(Both the published version and the arXiv version are available online.)

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. Let h(z) be a rational function in K(z) of degree d ≥ 2, and suppose that h(z) is not a d-th power in K(z). Applying Siegel's theorem to h(z)+1/h(z) shows that the image set h(K) contains only finitely many S-units in K. We conjecture that the number of such S-units is bounded by a function of d and the size of S (independently of h and K). 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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