Uniform Boundedness of S-Units in Arithmetic Dynamics

TL;DR

This paper investigates the boundedness of S-units in the images of rational functions over number fields, proving the conjecture for certain classes and linking the general case to the Bombieri--Lang conjecture.

Contribution

It proves the boundedness conjecture for specific classes of rational functions and connects the general case to a major conjecture in Diophantine geometry.

Findings

Proved the conjecture for several classes of rational functions.

Showed the full conjecture follows from the Bombieri--Lang conjecture.

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.