### Abstract

Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over double-struck 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 ∈ double-struck 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 ∈ double-struck Z_{≥2} are the unknowns and give an effective upper bound for M. Our results extend earlier work of Gyo{combining double acute accent}ry, Brindza and Végso{combining double acute accent}, where the equations mentioned above were considered only for a restricted class of finitely generated domains.

Original language | English |
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Pages (from-to) | 71-100 |

Number of pages | 30 |

Journal | Acta Arithmetica |

Volume | 163 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2014 |

### Keywords

- Diophantine equations over finitely generated domains
- Effective results
- Hyperelliptic equations
- Schinzel-Tijdeman equation
- Superelliptic equations
- Thue equations

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Acta Arithmetica*,

*163*(1), 71-100. https://doi.org/10.4064/aa163-1-6