论文标题
Mahler方法中Nishioka定理的新证明
A new proof of Nishioka's theorem in Mahler's method
论文作者
论文摘要
在最近的一项工作[3]中，作者从先验数理论的角度（例如Nishioka定理的多元扩展）建立了有关通用线性Mahler系统的新结果。使用多个变量的功能和不同的Mahler转换的功能会导致许多并发症，包括需要证明一般消失的定理，并使用来自Ergodic Ramsey Theory和Diophantine近似的工具（例如，$ P $ -P $ -ADIC SCHMIDT suppace sispace aspace Theorem）。这些并发症使得[3]中证明了主要结果的证明相当复杂。在本文中，我们将在一个变量的线性Mahler系统的特殊情况下描述我们的新方法。这导致了Nishioka定理的新，基础和独立的证据，以及菲利普[22]和作者[1]获得的提升定理的提升定理。尽管一般策略与[3]中的一般策略保持不变，但证明被大大简化。除了其自身的兴趣之外，我们希望阅读本文将有助于理解[3]中获得的主要结果的证明。
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 [22] 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].
