- Title
- Algebraic independence of Mahler functions via radial asymptotics
- Creator
- Brent, Richard P.; Coons, Michael; Zudilin, Wadim
- Relation
- ARC.DP140101417 | ARC|DE140100223 | ARC|DP140101186
- Relation
- International Mathematics Research Notices Vol. 2016, Issue 2, p. 571-603
- Publisher Link
- http://dx.doi.org/10.1093/imrn/rnv139
- Publisher
- Oxford University Press
- Resource Type
- journal article
- Date
- 2016
- Description
- 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.
- Subject
- algebraic independence; Mahler functions; radial asymptotics
- Identifier
- http://hdl.handle.net/1959.13/1319805
- Identifier
- uon:23973
- Identifier
- ISSN:1073-7928
- Rights
- 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.
- Language
- eng
- Full Text
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