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, we used a theorem of Philippon to show that given a Mahler function$f(z)$ in \ k\\\, where \ k denotes a number field, and an number \α in the domain of holomorphy of f, the number(\α) is either transcendental or belongs to \ k(\α). We here an effective procedure to decide if such a number is or not. More generally, given several Mahler functions₁(z),\⋯,fᵣ(z) and an algebraic number \α in the domain of of these functions, we show how to effectively determine a basis of vector space of \\ Q̄-linear relations between₁(\α),\⋯,fᵣ(\α).