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arXiv · 1406.1990

Uniform Boundedness of S-Units in Arithmetic Dynamics

Holly Krieger·Aaron Levin·Zachary Scherr·Thomas J. Tucker·Yu Yasufuku·Michael Zieve
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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 f(z) in K(z) of degree d at least 2 which is not a d-th power in \bar{K}(z), Siegel's theorem implies that the image set f(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 and f). We prove this conjecture for several classes of rational functions, and show that the full conjecture follows from the Bombieri--Lang conjecture.

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Holly Krieger, Aaron Levin, Zachary Scherr, Thomas J. Tucker, Yu Yasufuku, Michael Zieve. 2014-06-08. Uniform Boundedness of S-Units in Arithmetic Dynamics. https://doi.org/10.2140/pjm.2015.274.97

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