Méthode de Mahler, transcendance et relations linéaires : aspects effectifs

Adamczewski, Boris; Faverjon, Colin

doi:10.5802/jtnb.1039

Cited by 16 publications

(20 citation statements)

References 4 publications

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“…Finally, we mention that analogues of all the above mentioned theorems, Theorem included, have been recently proved in the setting of linear Mahler equations (see for statements and references). On the other hand, such results are far from being true for

G

‐functions, also defined and studied by Siegel ; see the introduction of for an historical survey.…”

Section: Introductionmentioning

confidence: 95%

Exceptional values of E-functions at algebraic points

Adamczewski

Rivoal

2018

Bull. London Math. Soc.

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E‐functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with coefficients in double-struckQ¯false(zfalse). They were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function, and studied further by Shidlovskii in 1956. The celebrated Siegel–Shidlovskii theorem deals with the algebraic (in)dependence of values at algebraic points of E‐functions solutions of a differential system. However, somewhat paradoxically, this deep result may fail to decide whether a given E‐function assumes an algebraic or a transcendental value at some given algebraic point. Building upon André's theory of E‐operators, Beukers refined in 2006 the Siegel–Shidlovskii theorem in an optimal way. In this paper, we use Beukers' work to prove the following result: there exists an algorithm which, given a transcendental E‐function f(z) as input, outputs the finite list of all exceptional algebraic points α such that f(α) is also algebraic, together with the corresponding list of values f(α). This result solves the problem of deciding whether values of E‐functions at algebraic points are transcendental.

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G

‐functions, also defined and studied by Siegel ; see the introduction of for an historical survey.…”

Section: Introductionmentioning

confidence: 95%

Exceptional values of E-functions at algebraic points

Adamczewski

Rivoal

2018

Bull. London Math. Soc.

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“…We shall simply mention that, quite recently, Philippon [Phi15] proved a refinement of Nishioka's analogue of the Siegel-Shidlovski theorem, in the spirit of Beukers' refinement of the Siegel-Shidlovski theorem [Beu06]. See also [AF16,AF17]. Roughly speaking, it says that the algebraic relations over Q between the above-mentioned special values come from algebraic relations over Q(z) between the functions themselves.…”

Section: Introductionmentioning

confidence: 99%

Hypertranscendence of solutions of Mahler equations

2018

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The last years have seen a growing interest from mathematicians in Mahler functions. This class of functions includes the generating series of the automatic sequences. The present paper is concerned with the following problem, which is rather frequently encountered in combinatorics: a set of Mahler functions u 1 , ..., un being given, are u 1 , ..., un and their successive derivatives algebraically independent? In this paper, we give general criteria ensuring an affirmative answer to this question. We apply our main results to the generating series attached to the so-called Baum-Sweet and Rudin-Shapiro automatic sequences. In particular, we show that these series are hyperalgebraically independent, i.e., that these series and their successive derivatives are algebraically independent. Our approach relies of the parametrized difference Galois theory (in this context, the algebro-differential relations between the solutions of a given Mahler equation are reflected by a linear differential algebraic group). 14 3.

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“…Any d-Mahler function is a coordinate of a vector solution of the d-Mahler system associated with the companion matrix of (1). Reciprocally, every coordinate of a vector solution of a d-Mahler system is a d-Mahler function.…”

Section: Introductionmentioning

confidence: 99%

“…. , f n (z) can be explicitly computed [2,1]. The arguments used by B. Adamczewski and C. Faverjon to obtain this result belong to linear algebra and might fit for function fields.…”

Section: Introductionmentioning

confidence: 99%

Regular extensions and algebraic relations between values of Mahler functions in positive characteristic

Fernandes¹

2019

Trans. Amer. Math. Soc.

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Let K be a function field of characteristic p > 0. We recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over K(z). This paper is dedicated to proving the following refinement of this theorem. Let f 1 (z), . . . fn(z) be d-Mahler functions such that K(z) (f 1 (z), . . . , fn(z)) is a regular extension over K(z). Then, every homogeneous algebraic relation over K between their values at a regular algebraic point arises as the specialization of a homogeneous algebraic relation over K(z) between these functions themselves.If K is replaced by a number field, this result is due to B. Adamczewski and C. Faverjon, as a consequence of a theorem of P. Philippon. The main difference is that in characteristic zero, every d-Mahler extension is regular, whereas, in characteristic p, non-regular d-Mahler extensions do exist. Furthermore, we prove that the regularity of the field extension K(z) (f 1 (z), . . . , fn(z)) is also necessary for our refinement to hold. Besides, we show that, when p d, d-Mahler extensions over K(z) are always regular. Finally, we describe some consequences of our main result concerning the transcendence of values of d-Mahler functions at algebraic points. arXiv:1808.00719v1 [math.NT] 2 Aug 2018 1. E is separable over k. That is, there exists a transcendence basis F of E over k such that E is a separable algebraic extension of k(F ) (see [9, Appendix A1.2] and also [17]).2. Every element of E that is algebraic over k belongs to k.

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Méthode de Mahler, transcendance et relations linéaires : aspects effectifs

Cited by 16 publications

References 4 publications

Exceptional values of E-functions at algebraic points

Exceptional values of E-functions at algebraic points

Hypertranscendence of solutions of Mahler equations

Regular extensions and algebraic relations between values of Mahler functions in positive characteristic

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