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Contribution à la méthode de Mahler : équations linéaires et automates finis

Abstract : In this thesis, we investigate topics belonging to number theory, and especially to transcendental number theory. We discuss the results we have obtained in the papers [13, 14, 15, 16]. Our main aim is to study the transcendence and algebraic independence of the values, at algebraic points, of the so-called Mahler functions. The latter are convergent power series satisfying difference equations for the operator z → z^q. In order to study these power series, we develop what is known as Mahler's method, a method initiated by Mahler in the late 1920's. Besides its theoretical interest, Mahler's method has applications regarding the complexity of expansions of real numbers in integer or algebraic bases. Indeed, Mahler functions whose coefficients belong to a finite set are precisely the generating series of automatic sequences. We broaden the scope of Mahler's method, completing the already well-advanced study of the transcendence and linear relations between q-Mahler functions evaluated at a given algebraic point. In particular, we prove a conjecture due to Cobham in 1968, stating that a Mahler function with rational coefficients cannot take algebraic irrational values at rational points. We also show that the algebraic relations between the values of q-Mahler functions at algebraic points all come from specializations of q-orbital functional relations, that is relations between these functions and their images under the iterated action of the map z → z^q. In addition, we establish an algorithm that allows us to determine whether or not an arbitrary Mahler function takes a transcendental value at a given algebraic point. In the second part of the thesis, we develop the theory of multivariate regular singular Mahler systems, a generic class of linear Mahler systems. We obtain a general criterion of algebraic independence for the values at algebraic points of Mahler functions associated with such systems. We could summarize this criterion in the following way: transcendental values of Mahler functions associated with operators having pairwise multiplicatively independent spectral radius, or with multiplicatively independent algebraic points, are always algebraically independent.
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Colin Faverjon. Contribution à la méthode de Mahler : équations linéaires et automates finis. Théorie des nombres [math.NT]. Université Claude Bernard Lyon 1, 2020. Français. ⟨tel-02977792⟩

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