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Integral points of bounded degree on the projective line and in dynamical orbits


Authors: Joseph Gunther and Wade Hindes
Journal: Proc. Amer. Math. Soc. 145 (2017), 5087-5096
MSC (2010): Primary 11D45, 37P15; Secondary 11G50, 11R04, 14G05
DOI: https://doi.org/10.1090/proc/13653
Published electronically: June 8, 2017
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Abstract: Let $ D$ be a non-empty effective divisor on $ \mathbb{P}^1$. We show that when ordered by height, any set of $ (D,S)$-integral points on $ \mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $ \varphi (z)\in \overline {\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $ \varphi ^2(z)$ is not a polynomial. We show that as we vary over points $ P\in \mathbb{P}^1(\overline {\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $ P$ is absolutely bounded and zero on average.


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Additional Information

Joseph Gunther
Affiliation: Department of Mathematics, The Graduate Center, City University of New York (CUNY), 365 Fifth Avenue, New York, New York 10016
Email: JGunther@gradcenter.cuny.edu

Wade Hindes
Affiliation: Department of Mathematics, The Graduate Center, City University of New York (CUNY), 365 Fifth Avenue, New York, New York 10016
Email: whindes@gc.cuny.edu

DOI: https://doi.org/10.1090/proc/13653
Keywords: Arithmetic dynamics, integral points
Received by editor(s): August 4, 2016
Received by editor(s) in revised form: December 5, 2016, and January 3, 2017
Published electronically: June 8, 2017
Additional Notes: The first author was partially supported by National Science Foundation grant DMS-1301690
Communicated by: Mathew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society

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