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Article Dans Une Revue Ann. Sci. Éc. Norm. Supér. Année : 2013

Diagonalization and Rationalization of algebraic Laurent series

Boris Adamczewski
Jason P. Bell
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Résumé

We prove a quantitative version of a result of Furstenberg and Deligne stating that the 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 $A^2p^{A+1}$, 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.
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hal-00698706 , version 1 (17-05-2012)

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Boris Adamczewski, Jason P. Bell. Diagonalization and Rationalization of algebraic Laurent series. Ann. Sci. Éc. Norm. Supér., 2013, 46 (6), 49 pp. ⟨hal-00698706⟩
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