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Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures


Author: Evan Chen
Journal: Proc. Amer. Math. Soc. 146 (2018), 4189-4198
MSC (2010): Primary 11R18, 37F10
DOI: https://doi.org/10.1090/proc/14115
Published electronically: July 13, 2018
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Abstract: Let $ k$ be a number field with cyclotomic closure $ k^{\mathrm {c}}$, and let $ h \in k^{\mathrm {c}}(x)$. For $ A \ge 1$ a real number, we show that

$\displaystyle \{ \alpha \in k^{\mathrm {c}} : h(\alpha ) \in \overline {\mathbb{Z}}$$\displaystyle \text { has house at most } A \} $

is finite for many $ h$. We also show that for many such $ h$ the same result holds if $ h(\alpha )$ is replaced by orbits $ h(h(\cdots h(\alpha )))$. This generalizes a result proved by Ostafe that concerns avoiding roots of unity, which is the case $ A=1$.

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Additional Information

Evan Chen
Affiliation: Department of Mathematics, Massachusetts Institute of Technology
Email: evanchen@mit.edu

DOI: https://doi.org/10.1090/proc/14115
Keywords: Cyclotomic closure, orbits, rational function
Received by editor(s): October 4, 2016
Received by editor(s) in revised form: November 3, 2017, January 7, 2018, and February 1, 2018
Published electronically: July 13, 2018
Additional Notes: This research was funded by NSF grant 1358659 and NSA grant H98230-16-1-0026 as part of the 2016 Duluth Research Experience for Undergraduates (REU)
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2018 American Mathematical Society

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