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Article Dans Une Revue Journal of the European Mathematical Society Année : 2018

HYPERTRANSCENDENCE OF SOLUTIONS OF MAHLER EQUATIONS

Julien Roques

Résumé

The last years have seen a growing interest from mathematicians in Mahler functions. This class of functions includes the generating series of the automatic sequences. The present paper is concerned with the following problem, which is rather frequently encountered in combinatorics: a set of Mahler functions u 1 , ..., un being given, are u 1 , ..., un and their successive derivatives algebraically independent? In this paper, we give general criteria ensuring an affirmative answer to this question. We apply our main results to the generating series attached to the so-called Baum-Sweet and Rudin-Shapiro automatic sequences. In particular, we show that these series are hyperalge-braically independent, i.e., that these series and their successive derivatives are algebraically independent. Our approach relies of the parametrized difference Galois theory (in this context, the algebro-differential relations between the solutions of a given Mahler equation are reflected by a linear differential algebraic group).
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Dates et versions

hal-01897318 , version 1 (17-10-2018)

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Thomas Dreyfus, Charlotte Hardouin, Julien Roques. HYPERTRANSCENDENCE OF SOLUTIONS OF MAHLER EQUATIONS. Journal of the European Mathematical Society, 2018, 20 (9), pp.2209 - 2238. ⟨10.4171/JEMS/810⟩. ⟨hal-01897318⟩
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