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

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