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

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 ϕ(z) K(z) of degree d 2 which is not a d-th power in K¯(z), Siegel’s theorem implies that the image set ϕ(K) 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 ϕ). We prove this conjecture for several classes of rational functions, and show that the full conjecture follows from the Bombieri–Lang conjecture.

arithmetic dynamics, S-units, uniform boundedness
Mathematical Subject Classification 2010
Primary: 37P05, 37P15
Secondary: 11G99, 11R99
Received: 19 June 2014
Revised: 27 August 2014
Accepted: 28 August 2014
Published: 2 March 2015
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
Michael E. Zieve
Mathematical Sciences Center
Tsinghua University
Beijing, 100084
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109–1043
United States