Effective results for Diophantine equations over finitely generated domains
Volume 163 / 2014
Abstract
Let $A$ be an arbitrary integral domain of characteristic $0$ that is finitely generated over $\mathbb {Z}$. We consider Thue equations $F(x,y)=\delta $ in $x,y\in A$, where $F$ is a binary form with coefficients from $A$, and $\delta $ is a non-zero element from $A$, and hyper- and superelliptic equations $f(x)=\delta y^m$ in $x,y\in A$, where $f\in A[X]$, $\delta \in A\setminus \{ 0\}$ and $m\in \mathbb {Z}_{\geq 2}$.
Under the necessary finiteness conditions we give effective upper bounds for the sizes of the solutions of the equations in terms of appropriate representations for $A$, $\delta $, $F$, $f$, $m$. These results imply that the solutions of these equations can be determined in principle. Further, we consider the Schinzel–Tijdeman equation $f(x)=\delta y^m$ where $x,y\in A$ and $m\in \mathbb {Z}_{\geq 2}$ are the unknowns and give an effective upper bound for $m$.
Our results extend earlier work of Győry, Brindza and Végső, where the equations mentioned above were considered only for a restricted class of finitely generated domains.
