Solving explicitly $F(x,y)=G(x,y)$ over function fields

Abstract

Consider binary forms $F(x,y), G(x,y)$ with coefficients in $\mathbb{Q}[t]$, assume that $F$ is irreducible. We give effective upper bounds for the heights of the solutions and an efficient algorithm to solve \[ w\cdot F(x,y)=z\cdot G(x,y) \] \[ in x,y\in \mathbb{Q}[t], w,z\in \mathbb{Q}[t]\cap U_S, \gcd(x,y)=1, \gcd(w,z)=1, \] where $U_S$ denotes a group of $S$-units in $\mathbb{Q}(t)$. We derive that there are only finitely many solutions up to constant factors. We also show that this is not true for global function fields. This is a generalization of the well known Thue equations. Effective upper bounds for the solutions of this general equation were given over number fields but it was not yet considered over function fields. We illustrate our method with a detailed numerical example.