posted on 2025-05-11, 11:33authored byRichard P. Brent, Michael Coons, Wadim Zudilin
We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behavior 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+z2)F(z4) +z4F(z16)=0. Specifically, we prove that the functions F(z), F(z4), F′(z), and F′(z4) are algebraically independent over ℂ(z). An application of a celebrated result of Ku. Nishioka then allows one to replace ℂ(z) by ℚ when evaluating these functions at a nonzero algebraic number α in the unit disc.
Funding
ARC
DP140101417
DE140100223
DP140101186
History
Journal title
International Mathematics Research Notices
Volume
2016
Issue
2
Pagination
571-603
Publisher
Oxford University Press
Place published
Oxford
Language
en, English
College/Research Centre
Faculty of Science and Information Technology
School
School of Mathematical and Physical Sciences
Rights statement
This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Brent, Richard P.; Coons, Michael; Zudilin, Wadim. “Algebraic independence of Mahler functions via radial asymptotics”, International Mathematics Research Notices Vol. 2016, Issue 2, p. 571-603 (2016) is available online at: http://dx.doi.org/10.1093/imrn/rnv139.