Bounds for the size of integral points on curves of genus zero
type: Item
instance of: scholarly article
title: Bounds for the size of integral points on curves of genus zero
zbMATH Open document ID: 0997.11047
DOI: 10.1023/A:1017971406157
published in: Acta Mathematica Hungarica
publication date: 2002-11-03T00:00:00Z
review text: The author gives improved bounds for the size of integral points on curves of genus zero defined over an algebraic number field in the case when there are at least three points at infinity on the curve. The bounds are of the form \[ \max \{ H_K(x),H_K(y) \} < \exp \{ \Omega \cdot D^A \cdot H_K(F)^B \} \] where \(K\) is the number field and \(D\) is the absolute value of its discriminant, \(F(X,Y)=0\) is the equation of the curve, and \(H_K\) denotes the height function. There are five cases, the simplest to state is when the curve has an ordinary singular point at infinity of multiplicity \(\geq 3\). Then \(\Omega = d^{14dN^4} N^{52dN^7}\), where \(d\) is the degree of \(K\) over the rationals and \(N\) is the degree of \(F\), \(A = 4N^6\) and \(B =13N^6\). In the proof the author uses his earlier results on the size of integral solutions of Thue equations [J. Aust. Math. Soc., Ser. A 63, 145-164 (1997; Zbl 0893.11010)] and on effective parametrization of curves of genus zero [Acta Arith. 86, 51-90 (1998; Zbl 0934.11035)].
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 1822360
zbMATH Keywords: algebraic curve of genus 0
zbMATH Keywords: integer point
zbMATH Keywords: valuation
author: Dimitros Poulakis
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
full work available at URL: a:1017971406157
OpenAlex ID: W1597758046
description: scientific article; zbMATH DE number 1822360