Effective results for Diophantine equations over finitely generated domains

Attila Bérczes ; Jan-Hendrik Evertse ; Kálmán Győry

Acta Arithmetica, Tome 166 (2014), p. 71-100 / Harvested from The Polish Digital Mathematics Library

Résumé

Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over ℤ. We consider Thue equations F(x,y) = δ in x,y ∈ A, where F is a binary form with coefficients from A, and δ is a non-zero element from A, and hyper- and superelliptic equations $f(x) = δy^m$ in x,y ∈ A, where f ∈ A[X], δ ∈ A∖0 and $m ∈ ℤ_{≥ 2}$. 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, δ, F, f, m. These results imply that the solutions of these equations can be determined in principle. Further, we consider the Schinzel-Tijdeman equation $f(x) = δy^m$ where x,y ∈ A and $m ∈ ℤ_{≥2}$ 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

Publié le : 2014-01-01

Zbl 1312.11019

EUDML-ID : urn:eudml:doc:279401