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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 -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].
Adamczewski, Boris 1 ; Faverjon, Colin 1
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@article{CRMATH_2023__361_G6_1011_0,
author = {Adamczewski, Boris and Faverjon, Colin},
title = {A new proof of {Nishioka{\textquoteright}s} theorem in {Mahler{\textquoteright}s} method},
journal = {Comptes Rendus. Math\'ematique},
pages = {1011--1028},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
number = {G6},
year = {2023},
doi = {10.5802/crmath.458},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/crmath.458/}
}
TY - JOUR AU - Adamczewski, Boris AU - Faverjon, Colin TI - A new proof of Nishioka’s theorem in Mahler’s method JO - Comptes Rendus. Mathématique PY - 2023 SP - 1011 EP - 1028 VL - 361 IS - G6 PB - Académie des sciences, Paris UR - http://geodesic.mathdoc.fr/articles/10.5802/crmath.458/ DO - 10.5802/crmath.458 LA - en ID - CRMATH_2023__361_G6_1011_0 ER -
%0 Journal Article %A Adamczewski, Boris %A Faverjon, Colin %T A new proof of Nishioka’s theorem in Mahler’s method %J Comptes Rendus. Mathématique %D 2023 %P 1011-1028 %V 361 %N G6 %I Académie des sciences, Paris %U http://geodesic.mathdoc.fr/articles/10.5802/crmath.458/ %R 10.5802/crmath.458 %G en %F CRMATH_2023__361_G6_1011_0
Adamczewski, Boris; Faverjon, Colin. A new proof of Nishioka’s theorem in Mahler’s method. Comptes Rendus. Mathématique, Tome 361 (2023) no. G6, pp. 1011-1028. doi: 10.5802/crmath.458
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