Bounds for the solutions of superelliptic equations

Extract

In this work, we study the diophantine equation\renewcommand{\theequation}{1.1}\begin{equation}f(X) = Y^m,\end{equation} where $m \geqslant 2$ is an integer and $f(X)$ is a polynomial with coefficients in a number field ${\bf K}$. 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 $m \geqslant 3$ 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.