Bounds for the size of integral solutions to Y-m=f(X)

Let K be an algebraic number field with ring of integers O-K and f(X) is an element of O-K[X]. In this paper we establish improved explicit upper boun ds for the size of solutions in O-K, of diophantine equations Y-2 = f(X), w here f(X) has at least three roots of odd order, and Y-m = f(X), where m is an integer greater than or equal to 3 and f(X) has at least two roots of o rder prime to m.