Uniform bounds for the number of solutions to Yn = f(X)

Abstract

Let K be an algebraic number field and f(X) ∈ K[X]. The Diophantine problem of describing the solutions to equations of the formhas attracted considerable interest over the past 60 years. Siegel [12], [13] was the first to show that under suitable non-degeneracy conditions, the equation (+) has only finitely many integral solutions in K. LeVeque[7] proved the following, more explicit, result. Letwhere a ∈ K* and αl,…,αk are distinct and algebraic over K. Then (+) has only finitely many integral solutions unless (nl,…,nk) is a permutation of one of the n-tuples