M\'ethode de Mahler: relations lin\'eaires, transcendance et applications aux nombres automatiques

TL;DR

This paper advances Mahler's method by analyzing linear relations between Mahler function values at algebraic points, providing criteria for transcendence, and applying results to digit sequences of algebraic numbers.

Contribution

It refines and simplifies Philippon's theorem on Mahler functions, establishing criteria for transcendence and algebraic relations at algebraic points.

Findings

Determines when Mahler function values are transcendental

Shows Mahler function values at algebraic points belong to specific number fields

Applies results to digit sequences of algebraic numbers

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.