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