Diagonalization and rationalization of algebraic Laurent series
Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 46 (2013) no. 6, p. 963-1004
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].
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].
DOI : https://doi.org/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
@article{ASENS_2013_4_46_6_963_0,
     author = {Adamczewski, Boris and Bell, Jason P.},
     title = {Diagonalization and rationalization of algebraic Laurent series},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
     number = {6},
     year = {2013},
     pages = {963-1004},
     doi = {10.24033/asens.2207},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2013_4_46_6_963_0}
}
Adamczewski, Boris; Bell, Jason P. Diagonalization and rationalization of algebraic Laurent series. Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 46 (2013) no. 6, pp. 963-1004. doi : 10.24033/asens.2207. http://www.numdam.org/item/ASENS_2013_4_46_6_963_0/

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