Mahler’s method, transcendence and linear relations: effective aspects. (Méthode de Mahler, transcendance et relations linéaires: aspects effectifs.) (French. English summary) Zbl 1441.11179

Summary: This note is concerned with algorithmic aspects of Mahler’s method. In a recent paper, we used a result of Philippon to prove that, given a \(q\)-mahler function \(f(z)\) in \(\mathbf{k} \{z \}\), where \(\mathbf{k}\) is a number field, and an algebraic number \(\alpha\) in the domain of holomorphy of \(f\), the number \(f(\alpha)\) either belongs to the number field \(\mathbf{k}(\alpha)\) or is transcendental. We describe here an algorithm which allows one to decide between these alternative facts. More generally, given several \(q\)-mahler functions \(f_1(z), \cdots, f_r(z)\) and an algebraic number \(\alpha\) lying in the domain of holomorphy of each \(f_i\), we show how to explicitly compute a basis of the vector space of the linear dependence relations over \(\mathbb Q\) between the numbers \(f_1(\alpha), \cdots, f_r(\alpha)\).


11J81 Transcendence (general theory)
11J72 Irrationality; linear independence over a field
Full Text: DOI


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