×

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 \(f_1(\alpha), \cdots, f_r(\alpha)\).

MSC:

11J81 Transcendence (general theory)
11J72 Irrationality; linear independence over a field

References:

[1] Boris Adamczewski & Jason P. Bell, A problem about Mahler functions, Ann. Sc. Norm. Super. Pisa, Cl. Sci.17 (2017), p. 1301-1355 · Zbl 1432.11086
[2] Boris Adamczewski & Colin Faverjon, Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques, Proc. Lond. Math. Soc.115 (2017), p. 55-90 · Zbl 1440.11132
[3] Jason P. Bell & Michael Coons, Transcendence tests for Mahler functions, Proc. Am. Math. Soc.145 (2017), p. 1061-107 · Zbl 1365.11092
[4] Philippe Dumas, Récurrences mahlériennes, suites automatiques, études asymptotique Mathématiques, Ph. D. Thesis, Université de Bordeaux I (France), 1993
[5] Patrice Philippon, Groupes de Galois et nombres automatiques, J. Lond. Math. Soc.92 (2015), p. 596-614 · Zbl 1391.11087
[6] Bernard Randé, Equations fonctionnelles de Mahler et applications aux suites \(p\)-régulières, Ph. D. Thesis, Université de Bordeaux I (France), 1992 · Zbl 0795.11010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.