Bounds for the solutions of superelliptic equations

In this work, we study the diophantine equation {}{1.1} {equation} f(X) = Y^m, {equation} where m 2 is an integer and $f(X)$ is a polynomial with coefficients in a number field K . The first important result on this topic is due to Siegel [19], who showed that if $m = 2$ and f has at least three simple roots or if m 3 and f has at least two simple roots, then (1.1) has only finitely many integral solutions. Three years later, he proved [20] that if the algebraic curve defined by (1.1) is of positive genus, then (1.1) has only finitely many integral solutions. The p -adic analogue of this theorem was established independently by Lang [9] and LeVeque [12], who showed that, under the same conditions, (1.1) has only finitely many S -integral solutions. After that, LeVeque [13] gave a necessary and sufficient condition for the algebraic curve defined by (1.1) to have positive genus. However, all these results are based on Thue's method, and hence are ineffective.