## Automata and the arithmetic of formal power series.(English)Zbl 0599.12020

Let $$f=f(X)=\sum^{\infty}_{h=0}f_ h X^ h$$ be a formal power series with coefficients in a field $${\mathcal F}$$ of characteristic p, and let $$\lambda =\sum^{\infty}_{k=0}\lambda_ k p^ k$$, $$\lambda_ k\in \{0,1,...,p-1\}$$, be a p-adic integer. Then we set, by definition, $f^{\lambda}=\prod^{\infty}_{k=0}(1+(f-1)^{p^ k})^{\lambda_ k}=\sum^{\infty}_{n=0}\left( \begin{matrix} \lambda \\ n\end{matrix} \right) (f-1)^ n,$ where for each rational integer $$n=\sum^{\infty}_{k=0}n_ k p^ k,\left( \begin{matrix} \lambda \\ n\end{matrix} \right)=\prod^{\infty}_{k=0}\left( \begin{matrix} \lambda_ k\\ n_ k\end{matrix} \right).$$The authors give an elegant proof of the result, which they conjectured several years ago, that if $$f_ 0=1$$ and $$f\neq f_ 0$$ and if both f and $$f^{\lambda}$$ are algebraic over the field $${\mathcal F}(X)$$ of rational functions in X with coefficients in $${\mathcal F}$$, then $$\lambda$$ is rational. A significant feature of the proof is in the fact that this result is established by appealing to the theory of finite automata.
Reviewer: S.Uchiyama

### MSC:

 11S85 Other nonanalytic theory 13F25 Formal power series rings 68Q70 Algebraic theory of languages and automata

### Keywords:

formal power series; finite automata
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