Effective results for Diophantine equations over finitely generated domains

• Autorzy: Attila Bérczes , Jan-Hendrik Evertse , Kálmán Győry
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11D61: Exponential equations
• 11D41: Higher degree equations; Fermat's equation
• 11D59: Thue-Mahler equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 163
• Numer: 1
• Strony: 71-100

Daty

wydano

2014

Twórcy

autor

Attila Bérczes

• Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary

autor

Jan-Hendrik Evertse

• Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands

autor

Kálmán Győry

• Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary

Identyfikatory

DOI

10.4064/aa163-1-6