Comptes Rendus
Number theory
A new proof of Nishioka’s theorem in Mahler’s method
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1011-1028.

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].

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DOI: 10.5802/crmath.458
Boris Adamczewski 1; Colin Faverjon 1

1 Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 69622 Villeurbanne Cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/

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