Diagonalization and rationalization of algebraic Laurent series

Adamczewski, Boris; Bell, Jason P.

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 $A{p}^A$, 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 $A{p}^A$, 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].

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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