Algebraic power series and diagonals. (English) Zbl 0609.12020

This paper contains two parts. In the first it is proved that a power series in several variables over the \(p\)-adic integers \(\mathbb Z_p\) is congruent mod \(p^s\) to an algebraic power series if and only if its coefficients satisfy certain congruences mod \(p^s\). These congruences can be expressed in terms of finite automata. The deepest result is proved in the second part: any algebraic power series in \(m\) variables over a field can be written as the diagonal of a rational power series in \(2m\) variables. The proof uses Furstenberg technique.
As an application one gets an elementary proof of a result of Deligne: the diagonal of an algebraic power series in several variables over a field of non-zero characteristic is algebraic. It is also proved that the diagonal of an algebraic power series over \(\mathbb Z_p\) satisfies the congruences of the first part for all \(s\).


11S85 Other nonanalytic theory
13J05 Power series rings
11R04 Algebraic numbers; rings of algebraic integers
68Q45 Formal languages and automata
13J15 Henselian rings
11J81 Transcendence (general theory)
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