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Algebraic independence of local conjugacies and related questions in polynomial dynamics


Author: Khoa D. Nguyen
Journal: Proc. Amer. Math. Soc. 143 (2015), 1491-1499
MSC (2010): Primary 11J91, 37F10; Secondary 37P30
DOI: https://doi.org/10.1090/S0002-9939-2014-12438-5
Published electronically: December 9, 2014
MathSciNet review: 3314064
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Abstract: Let $ K$ be an algebraically closed field of characteristic 0 and $ f\in K[t]$ a polynomial of degree $ d\geq 2$. There exists a local conjugacy $ \psi _f(t)\in tK[[1/t]]$ such that $ \psi _f(t^d)=f(\psi _f(t))$. It has been known that $ \psi _f$ is transcendental over $ K(t)$ if $ f$ is not conjugate to $ t^d$ or a constant multiple of the Chebyshev polynomial. In this paper, we study the algebraic independence of $ \psi _{f_1}$,..., $ \psi _{f_n}$ using a recent result of Medvedev and Scanlon. Related questions in transcendental number theory and canonical heights in arithmetic dynamics are also discussed.


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Additional Information

Khoa D. Nguyen
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, University of British Columbia, and Pacific Institute for the Mathematical Sciences, Vancouver, British Columbia V6T 1Z2, Canada
Email: khoanguyen2511@gmail.com, dknguyen@math.ubc.ca

DOI: https://doi.org/10.1090/S0002-9939-2014-12438-5
Keywords: Local conjugacies, algebraic independence, B\"ottcher coordinates, canonical heights
Received by editor(s): October 8, 2013
Published electronically: December 9, 2014
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society