Effective results for hyper- and superelliptic equations over number fields

Let f be a polynomial with coefficients in the ring O-s of S-integers of a given number field K, b a non-zero S-integer, and m an integer >= 2. Suppose that f has no multiple zeros. We consider the equation (*) f(x) = by(m) in x,y is an element of O-s. In the present paper we give explicit upper bounds in terms of K, S, b, f, m for the heights of the solutions of (*). Further, we give an explicit bound C in terms of K, S, b, f such that if m > C then (*) has only solutions with y = 0 or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, and Shorey and Tijdeman. The results in the present paper are needed in a forthcoming paper of ours on Diophantine equations over integral domains which are finitely generated over Z.