On the number of solutions to $S$-unit equations

Abstract

Let $K$ be a number field and $S$ a finite set of places including the archimedian ones. Let $(a_1,a_2,a_3)\in K^*$ we consider the $S$-unit equation $a_1x+a_2y=a_3$ with $x,y\in U_S$. In this paper we prove that for fixed $a_1,a_2$ the equation has at most $\frac{|S|-1}2$ solutions provided that the height of $a_3$ is sufficiently large. We want to emphasize that our results are effective.