# 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 $F(z)-(1+z+z^2)F(z^4)+z^4F(z… 15 Citations ## Figures and Tables from this paper • Mathematics Transactions of the American Mathematical Society • 2019 Becker's conjecture is proved in the best-possible form; it is shown that the rational function R(z) can be taken to be a polynomial for some explicit non-negative integer$\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 ## References SHOWING 1-10 OF 38 REFERENCES Using Mahler’s transcendence method, results on the algebraic independence over $$\mathbb{Q}$$ of the numbers A q (z) are proved at algebraic points α with 0 < | α | < 1. It is shown that if the zeros λ 1 , λ 2 ,..., λ n of the polynomial q(λ) = λ n + a 1 λ n−1 +... + a n are distinct and r is an integer in {1, 2,..., n} such that |λ s | ¬= |λ r | if s ¬= r, then the • Mathematics • 2013 Very recently, the generating function A(z) of the Stern sequence (an)n≥0, defined by a0 := 0, a1 := 1, and a2n := an, a2n+1 := an + an+1 for any integer n > 0, has been considered from the denoted subsequently as usual by [a, a2, a4, . . . , a2 n , . . .], is transcendental. This is a consequence of Roth’s theorem and follows directly from a result of Davenport and Roth [5] concerning • Mathematics • 2013 Let$K$be a field of characteristic zero and$k$and$l\$ be two multiplicatively independent positive integers. We prove the following result that was conjectured by Loxton and van der Poorten
In this same journal, Coons published recently a paper [The transcendence of series related to Stern's diatomic sequence, Int. J. Number Theory6 (2010) 211–217] on the function theoretical
One of the most important motivations to write this paper originates the paper [3] by Loxton and van der Poorten in which they study algebraic independence of the values of Mahler functions. The
• E. N.
• Mathematics
Nature
• 1934
THE last edition of Boole's “Finite Differences” appeared in 1880, and was in fact a reprint of the edition of 1872. The interval of sixty years has seen in the elementary field Sheppard's
(1.1) y(n + 1) = Jy(n) + f(n, y(n)), where y is a d-vector, J is a constant d x d matrix and f(n, y) is a vector-valued function which is continuous in y for fixed n and becomes "small" in some sense
• Mathematics
• 2013
— Let K be a field of characteristic zero and k and l be two multiplicatively independent positive integers. We prove the following result that was conjectured by Loxton and van der Poorten during