Méthode de Mahler: relations linéaires, transcendance et applications aux nombres automatiques

Abstract: This paper is concerned with Mahler's method. We study in detail the structure of linear relations between values of Mahler functions at algebraic points. In particular, given a field ${\bf k}$, a Mahler function $f(z)\in{\bf k}{z}$, and an algebraic number $\alpha$, $0<\vert \alpha\vert <1$, that is not a pole for $f$, we show that one can always determined whether the number $f(\alpha)$ is transcendental or not. In the latter case, we obtain that $f(\alpha)$ belong to the number fields ${\bf k}(\alpha)$. We also consider some consequences of such results to a classical number theoretical problem: the study of sequences of digits of algebraic numbers in an integer (or, more generally, algebraic) base. Our results are based on a theorem of Philippon [31] that we refine. We also simplify his proof.