M\'ethode de Mahler en caract{\'e}ristique non nulle : un analogue du Th\'eor\`eme de Ku. Nishioka

TL;DR

This paper extends Ku. Nishioka's Mahler's method theorem to positive characteristic function fields, using Denis's approach and Philippon's algebraic independence criterion, revealing new links between Mahler functions and special values like pi and zeta values.

Contribution

It generalizes Nishioka's theorem for Mahler functions to positive characteristic fields, introducing a new proof approach and connecting Mahler values to classical constants.

Findings

Extension of Nishioka's theorem to positive characteristic

Use of Denis's approach and Philippon's criterion

Identification of Mahler function values with classical constants

Abstract

In 1990, Ku. Nishioka proved a fundamental theorem for Mahler's method, which is the analog of the Siegel-Shidlovskii theorem for Mahler functions. In this article, we establish a version of the theorem of Ku. Nishioka which is also valid for Mahler systems over function fields of positive characteristic. We follow the approach introduced by Denis in 1999 in a particular case. It is based on an algebraic independence criterion from Philippon. The main motivation of this work is built on the following remarkable fact discovered by Denis. Over function fields of positive characteristic, analogs of periods such as \pi or the values at integer points of the Zeta Riemann function can be obtained as values of Mahler functions at algebraic points.