Effective results for Diophantine equations over finitely generated domains

Attila Bérczes¹ ; Jan-Hendrik Evertse² ; Kálmán Győry¹

¹ Institute of Mathematics University of Debrecen H-4010 Debrecen, P.O. Box 12, Hungary
² Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden, The Netherlands

Acta Arithmetica, Tome 163 (2014) no. 1, pp. 71-100.

Résumé

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.

Détail

DOI : 10.4064/aa163-1-6

Keywords: arbitrary integral domain characteristic finitely generated mathbb consider thue equations delta where binary form coefficients delta non zero element hyper superelliptic equations delta where delta setminus mathbb geq under necessary finiteness conditions effective upper bounds sizes solutions equations terms appropriate representations delta these results imply solutions these equations determined principle further consider schinzel tijdeman equation delta where mathbb geq unknowns effective upper bound results extend earlier work brindza where equations mentioned above considered only restricted class finitely generated domains