{
  "abstract": "We study the nonconstant rational solutions of generalized Abel equations \\[ x'=\\sum_{i=1}^s A_i(t)x^{n_i},\\qquad s\\geq3,\\qquad 1\\leq n_1<\\cdots<n_s, \\] with polynomial coefficients over $\\mathbb{R}$ or $\\mathbb{C}$. Using the Newton diagram at infinity and the associated edge polynomials, we obtain, under simplicity and nonresonance conditions on the edge polynomials, the global upper bound $n_s-1$ for the number of nonconstant rational solutions. Over $\\mathbb{R}$, the same approach yields the additional bound $\\min\\{n_s-1,2s\\}$. We construct a family of equations showing that both bounds are sharp for every $s\\geq3$. When $n_1=1$, the real bound improves to $\\min\\{n_s-1,2(s-1)\\}$. We then specialize to the three-term case with $1<n_1<n_2<n_3$, where we obtain a finer classification of the rational solutions. Under an additional antipodal separation condition, we obtain the real bound $\\min\\{n_3-1,5\\}$, and show that the absolute constant $5$ is sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.10591",
  "author_total": 2,
  "authors": [
    {
      "name": "L. A. Calderon"
    },
    {
      "name": "I. Ojeda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0t2z3dzhjjjs3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.10591",
  "fulltext_url": null,
  "id": "0t2z3dzhjjjs3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.10591",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7160914185",
  "pdf_url": "https://arxiv.org/pdf/2605.10591",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0t2z3dzhjjjs3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2605.10591",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.10591"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the rational solutions of generalized Abel equations",
  "title_text": "On the rational solutions of generalized Abel equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/0t2z3dzhjjjs3",
  "venue": "arXiv",
  "versions": []
}