On the equation \(f(x)=y^ m\) over finitely generated domains
type: Item
instance of: scholarly article
title: On the equation \(f(x)=y^ m\) over finitely generated domains
zbMATH Open document ID: 0691.10006
DOI: 10.1007/BF01953374
published in: Acta Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: The purpose of the paper is to give the first effective upper bounds for the solutions of hyperelliptic equations considered over integral domains, finitely generated over \(\mathbb Z\). This type of equations plays an essential role in the theory and applications of diophantine equations. Ineffective theorems on hyperelliptic equations over number fields were obtained by C. L. Siegel and W. J. LeVeque. These results were extended to the case of integral domains of finite type over \(\mathbb Z\) by S. Lang. The first effective bounds for the solutions of hyperelliptic equations over number fields were given by A. Baker, whose result was extended and improved by several authors. In the function field case W. M. Schmidt, R. C. Mason and the author obtained effective theorems. In case of integral domains, finitely generated over \(\mathbb Z\), hyperelliptic equations can not be reduced to Thue equations directly. Hence the proof of the main theorem combines the effective result of \textit{R. C. Mason} and the author [Acta Arith. 47, No. 2, 167--173 (1986; Zbl 0561.10009)] on the solutions of hyperelliptic equations over function fields with the result of the author [Acta Math. Hung. 44, No. 1--2, 133--139 (1984; Zbl 0552.10009)] on S-integral solutions of hyperelliptic equations over number fields by applying the technics developed by \textit{K. Györy} [Acta Math. Hung. 42, No. 1--2, 45--80 (1983; Zbl 0528.10014)].
reviewed by: István Gaál
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 4130455
zbMATH Keywords: effective upper bounds
zbMATH Keywords: hyperelliptic equations
zbMATH Keywords: integral domains
author: Bela Brindza
MaRDI profile type: Publication
cites work: Q5562698
cites work: On \(S\)-integral solutions of the equation \(y^ m=f(x)\)
cites work: Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains
cites work: Effective finiteness theorems for polynomials with given discriminant and integral elements with given discriminant over finitely domains.
cites work: Q5724382
cites work: On the equation $y^m=f(x)$
cites work: The hyperelliptic equation over function fields
cites work: Le Veque's superelliptic equation over function fields
cites work: Q4170806
cites work: Q4177652
cites work: A lower bound for linear forms in logarithms
description: scientific article; zbMATH DE number 4130455