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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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https://tel.archives-ouvertes.fr/tel-00009653
Contributor : Samy Khémira <>
Submitted on : Sunday, July 3, 2005 - 3:58:28 PM
Last modification on : Friday, May 29, 2020 - 3:59:51 PM
Long-term archiving on: : Friday, April 2, 2010 - 10:30:59 PM

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• HAL Id : tel-00009653, version 1

### Citation

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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