Mahler's method in several variables II: Applications to base change problems and finite automata

TL;DR

This paper extends Mahler's method to several variables, providing new results on algebraic independence of Mahler functions at algebraic points and applying these to number base representations, automata theory, and special functions.

Contribution

It introduces a general algebraic independence result for multivariable Mahler functions with dependent spectral radii and applies it to number representation and automata problems.

Findings

Algebraic independence of Mahler functions at algebraic points under certain conditions

Application to base change problems and automata theory in number representation

Results on algebraic independence of Mahler functions and their specializations

Abstract

This is the second part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. From the lifting theorem obtained in the first part, we first derive a general result, showing that Mahler functions in several variables, associated with transformations having multiplicatively dependent spectral radii, take algebraic independent values at algebraic points provided that these points are sufficiently independent. Then, we focus on applications of this result and of the two main results of Part I of this work. Our main application concerns problems about the representation of natural and real numbers in integer bases involving automata theory. These can be translated in terms of algebraic relations over $\overline{\mathbb Q}$ between values of Mahler functions in one variable. We also apply our results to the algebraic independence of Mahler functions and their specializations, and to the study of the values of Hecke-Mahler series.