### Abstract

Let f be a polynomial with coefficients in the ring OS of S-integers of a given number field K, b a non-zero S-integer, and m an integer ≥ 2. Suppose that f has no multiple zeros. We consider the equation (*) f(x) = by ^{m} in x; y ∈ O_{S}. In the present paper we give explicit upper bounds in terms of K, S, b, f, m for the heights of the solutions of (*). Further, we give an explicit bound C in terms of K, S, b, f such that if m > C then (*) has only solutions with y = 0 or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, and Shorey and Tijdeman. The results in the present paper are needed in a forthcoming paper of ours on Diophantine equations over integral domains which are finitely generated over.

Original language | English |
---|---|

Pages (from-to) | 727-756 |

Number of pages | 30 |

Journal | Publicationes Mathematicae |

Volume | 82 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2013 |

### Fingerprint

### Keywords

- Baker's method
- Hyperelliptic equations
- Schinzel-tijdeman theorem
- Superelliptic equations

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*82*(3-4), 727-756. https://doi.org/10.5486/PMD.2013.5748

**Effective results for hyper- and superelliptic equations over number fields.** / Bérczes, Attila; Evertse, Jan Hendrik; Györy, K.

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 82, no. 3-4, pp. 727-756. https://doi.org/10.5486/PMD.2013.5748

}

TY - JOUR

T1 - Effective results for hyper- and superelliptic equations over number fields

AU - Bérczes, Attila

AU - Evertse, Jan Hendrik

AU - Györy, K.

PY - 2013

Y1 - 2013

N2 - Let f be a polynomial with coefficients in the ring OS of S-integers of a given number field K, b a non-zero S-integer, and m an integer ≥ 2. Suppose that f has no multiple zeros. We consider the equation (*) f(x) = by m in x; y ∈ OS. In the present paper we give explicit upper bounds in terms of K, S, b, f, m for the heights of the solutions of (*). Further, we give an explicit bound C in terms of K, S, b, f such that if m > C then (*) has only solutions with y = 0 or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, and Shorey and Tijdeman. The results in the present paper are needed in a forthcoming paper of ours on Diophantine equations over integral domains which are finitely generated over.

AB - Let f be a polynomial with coefficients in the ring OS of S-integers of a given number field K, b a non-zero S-integer, and m an integer ≥ 2. Suppose that f has no multiple zeros. We consider the equation (*) f(x) = by m in x; y ∈ OS. In the present paper we give explicit upper bounds in terms of K, S, b, f, m for the heights of the solutions of (*). Further, we give an explicit bound C in terms of K, S, b, f such that if m > C then (*) has only solutions with y = 0 or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, and Shorey and Tijdeman. The results in the present paper are needed in a forthcoming paper of ours on Diophantine equations over integral domains which are finitely generated over.

KW - Baker's method

KW - Hyperelliptic equations

KW - Schinzel-tijdeman theorem

KW - Superelliptic equations

UR - http://www.scopus.com/inward/record.url?scp=84879076907&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879076907&partnerID=8YFLogxK

U2 - 10.5486/PMD.2013.5748

DO - 10.5486/PMD.2013.5748

M3 - Article

AN - SCOPUS:84879076907

VL - 82

SP - 727

EP - 756

JO - Publicationes Mathematicae

JF - Publicationes Mathematicae

SN - 0033-3883

IS - 3-4

ER -