture for the First Kind of Zeros",
  "title_text": "Comprehensive Resolution of the Gauss-Fourier Conjecture for the First Kind of Zeros",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/dgkfwskj0nk6b",
  "venue": "arXiv",
  "versions": []
}