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TRANSCENDENCE OVER MEROMORPHIC FUNCTIONS

Part of: Series expansions Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  13 March 2017

MICHAEL COONS  and
YOHEI TACHIYA
MICHAEL COONS*
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, Australia email Michael.Coons@newcastle.edu.au
YOHEI TACHIYA
Affiliation:
Graduate School of Science and Technology, Hirosaki University, Hirosaki, Japan email tachiya@hirosaki-u.ac.jp
*
✉ Michael.Coons@newcastle.edu.au
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Abstract

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In this short note, considering functions, we show that taking an asymptotic viewpoint allows one to prove strong transcendence statements in many general situations. In particular, as a consequence of a more general result, we show that if $F(z)\in \mathbb{C}[[z]]$ is a power series with coefficients from a finite set, then $F(z)$ is either rational or it is transcendental over the field of meromorphic functions.

Keywords

transcendencealgebraic independenceradial asymptotics

MSC classification

Primary: 11J81: Transcendence (general theory)
Secondary: 11J91: Transcendence theory of other special functions 30B30: Boundary behavior of power series, over-convergence

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 3 , June 2017 , pp. 393 - 399
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The research of M. Coons was supported by ARC grant DE140100223 and the research of Y. Tachiya was supported by JSPS, Grant-in-Aid for Young Scientists (B), 15K17504.

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