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Algebraic independence of Mahler functions via radial asymptotics

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posted on 2025-05-11, 11:33 authored by Richard 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.

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