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 ${\bf k}\{z\}$, where ${\bf k}$ 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 ${\bf k}(\alpha)$. 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 $\overline{\mathbb Q}$-linear relations between $f_1(\alpha),\cdots,f_r(\alpha)$.