Algebraic power series and diagonals (Q1086624)

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 \(s\).