Diophantine equations over global function fields IV: S-unit equations in several variables with an application to norm form equations

We describe an efficient algorithm to calculate all solutions of unit equations in several variables over global function fields. Note that using the present tools it is not possible to solve completely unit equations in more than two variables over number fields. In the function field case such equations are completely solved here for the first time. As a typical application we determine all solutions of norm form equations.