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arXiv · 2404.09175

Expansions in completions of global function fields

Chunlin Wang
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Abstract

It is well known that any power series over a finite field represents a rational function if and only if its sequence of coefficients is ultimately periodic. The famous Christol's Theorem states that a power series over a finite field is algebraic if and only if its sequence of coefficients is $p$-automatic. In this paper, we extend these two results to expansions of elements in the completion of a global function field under a nontrivial valuation. As application of our generalization of Christol's theorem, we answer some questions about $\beta$-expansions of formal Laurent series over finite fields.

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Chunlin Wang. 2024-04-14. Expansions in completions of global function fields. https://arxiv.org/abs/2404.09175

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