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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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[1] Artin, M, Algebraic approximation of structures over complete local rings, Inst. hautes etudes sci. publ. math., 36, 23-58, (1968) · Zbl 0181.48802
[2] Christol, G, Eléments analytiques uniformes et multiformes, (), No. 6 · Zbl 0328.12013
[3] Christol, G, Fonctions et éléments algébriques, Pacific J. math., 125, 1-37, (1986) · Zbl 0591.12018
[4] Christol, G; Kamae, T; Mendès France, M; Rauzy, G, Suites algébriques, automates et substitutions, Bull. soc. math. France, 108, 401-409, (1980) · Zbl 0472.10035
[5] Deligne, P, Intégration sur un cycle évanescant, Invent. math., 76, 129-143, (1984) · Zbl 0538.13007
[6] Furstenberg, H, Algebraic functions over finite fields, J. algebra, 7, 271-277, (1967) · Zbl 0175.03903
[7] Greenberg, M, Lectures on forms in many variables, (1969), Benjamin New York · Zbl 0185.08304
[8] Lang, S, Algebra, (1965), Addison-Wesley Reading, Mass · Zbl 0193.34701
[9] Matsumura, ()
[10] {\scM. Mendès-France and A. van der Poorten}, Automata and the arithmetic of formal power series, preprint.
[11] Minsky, M, Computation: finite and infinite machines, (1967), Prentice-Hall Englewood Cliffs, N.J · Zbl 0195.02402
[12] Raynaud, M, (), Lecture Notes in Mathematics, No. 169
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