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Suites automatiques à multi-indices et algébricité. (Automatic multi-sequences and algebraicity). (French) Zbl 0628.10007
From the author’s abstract: “We give a generalization of the theorem of G. Christol, T. Kamae, M. Mendes-France and G. Rauzy [Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)]: A multisequence with k indices and with values in a finite field K with characteristic p is p-automatic if and only if the associated formal power series is algebraic over the field $$K(X_ 1,...,X_ k)$$. This leads to an easy proof of a theorem of Deligne: If a formal double power series is algebraic over K(X,Y), then its diagonal is algebraic over K(x).”
Proofs appeared in “Suites automatiques à multi-indices, Sémin. Théorie Nombres, Bordeaux 1986/87, Exposé No.4”.
Reviewer: G.Christol

##### MSC:
 11A63 Radix representation; digital problems 68Q45 Formal languages and automata