FAMILIES OF THUE EQUATIONS ASSOCIATED WITH A RANK ONE SUBGROUP OF THE UNIT GROUP OF A NUMBER FIELD

Let be an algebraic number field of degree , the embeddings of into , a non-zero element in , , and Let be a unit in . For , we twist the binary form by the powers ( ) of by setting Given , our main result is an effective upper bound for the size of solutions of the Diophantine inequalities for which and . Our estimate is explicit in terms of its dependence on , the regulator of and the heights of and of ; it also involves an effectively computable constant depending only on .