# Algebraic independence of Mahler functions via radial asymptotics

@article{Brent2014AlgebraicIO,
title={Algebraic independence of Mahler functions via radial asymptotics},
author={Richard P. Brent and Michael Coons and Wadim Zudilin},
journal={arXiv: Number Theory},
year={2014}
}
• Published 26 December 2014
• Mathematics
• arXiv: Number Theory
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 $F(z)-(1+z+z^2)F(z^4)+z^4F(z… 15 Citations • Mathematics Transactions of the American Mathematical Society • 2019 Becker's conjecture is proved in the best-possible form; it is shown that the rational function R(z) can be taken to be a polynomial for some explicit non-negative integer$\gamma$and such that$1/Q (z)$is$k$-regular. We provide a general result for the algebraic independence of Mahler functions by a new method based on asymptotic analysis. As a consequence of our method, these results hold not only over • Mathematics • 2015 We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue$\lambda_F$of a Mahler function$F(z)$, and develop a quick test for the • Mathematics • 2015 This paper is devoted to the so‐called Mahler method. We precisely describe the structure of linear relations between values at algebraic points of Mahler functions. Given a number field k , a Mahler • Mathematics, Philosophy • 2018 a(z)b(z) = p(z)b(z)a(z). Since a(z) and b(z) are coprime, it follows that a(z) | a(z) giving a(z) = c ∈ C×. Thus p(z) = b(z)/b(z). On the other hand, if p(z) is of this form, then F (z) = 1/b(z) is a • Mathematics • 2020 We show that missing$q$-ary digit sets$F\subseteq[0,1]$have corresponding naturally associated countable binary$q$-automatic sequence$f\$. Using this correspondence, we show that the Hausdorff
This paper associates a regular sequence---in the sense of Allouche and Shallit---and establishes various properties and results concerning the generating function of the regular sequence.
• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2018
We estimate the linear independence measures for the values of a class of Mahler functions of degrees 1 and 2. For this purpose, we study the determinants of suitable Hermite–Padé approximation
α−4 + 1 α−8 + · · · and with the algebraic independence of the numbers f(α), f′(α), f”(α), . . .. Here, α denotes again an algebraic number with 0 < |α| < 1. Moreover, examples of this kind can be
• Marina Poulet
• Mathematics
International Mathematics Research Notices
• 2021
The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices