On the algebraic relations between Mahler functions

Roques, Julien

doi:10.1090/tran/6945

Cited by 19 publications

(18 citation statements)

References 17 publications

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“…Similar, the diagonal blocks in Thus we have shown that y(x p ) = A(x)y(x) is equivalent overK to w(x p ) =Ǎ(x)w(x) where all entries of the coefficient matrixǍ(x) have positive valuation. As shown in the beginning of the proof of Corollary 25 and stated in Proposition 34 of [39], this implies that x = 0 is a regular singular point of the latter system and hence also of the given system y(x p ) = A(x)y(x). This completes the proof of Proposition 26.…”

Section: Proof Of Theorem 13: Case 2mmentioning

confidence: 65%

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Consistent systems of linear differential and difference equations

Schäfke

Singer

2019

J. Eur. Math. Soc.

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We consider systems of linear differential and difference equationswith δ = d dx , σ a shift operator σ(x) = x + a, q-dilation operator σ(x) = qx or Mahler operator σ(x) = x p and systems of two linear difference equationswith (σ 1 , σ 2 ) a sufficiently independent pair of shift operators, pair of q-dilation operators or pair of Mahler operators. Here A(x) and B(x) are n × n matrices with rational function entries. Assuming a consistency hypothesis, we show that such systems can be reduced to a system of a very simple form. Using this we characterize functions satisfying two linear scalar differential or difference equations with respect to these operators. We also indicate how these results have consequences both in the theory of automatic sets, leading to a new proof of Cobham's Theorem, and in the Galois theories of linear difference and differential equations, leading to hypertranscendence results.

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Section: Proof Of Theorem 13: Case 2mmentioning

confidence: 65%

“…We now prove the Proposition by induction. In case n = 1, it has been shown in Proposition 23 of [39] So suppose that the statement has been proved for all dimensions smaller than n. Given A(x), B(x) as in the hypothesis, we now invoke Lemma 27. WhenÃ(x) = dI, there is nothing to do.…”

Section: Proof Of Theorem 13: Case 2mmentioning

confidence: 95%

Consistent systems of linear differential and difference equations

Schäfke

Singer

2019

J. Eur. Math. Soc.

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“…There has been a flurry of recent activity involving the study of Mahler seriessee, e.g. [2,6,7,8,9,13,19,20, 21]-in large part due to the fact that one can often deduce transcendence of special values of Mahler series by knowing transcendence of the series itself, and also due to the guiding principle that much of the theory of Mahler series should mirror the much better developed theory of solutions to homogeneous differential equations.…”

Section: Introductionmentioning

confidence: 99%

Becker’s conjecture on Mahler functions

Bell¹,

Chyzak²,

Coons³

et al. 2019

Trans. Amer. Math. Soc.

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In 1994, Becker conjectured that if F (z) is a k-regular power series, then there exists a k-regular rational function R(z) such that F (z)/R(z) satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies a 0 (z) = 1. In this paper, we prove Becker's conjecture in the best-possible form; we show that the rational function R(z) can be taken to be a polynomial z γ Q(z) for some explicit non-negative integer γ and such that 1/Q(z) is k-regular.

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“…Rappelons que décrire les relations de dépendance algébrique entre les solutions d'un système mahlerien est une tâche ardue, en témoigne le peu de résultats obtenus jusqu'à présent (voir par exemple [, Chap. 5] et plus récemment ). A contrario , nous montrerons que l'espace

{Rel}_{\bar{double-struckQ} false(z false)} false(f_{1} (z), \dots, f_{n} (z) false)

a une description simple.…”

Section: Introductionunclassified

Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques

Adamczewski

Faverjon²

2017

Proc. London Math. Soc.

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This paper is devoted to the so-called Mahler method. We precisely describe the structure of linear relations between values at algebraic points of Mahler functions. Given a number field k, a Mahler function f (z) ∈ k{z}, and an algebraic number α, 0 < |α| < 1, that is not a pole of f , we show that one can always determine whether the number f (α) is transcendental or not. In the latter case, we further obtain that f (α) belongs to the number field k(α). We also discuss some consequences of this theory concerning a classical number theoretical problem: the study of the sequence of digits of the expansion of algebraic numbers in integer bases, or, more generally in algebraic bases. Our results are derived from a recent theorem of Philippon ['Groupes de Galois et nombres automatiques ', J. Lond. Math. Soc. 95 (2015) 596-614] that we refine. We also simplify its proof. RésuméCet article est consacréà la méthode de Mahler. Nous décrivons en détail la structure des relations de dépendance linéaire entre les valeurs aux points algébriques de fonctions mahlériennes. Etant donnés un corps de nombres k, une fonction mahlérienne f (z) ∈ k{z} et α un nombre algébrique, 0 < |α| < 1, qui n'est pas un pôle de f , nous montrons notamment que l'on peut toujours déterminer si le nombre f (α) est transcendant ou non. Dans ce dernier cas, nous obtenons que f (α) appartient nécessairementà l'extension k(α). Nous considéronségalement les conséquences de cette théorie concernant un problème arithmétique classique : l'étude de la suite des chiffres des nombres algébriques dans une base entière ou, plus généralement, algébrique. Nos résultats sont obtenusà partir d'un théorème récent de Philippon ['Groupes de Galois et nombres automatiques ', J. Lond. Math. Soc. 95 (2015) 596-614] que nous raffinons et dont nous simplifions la démonstration.

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On the algebraic relations between Mahler functions

Cited by 19 publications

References 17 publications

Consistent systems of linear differential and difference equations

Consistent systems of linear differential and difference equations

Becker’s conjecture on Mahler functions

Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques

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