Diagonalization and rationalization of algebraic Laurent series
[Diagonalisation et rationalisation des séries algébriques de Laurent]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 6, pp. 963-1004.

Nous démontrons une version quantitative d’un résultat de Furstenberg [20] et Deligne [14] : la diagonale d’une série formelle algébrique de plusieurs variables à coefficients dans un corps de caractéristique non nulle est une série formelle algébrique d’une variable. Comme conséquence, nous obtenons que, pour tout nombre premier p, la réduction modulo p de la diagonale d’une série formelle algébrique de plusieurs variables f à coefficients entiers est une série formelle algébrique de degré au plus p A et de hauteur au plus Ap A , où A est une constante effective ne dépendant que du nombre de variables, du degré de f et de la hauteur de f. Cela répond à une question soulevée par Deligne [14].

We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime p the reduction modulo p of the diagonal of a multivariate algebraic power series f with integer coefficients is an algebraic power series of degree at most p A and height at most Ap A , where A is an effective constant that only depends on the number of variables, the degree of f and the height of f. This answers a question raised by Deligne [14].

DOI : 10.24033/asens.2207
Classification : 13F25, 11B85, 11J85, 11T99, 34M99, 05A15, 33E99
Keywords: diagonals of algebraic functions, formal power series, multivariate Laurent series, G-functions, reduction modulo $p$
Mot clés : diagonales de fonctions algébriques, séries formelles, séries de Laurent à plusieurs variables, G-fonctions, réduction modulo $p$
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     author = {Adamczewski, Boris and Bell, Jason P.},
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Adamczewski, Boris; Bell, Jason P. Diagonalization and rationalization of algebraic Laurent series. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 6, pp. 963-1004. doi : 10.24033/asens.2207. http://www.numdam.org/articles/10.24033/asens.2207/

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