A new proof of Nishioka’s theorem in Mahler’s method

Boris Adamczewski; Colin Faverjon

doi:10.5802/crmath.458

Théorie des nombres

A new proof of Nishioka’s theorem in Mahler’s method

Boris Adamczewski ¹ ; Colin Faverjon ¹

¹ Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 69622 Villeurbanne Cedex, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1011-1028.

Résumé

In a recent work [3], the authors established new results about general linear Mahler systems in several variables from the perspective of transcendental number theory, such as a multivariate extension of Nishioka’s theorem. Working with functions of several variables and with different Mahler transformations leads to a number of complications, including the need to prove a general vanishing theorem and to use tools from ergodic Ramsey theory and Diophantine approximation (e.g., a variant of the $p$ -adic Schmidt subspace theorem). These complications make the proof of the main results proved in [3] rather intricate. In this article, we describe our new approach in the special case of linear Mahler systems in one variable. This leads to a new, elementary, and self-contained proof of Nishioka’s theorem, as well as of the lifting theorem more recently obtained by Philippon [23] and the authors [1]. Though the general strategy remains the same as in [3], the proof turns out to be greatly simplified. Beyond its own interest, we hope that reading this article will facilitate the understanding of the proof of the main results obtained in [3].

Reçu le : 2022-10-25
Révisé le : 2022-12-15
Accepté le : 2022-12-19
Publié le : 2023-09-07

DOI : 10.5802/crmath.458

Affiliations des auteurs :

Boris Adamczewski ¹ ; Colin Faverjon ¹

¹ Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 69622 Villeurbanne Cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Boris Adamczewski; Colin Faverjon. A new proof of Nishioka’s theorem in Mahler’s method. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1011-1028. doi : 10.5802/crmath.458. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.458/

Bibliographie
Cité par

[1] Boris Adamczewski; Colin Faverjon Méthode de Mahler: relations linéaires, transcendance et applications aux nombres automatiques, Proc. Lond. Math. Soc., Volume 115 (2017) no. 1, pp. 55-90 | DOI | Zbl

[2] Boris Adamczewski; Colin Faverjon Méthode de Mahler, transcendance et relations linéaires: aspects effectifs, J. Théor. Nombres Bordeaux, Volume 30 (2018) no. 2, pp. 557-573 | DOI | Numdam | Zbl

[3] Boris Adamczewski; Colin Faverjon Mahler’s method in several variables and finite automata (2020) (https://arxiv.org/abs/2012.08283)

[4] Boris Adamczewski; Colin Faverjon A new proof of Nishioka’s theorem in Mahler’s method (2022) (https://arxiv.org/abs/2210.14528)

[5] Yves André Séries Gevrey de type arithmétique I. Théorèmes de pureté et de dualité, Ann. Math., Volume 151 (2000) no. 2, pp. 705-740 | DOI | Zbl

[6] Yves André Séries Gevrey de type arithmétique II. Transcendance sans transcendance, Ann. Math., Volume 151 (2000) no. 2, pp. 741-756 | DOI | Zbl

[7] Yves André Solution algebras of differential equations and quasi-homogeneous varieties $:$ a new differential Galois correspondence, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 2, pp. 449-467 | DOI | MR | Zbl

[8] Frits Beukers A refined version of the Siegel–Shidlovskii theorem, Ann. Math., Volume 163 (2006) no. 1, pp. 369-379 | DOI | MR | Zbl

[9] Philippe Dumas Récurrences mahlériennes, suites automatiques, études asymptotiques, Ph. D. Thesis, Université Bordeaux I, Talence (1993) (https://theses.hal.science/tel-00614660) | MR

[10] Naum Il’ich Fel’dman; Yuriĭ V. Nesterenko Transcendental numbers. Number theory IV, Encyclopaedia of Mathematical Sciences, 44, Springer, 1998

[11] Gwladys Fernandes Méthode de Mahler en caractéristique non nulle: un analogue du théorème de Ku. Nishioka, Ann. Inst. Fourier, Volume 68 (2018) no. 6, pp. 2553-2580 | DOI | Numdam | MR | Zbl

[12] Roger A. Horn; Charles R. Johnson Topics in matrix analysis, Cambridge University Press, 1994

[13] Kenneth K. Kubota On the algebraic independence of holomorphic solutions of certain functional equations and their values, Math. Ann., Volume 227 (1977), pp. 9-50 | DOI | MR | Zbl

[14] Serge Lang Algebra, 3rd revised ed, Graduate Texts in Mathematics, 211, Springer, 2002 | Numdam

[15] John H. Loxton; Alfred J. van der Poorten Arithmetic properties of the solutions of a class of functional equations, J. Reine Angew. Math., Volume 330 (1982), pp. 159-172 | MR | Zbl

[16] Kurt Mahler Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen, Math. Z., Volume 32 (1930), pp. 545-585 | DOI | MR | Zbl

[17] Levente Nagy; Tamás Szamuely A general theory of André’s solution algebras, Ann. Inst. Fourier, Volume 70 (2020) no. 5, pp. 2003-2129 | Numdam | Zbl

[18] Yuriĭ V. Nesterenko Estimate of the orders of the zeroes of functions of a certain class, and their application in the theory of transcendental numbers, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 41 (1977), pp. 253-284 | MR

[19] Kumiko Nishioka New approach in Mahler’s method, J. Reine Angew. Math., Volume 407 (1990), pp. 202-219 | MR | Zbl

[20] Kumiko Nishioka Algebraic independence by Mahler’s method and S-unit equations, Compos. Math., Volume 92 (1994) no. 1, pp. 87-110 | Numdam | MR | Zbl

[21] Kumiko Nishioka Algebraic independence of Mahler functions and their values, Tôhoku Math. J., Volume 48 (1996) no. 1, pp. 51-70 | MR | Zbl

[22] Patrice Philippon Critères pour l’indépendance algébrique, Publ. Math., Inst. Hautes Étud. Sci., Volume 64 (1986), pp. 5-52 | DOI | Numdam | Zbl

[23] Patrice Philippon Groupes de Galois et nombres automatiques, J. Lond. Math. Soc., Volume 92 (2015) no. 3, pp. 596-614 | DOI | MR | Zbl

[24] Bernard Randé Équations fonctionnelles de Mahler et applications aux suites $p$ -régulières, Ph. D. Thesis, Université Bordeaux I, Talence (1992) (https://theses.hal.science/tel-01183330)

[25] A. B. Shidlovskij Transcendental numbers, De Gruyter Studies in Mathematics, 12, Walter de Gruyter, 1989 | DOI

[26] Michel Waldschmidt Diophantine approximation on linear algebraic groups. Transcendence properties of the exponential function in several variables, Grundlehren der Mathematischen Wissenschaften, 326, Springer, 2000 | DOI | Numdam

[27] Oscar Zariski; Pierre Samuel Commutative algebra II, Graduate Texts in Mathematics, 29, Springer, 1976

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