#### Vol. 274, No. 1, 2015

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Uniform boundedness of $S$-units in arithmetic dynamics

### Holly Krieger, Aaron Levin, Zachary Scherr, Thomas Tucker, Yu Yasufuku and Michael E. Zieve

Vol. 274 (2015), No. 1, 97–106
##### Abstract

Let $K$ be a number field and let $S$ be a finite set of places of $K$ which contains all the archimedean places. For any $\varphi \left(z\right)\in K\left(z\right)$ of degree $d\ge 2$ which is not a $d$-th power in $\overline{K}\left(z\right)$, Siegel’s theorem implies that the image set $\varphi \left(K\right)$ contains only finitely many $S$-units. We conjecture that the number of such $S$-units is bounded by a function of $|S|$ and $d$ (independently of $K$, $S$ and $\varphi$). We prove this conjecture for several classes of rational functions, and show that the full conjecture follows from the Bombieri–Lang conjecture.

##### Keywords
arithmetic dynamics, S-units, uniform boundedness
##### Mathematical Subject Classification 2010
Primary: 37P05, 37P15
Secondary: 11G99, 11R99
##### Milestones
Revised: 27 August 2014
Accepted: 28 August 2014
Published: 2 March 2015
##### Authors
 Holly Krieger Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 United States Aaron Levin Department of Mathematics Michigan State University East Lansing, MI 48824 United States Zachary Scherr Department of Mathematics University of Pennsylvania David Rittenhouse Lab Philadelphia, PA 19104–6395 United States Thomas Tucker Department of Mathematics University of Rochester Rochester, NY 14627 United States Yu Yasufuku Department of Mathematics College of Science and Technology Nihon University Chiyoda-ku, Tokyo 101-8308 Japan Michael E. Zieve Mathematical Sciences Center Tsinghua University Beijing, 100084 China Department of Mathematics University of Michigan Ann Arbor, MI 48109–1043 United States