{
  "abstract": "Fix a finite field $\\mathbf{F}_q$, an integer $m\\ge2$ prime to $q$, and $2\\leκ\\le m$. For a uniformly chosen $κ$-power-free polynomial $F\\in\\mathbf{F}_q[x]$ of degree $D$, we determine the limiting joint distribution of the affine point counts of $y^m=F(x)$ over $\\mathbf{F}_q,\\ldots,\\mathbf{F}_{q^n}$ as $D\\to\\infty$, with $q$ and $n$ fixed. The probability generating function is a finite Euler product with explicit local factors. Equivalently, the limiting vector is a sum of independent closed-point contributions, each of which may affect several extension degrees. We obtain the formula by specializing the equidistribution and motivic Euler product formalism of Bertucci-Howe. We then compute means, covariances, and moment expansions as $q$ varies in a fixed reduced residue class modulo $m$, after the degree limit has been taken. The one-field specialization recovers the power-free affine point-count distribution of Cheong-Wood-Zaman.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33847",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuxiang Yao"
    },
    {
      "name": "Xu Zhuang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/brghsyhaqrp45.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33847",
  "fulltext_url": null,
  "id": "brghsyhaqrp45",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33847",
  "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.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223988036",
  "pdf_url": "https://arxiv.org/pdf/2610.33847",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/brghsyhaqrp45.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33847",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33847"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Joint Extension Field Point Counts for Superelliptic Curves",
  "title_text": "Joint Extension Field Point Counts for Superelliptic Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/brghsyhaqrp45",
  "venue": "arXiv",
  "versions": []
}