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Theses

Méthode de Mahler en caractéristique non nulle.

Abstract : This thesis is part of Number Theory. It deals with transcendence and algebraic independence of values of Mahler functions over function fields of characteristic p>0. The starting point of this thesis is to prove the equivalence between algebraic independence of values of Mahler functions at algebraic points and that of the functions themselves. One of our main motivations is the fruitful observation due to L. Denis that it is possible to reach special numbers (periods of Drinfeld modules) as values of Mahler functions in positive characteristic. We show that every homogeneous algebraic relation between values of solutions of Mahler systems, which generate regular extensions, at nonzero algebraic regular numbers, arises as specialization of an homogeneous algebraic relation between the functions themselves. This is the analogue of the work of P. Philippon and B. Adamczewski and C. Faverjon, and a refinement of a fundamental theorem from Ku. Nishioka, when K is a number field. Thus, the study of algebraic independence between values of Mahler functions turns into that between the functions themselves. But algebraic Mahler functions over function fields of positive characteristic are not necessarily rational, contrary to the number fields case. Transcendence of Mahler functions in this framework still remains mysterious. Nevertheless, we state that this dichotomy is still valid for d-Mahler functions, when p does not divide d. Moreover, we prove a Kolchin theorem that provides a sufficient condition for algebraic independence of inhomogeneous Mahler functions of order 1, along with a characterization of the transcendence of such functions. Finally, we are interested in algebraic independence measures of values of Mahler functions in positive characteristic. We suggest an approach, based on a recent work of E. Zorin in characteristic zero, which would give such qualitative results in our context
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https://tel.archives-ouvertes.fr/tel-02386667
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Submitted on : Friday, November 29, 2019 - 2:06:06 PM
Last modification on : Wednesday, July 8, 2020 - 12:43:25 PM

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  • HAL Id : tel-02386667, version 1

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Gwladys Fernandes. Méthode de Mahler en caractéristique non nulle.. Mathématiques générales [math.GM]. Université de Lyon, 2019. Français. ⟨NNT : 2019LYSE1078⟩. ⟨tel-02386667⟩

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