## 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 $$s$$.

### MSC:

 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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### References:

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