## 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

### Keywords:

Mahler’s method; transcendence; linear independence
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### References:

  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  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  Jason P. Bell & Michael Coons, Transcendence tests for Mahler functions, Proc. Am. Math. Soc.145 (2017), p. 1061-107 · Zbl 1365.11092  Philippe Dumas, Récurrences mahlériennes, suites automatiques, études asymptotique Mathématiques, Ph. D. Thesis, Université de Bordeaux I (France), 1993  Patrice Philippon, Groupes de Galois et nombres automatiques, J. Lond. Math. Soc.92 (2015), p. 596-614 · Zbl 1391.11087  Bernard Randé, Equations fonctionnelles de Mahler et applications aux suites $$p$$-régulières, Ph. D. Thesis, Université de Bordeaux I (France), 1992 · Zbl 0795.11010
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