Solving explicitly \(F(x,y)=G(x,y)\) over function fields
The authors consider the Diophantine equation \[ w F(x,y)=zG(x,y), \] where \(F\) and \(G\) are binary forms with coefficients in \(\mathbb Q[t]\) and moreover \(F\) is an irreducible form of degree \(\geq 3\). Further \(x,y,w,z \in \mathbb Q[t]\), \(\mathrm{gcd}(x,y)=\mathrm{gcd}(w,z)=1\) and \(w,z \in U_S\), where \(U_S\) denotes the group of \(S\)-units in \(\mathbb Q(t)\) for a finite set \(S\) of places. The authors prove the finiteness of solutions \((x,y,z,w)\) to such Diophantine equations by proving effective upper bounds for the heights of these solutions. Thus they also provide an algorithm to solve such Diophantine equations explicitly. Their proof is based on a result on \(S\)-unit equations of the form \(x+y=1\) over function fields established in an earlier paper of the authors [J. Number Theory 119, No. 1, 49--65 (2006; Zbl 1157.11011)]. The reviewer wants to emphasize that in most cases Diophantine equations over function fields are considered for function fields with algebraic closed constant field or global function fields.
scientific article; zbMATH DE number 5967689
scientific article