Transcendence tests for Mahler functions
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- by Jason P. Bell and Michael Coons PDF
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Abstract:
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 transcendence of $F(z)$ over $\mathbb {C}(z)$, which is determined by the value of the eigenvalue $\lambda _F$. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of $F(z)$. We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given.References
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Additional Information
- Jason P. Bell
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- MR Author ID: 632303
- Email: jpbell@uwaterloo.ca
- Michael Coons
- Affiliation: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, New South Wales 2308, Australia
- MR Author ID: 857151
- Email: Michael.Coons@newcastle.edu.au
- Received by editor(s): November 1, 2015
- Received by editor(s) in revised form: May 17, 2016
- Published electronically: September 15, 2016
- Additional Notes: The research of the first author was supported by NSERC grant 31-611456
The research of the second author was supported by ARC grant DE140100223. - Communicated by: Matthew A. Papanikolas
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1061-1070
- MSC (2010): Primary 11J91; Secondary 39A06, 30B30
- DOI: https://doi.org/10.1090/proc/13297
- MathSciNet review: 3589306