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

DOI: 10.2140/pjm.2015.274.97

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 $\left|S\right|$ 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

Received: 19 June 2014

Revised: 27 August 2014

Accepted: 28 August 2014

Published: 2 March 2015