{
  "abstract": "The precise origin of critical points in rational functions remains a longstanding open problem. This paper elucidates the mechanism underlying critical point generation in rational functions. For over two centuries, a conjecture attributed to Gauss and Fourier has remained unresolved: every real zero of $(p^{'}(s)/p(s))^{'}$ is associated with a pair of non-real zeros of the real polynomial $p(s)$. The real zeros of $(p^{'}(s)/p(s))^{'}$ that are multiple real zeros of $p^{'}(s)$ are termed second-kind zeros of $(p^{'}(s)/p(s))^{'}$. By developing the root locus method, we establish a necessary and sufficient condition for the emergence of second-kind zeros: a second-kind zero $M$ arises if and only if at least two root loci of $2qπ$ or $2qπ+π$ degree, emitted from at least one pair of non-real conjugate zeros of $p(s)$ (in the extended complex plane, excluding the real axis), directly intersect another root locus at $M$ on the real axis. Our results resolve one conjecture of the Gauss-Fourier conjectures and reveal the fundamental mechanism by which critical points of rational functions are generated.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29790",
  "author_total": 2,
  "authors": [
    {
      "name": "Lande Ma"
    },
    {
      "name": "Zhaokun Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dwx5heejkh4z6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29790",
  "fulltext_url": null,
  "id": "dwx5heejkh4z6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29790",
  "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": "W7223212581",
  "pdf_url": "https://arxiv.org/pdf/2610.29790",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dwx5heejkh4z6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29790",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29790"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Principle of generating critical points of rational functions and Gauss-Fourier conjecture for second-kind zeros",
  "title_text": "Principle of generating critical points of rational functions and Gauss-Fourier conjecture for second-kind zeros",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/dwx5heejkh4z6",
  "venue": "arXiv",
  "versions": []
}