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


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