Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
• Artykuł - szczegóły

Acta Arithmetica

2014 | 163 | 1 | 71-100

Effective results for Diophantine equations over finitely generated domains

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.

71-100

wydano
2014

Twórcy

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