On the generalization of Christol's theorem to domains of finite type over finite fields

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~𝔽_q. Let A be such a domain of dimension~d and a an ideal of depth~d. In the a-adic completion Â, choose an Â-regular sequence x={x₁,…,x_d}⊂â and a set of representatives E of Â/(x) that is an 𝔽_q-vector subspace consisting of algebraic elements. Every element y∈Â then admits a unique expansion y=∑_u∈ℕ^da_ux^u with coefficients a_u∈ E. We prove that y is algebraic over A if and only if the coefficient sequence {a_u}_u∈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 𝔽_q, answering questions posed by Hbaib--Mkaouar and Scheicher--Sirvent.

A Generalization of Sárközy's theorem in function fields
Pierre-Yves Bienvenu, Thái Hoàng Lê, Gauree Wathodkar
arXiv11 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem
Xiyu Hu
arXiv4 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Uniform generalization of Goodstein's theorem and Cichon's independence proof
Gunnar Wilken
arXiv12 Oct 2026PreprintCC BY-NC-ND 4.0Full textLogic & foundationsarXiv
Rural Readiness: Domain Generalization with Adversarially Chosen Domains
Cynthia Dwork, Andisheh Ghasemi, Lunjia Hu
arXiv29 Oct 2026PreprintCC BY 4.0Full textStatisticsarXiv
A generalization of Gordon's Theorem to arbitrary winding number
M. Fenucci
arXiv22 Oct 2026PreprintCC BY 4.0Full textDynamical systems Mathematical physicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{duan2026generalization,
  title = {{On the generalization of Christol's theorem to domains of finite type over finite fields}},
  author = {Lian Duan and Chunlin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14450},
  eprint = {2610.14450},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14450},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On the generalization of Christol's theorem to domains of finite type over finite fields
AU  - Lian Duan
AU  - Chunlin Wang
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14450
UR  - https://arxiv.org/abs/2610.14450
AB  - 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~𝔽_q. Let A be such a domain of dimension~d and a an ideal of depth~d. In the a-adic completion Â, choose an Â-regular sequence x={x_1,…,x_d}⊂â and a set of representatives E of Â/(x) that is an 𝔽_q-vector subspace consisting of algebraic elements. Every element y∈Â then admits a unique expansion y=∑_u∈ℕ^da_ux^u with coefficients a_u∈ E. We prove that y is algebraic over A if and only if the coefficient sequence {a_u}_u∈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 𝔽_q, answering questions posed by Hbaib--Mkaouar and Scheicher--Sirvent.
ER  -

Abstract

Related

Cite