M\'ethode de Mahler, transcendance et relations lin\'eaires : aspects effectifs

This note deals with some effective results in Mahler’s method. In a recent work, we used a theorem of Philippon to show that given a Mahler function $f(z)$ in $\mathbf{k}\{z\}$, where $\mathbf{k}$ denotes a number field, and an algebraic number $\alpha$ in the domain of holomorphy of $f$, the number $f(\alpha )$ is either transcendental or belongs to $\mathbf{k}(\alpha )$. We describe here an effective procedure to decide if such a number is transcendental or not. More generally, given several Mahler functions $f_1(z),\cdots ,f_r(z)$ and an algebraic number $\alpha$ in the domain of holomorphy of these functions, we show how to effectively determine a basis of the vector space of $\bar{\mathbb{Q}}$-linear relations between $f_1(\alpha ),\cdots ,f_r(\alpha )$.