SOLVING EXPLICITLY DECOMPOSABLE FORM EQUATIONS OVER GLOBAL FUNCTION FIELDS

In [I. Gaal and M. Pohst, Diophantine equations over global function fields I: The Thue equation, J. Number Theory, to appear] and [I. Gaal and M. Pohst, Diophantine equations over global function fields II: R-integral solutions of Thue equations, J. Experimental Math., to appear] we considered Thue equations over function fields of finite characteristics. Using the fundamental lemma proved in [I. Gaal and M. Pohst, Diophantine equations over global function fields I: The Thue equation, J. Number Theory, to appear], the improvements of [I. Gaal and M. Pohst, Diophantine equations over global function fields II: R-integral solutions of Thue equations, J. Experimental Math., to appear] and the formerly known construction of Gyory [Acta Math. Hung. 42(1-2) (1983), 45-80], in the present paper we develop a fast algorithm for solving wide classes of decomposable form equations. Our method is directly applicable for general discriminant form and index form equations, Thue equations and for certain classes of norm form equations. We illustrate our method by solving explicitly a specific equation.