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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures
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by Evan Chen PDF
Proc. Amer. Math. Soc. 146 (2018), 4189-4198 Request permission

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
  • Email: evanchen@mit.edu
  • 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
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 4189-4198
  • MSC (2010): Primary 11R18, 37F10
  • DOI: https://doi.org/10.1090/proc/14115
  • MathSciNet review: 3834649