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Approximants de Hermite-Padé, déterminants d'interpolation et approximation diophantienne

Abstract : This thesis deals with transcendence and diophantine approximation related to the values of exponential functions. In the first part, we prove various relationships between coefficients of Hermite-Padé approximants, Hermite polynomials and certain cofactors of a generalized Vandermonde determinant. We then use the concept of the height of a matrix (which we bound using the relationships established in the first part) to give a new proof of the transcendence of $e$. These results then enable us to obtain new results in diophantine approximation such as a lower bound on the distance between the exponential of an algebraic number to another algebraic number in terms of a bound on the respective absolute logarithmic Weil heights. Finally, we give some exceptional estimates, such as a minimum for $|e^(b)-a^(c)|$ ($b$ and $c$ non negative integers) for various rational numbers $a$.
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Contributor : Samy Khémira <>
Submitted on : Sunday, July 3, 2005 - 3:58:28 PM
Last modification on : Wednesday, December 9, 2020 - 3:10:41 PM
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  • HAL Id : tel-00009653, version 1


Samy Khémira. Approximants de Hermite-Padé, déterminants d'interpolation et approximation diophantienne. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2005. Français. ⟨tel-00009653⟩



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