Łukasiewicz language and diagonals of formal series (Q1919150)

\textit{H. Furstenberg} has shown [J. Algebra 7, 271-277 (1967; Zbl 0175.03903)] that every algebraic formal power series in one variable with coefficients in a finite field is the diagonal of a two-variable rational fraction which is moreover explicitly computable. The aim of the paper under review is to give a purely combinatorial proof of this result by using the notion of Łukasiewicz language. More precisely, the author first works with noncommutative series connected with Łukasiewicz languages, then translates the obtained results to algebraic commutative series: this translation gives, for any algebraic formal power series in one variable \(\varphi\) (with coefficients in a finite field and satisfying \(\varphi(0)=0\)), the expression, originally given by Furstenberg in his proof, of a two-variable rational function whose diagonal is \(\varphi\). Let us note that \textit{P. Deligne} has shown ``conversely'' that every diagonal of an algebraic formal power series in several interminates with coefficients in a field of positive characteristic is algebraic [Invent. Math. 76, 129-143 (1984; Zbl 0538.13007)]. For more information on the subject, see the survey of \textit{J.-P. Allouche} [Sémin. Théor. Nombres Bordx., Sér. II 1, No. 1, 163-187 (1989; Zbl 0714.12006)].