Effective results for Diophantine equations over finitely generated domains

Attila Bérczes, Jan-Hendrik Evertse, and Kálmán Győry. "Effective results for Diophantine equations over finitely generated domains." Acta Arithmetica 163.1 (2014): 71-100. <http://eudml.org/doc/279401>.

@article{AttilaBérczes2014,
abstract = {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.},
author = {Attila Bérczes, Jan-Hendrik Evertse, Kálmán Győry},
journal = {Acta Arithmetica},
keywords = {Thue equations; hyperelliptic equatons; superelliptic equations; Schinzel-Tijdeman Equation; effective results; Diophantine equations over finitely generated domains},
language = {eng},
number = {1},
pages = {71-100},
title = {Effective results for Diophantine equations over finitely generated domains},
url = {http://eudml.org/doc/279401},
volume = {163},
year = {2014},
}

TY - JOUR
AU - Attila Bérczes
AU - Jan-Hendrik Evertse
AU - Kálmán Győry
TI - Effective results for Diophantine equations over finitely generated domains
JO - Acta Arithmetica
PY - 2014
VL - 163
IS - 1
SP - 71
EP - 100
AB - 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.
LA - eng
KW - Thue equations; hyperelliptic equatons; superelliptic equations; Schinzel-Tijdeman Equation; effective results; Diophantine equations over finitely generated domains
UR - http://eudml.org/doc/279401
ER -