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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
Published electronically: July 13, 2018
MathSciNet review: 3834649
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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 \[ \{ \alpha \in k^{\mathrm {c}} : h(\alpha ) \in \overline {\mathbb Z} \text { has house at most } A \} \] 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
MR Author ID: 1158569

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