Algebraic independence of Mahler functions via radial asymptotics

TL;DR

The paper introduces a novel method leveraging radial asymptotics of Mahler functions to establish algebraic independence of their values at algebraic points, demonstrated on a specific degree two Mahler function.

Contribution

It develops a new approach using asymptotic analysis to prove algebraic independence in Mahler's method, applied to a canonical degree two Mahler function.

Findings

Proved algebraic independence of $F(z)$, $F(z^4)$, $F'(z)$, and $F'(z^4)$ over $ ext{C}(z)$.

Extended results to algebraic numbers in the unit disk using Nishioka's theorem.

Introduced a new technique based on radial asymptotics for Mahler functions.

Abstract

We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behaviour of a Mahler function $f(z)$ as $z$ goes radially to a root of unity to deduce algebraic independence results about the values of $f(z)$ at algebraic numbers. We apply our method to the canonical example of a degree two Mahler function; that is, we apply it to $F(z)$, the power series solution to the functional equation $F(z)-(1+z+z^2)F(z^4)+z^4F(z^{16})=0$. Specifically, we prove that the functions $F(z)$, $F(z^4)$, $F'(z)$, and $F'(z^4)$ are algebraically independent over $\mathbb{C}(z)$. An application of a celebrated result of Nishioka then allows one to replace $\mathbb{C}(z)$ by $\mathbb{Q}$ when evaluating these functions at a nonzero algebraic number $\alpha$ in the unit disc.