  In this article we study the three-variable unit equation $x + y + z = 1$ to
be solved in $x, y, z \in \mathcal{O}_S^\ast$, where $\mathcal{O}_S^\ast$ is
the $S$-unit group of some global function field. We give upper bounds for the
height of solutions and the number of solutions. We also apply these techniques
to study the Fermat surface $x^N + y^N + z^N = 1$.

Number Theory