Uniform boundedness of S-units in arithmetic dynamics

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