### 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 |
---|---|

Pages (from-to) | 71-100 |

Number of pages | 30 |

Journal | Acta Arithmetica |

Volume | 163 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### 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

### Cite this

*Acta Arithmetica*,

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

**Effective results for Diophantine equations over finitely generated domains.** / Bérczes, Attila; Evertse, Jan Hendrik; Gyory, Kálmán.

Research output: Contribution to journal › Article

*Acta Arithmetica*, vol. 163, no. 1, pp. 71-100. https://doi.org/10.4064/aa163-1-6

}

TY - JOUR

T1 - Effective results for Diophantine equations over finitely generated domains

AU - Bérczes, Attila

AU - Evertse, Jan Hendrik

AU - Gyory, Kálmán

PY - 2014

Y1 - 2014

N2 - 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) = δym 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) = δyM 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.

AB - 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) = δym 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) = δyM 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.

KW - Diophantine equations over finitely generated domains

KW - Effective results

KW - Hyperelliptic equations

KW - Schinzel-Tijdeman equation

KW - Superelliptic equations

KW - Thue equations

UR - http://www.scopus.com/inward/record.url?scp=84907413356&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907413356&partnerID=8YFLogxK

U2 - 10.4064/aa163-1-6

DO - 10.4064/aa163-1-6

M3 - Article

AN - SCOPUS:84907413356

VL - 163

SP - 71

EP - 100

JO - Acta Arithmetica

JF - Acta Arithmetica

SN - 0065-1036

IS - 1

ER -