Galois groups and automata

TL;DR

This paper explores the relationship between Galois groups, automata, and algebraic independence of solutions to functional equations, demonstrating how algebraic relations among functions influence the relations among their specialized values.

Contribution

It establishes that algebraic relations among function solutions are derived from relations between the functions themselves, advancing understanding in Mahler's method.

Findings

Algebraic relations among solutions derive from relations between functions.

Results on linear independence of q-regular functions.

Insights into algebraic independence in Mahler's method.

Abstract

In the frame of Mahler's method for algebraic independence we show that the algebraic relations over Q linking the values of functions solutions of a system of functional equations come from the algebraic relations between the functions themselves, by specialisation. We deduce some results on the linear independence of the q-regular functions.