Diagonalization and rationalization of algebraic Laurent series

Boris Adamczewski; Jason P. Bell

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 6, page 963-1004
  • ISSN: 0012-9593

Abstract

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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].

How to cite

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Adamczewski, Boris, and Bell, Jason P.. "Diagonalization and rationalization of algebraic Laurent series." Annales scientifiques de l'École Normale Supérieure 46.6 (2013): 963-1004. <http://eudml.org/doc/272194>.

@article{Adamczewski2013,
abstract = {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].},
author = {Adamczewski, Boris, Bell, Jason P.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {diagonals of algebraic functions; formal power series; multivariate Laurent series; G-functions; reduction modulo $p$},
language = {eng},
number = {6},
pages = {963-1004},
publisher = {Société mathématique de France},
title = {Diagonalization and rationalization of algebraic Laurent series},
url = {http://eudml.org/doc/272194},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Adamczewski, Boris
AU - Bell, Jason P.
TI - Diagonalization and rationalization of algebraic Laurent series
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 6
SP - 963
EP - 1004
AB - 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].
LA - eng
KW - diagonals of algebraic functions; formal power series; multivariate Laurent series; G-functions; reduction modulo $p$
UR - http://eudml.org/doc/272194
ER -

References

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