{
  "abstract": "We generalize Christol's theorem---which characterizes algebraic power series over a finite field in terms of automatic coefficient sequences---to completions of domains of finite type over a finite field~$\\mathbb{F}_{q}$. Let $A$ be such a domain of dimension~$d$ and $\\mathfrak{a}$ an ideal of depth~$d$. In the $\\mathfrak{a}$-adic completion $\\hat{A}$, choose an $\\hat{A}$-regular sequence $\\boldsymbol{x}=\\{x_{1},\\dots,x_{d}\\}\\subset\\hat{\\mathfrak{a}}$ and a set of representatives $E$ of $\\hat{A}/(\\boldsymbol{x})$ that is an $\\mathbb{F}_{q}$-vector subspace consisting of algebraic elements. Every element $y\\in\\hat{A}$ then admits a unique expansion $y=\\sum_{u\\in\\mathbb{N}^{d}}a_{u}x^{u}$ with coefficients $a_{u}\\in E$. We prove that $y$ is algebraic over $A$ if and only if the coefficient sequence $\\{a_{u}\\}_{u\\in\\mathbb{N}^{d}}$ is $q$-automatic. The proof relies on a careful study of henselizations, which are shown to coincide with the algebraic closures inside the completions under certain conditions. As an application, we obtain a complete characterization of the automaticity of $β$-expansions for formal Laurent series over $\\mathbb{F}_{q}$, answering questions posed by Hbaib--Mkaouar and Scheicher--Sirvent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14450",
  "author_total": 2,
  "authors": [
    {
      "name": "Lian Duan"
    },
    {
      "name": "Chunlin Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bbg9vjv2kwt9f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14450",
  "fulltext_url": null,
  "id": "bbg9vjv2kwt9f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14450",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221753985",
  "pdf_url": "https://arxiv.org/pdf/2610.14450",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bbg9vjv2kwt9f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14450",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14450"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the generalization of Christol's theorem to domains of finite type over finite fields",
  "title_text": "On the generalization of Christol's theorem to domains of finite type over finite fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/bbg9vjv2kwt9f",
  "venue": "arXiv",
  "versions": []
}