Hypertranscendence of solutions of Mahler equations

TL;DR

This paper establishes criteria for the algebraic independence of Mahler functions and their derivatives, demonstrating hyperalgebraic independence for specific automatic sequence generating series using parametrized difference Galois theory.

Contribution

It provides new criteria for algebraic independence of Mahler functions and applies them to important automatic sequence series, showing their hyperalgebraic independence.

Findings

Baum-Sweet and Rudin-Shapiro series are hyperalgebraically independent.

Criteria for algebraic independence of Mahler functions are established.

Application of parametrized difference Galois theory to Mahler equations.

Abstract

The last years have seen a growing interest from mathematicians in Mahler functions. This class of functions includes the generating series of the automatic sequences. The present paper is concerned with the following problem, which is omnipresent in combinatorics: a set of Mahler functions $u_{1},...,u_{n}$ being given, are $u_{1},...,u_{n}$ and their successive derivatives algebraically independent? In this paper, we give general criteria ensuring an affirmative answer to this question. We apply our main results to the generating series attached to the so-called Baum-Sweet and Rudin-Shapiro automatic sequences. In particular, we show that these series are hyperalgebraically independent, i.e., that these series and their successive derivatives are algebraically independent. Our approach relies of the parametrized difference Galois theory (in this context, the algebro-differential relations between the solutions of a given Mahler equation are reflected by a linear differential algebraic group).