Polynomial bounds for the solutions of a class of diophantine equations
type: Item
instance of: scholarly article
title: Polynomial bounds for the solutions of a class of diophantine equations
zbMATH Open document ID: 0894.11011
author: Dimitros Poulakis
published in: Journal of Number Theory
publication date: 1997-11-18T00:00:00Z
review text: Let \(K\) be an algebraic number field of degree \(d\) such that all embeddings of \(K\) into \(\mathbb{C}\) are real. Let \(F(X,Y)\in K[X,Y]-K[Y]\) be an irreducible polynomial of total degree \(N\) and of degree \(n>0\) in \(Y\). Denote by \(F_N(X,Y)\) the homogeneous part of degree \(N\) of \(F(X,Y)\) and assume that \(F_N(1,Y)\) is of degree \(n\) without real roots. Consider the equation \[ F(x,y)=0 \text{ in} (x,y)\in O_K\times K. \] In the main result of the paper the author proves that the solutions of this equation satisfy \[ \max(H_K(x),H_K(y))<(2^N N)^{4dN^3} H_K(F)^{2N^3}. \] This is maybe the first polynomial bound for the height of integer points of a class of algebraic curves over a number field. In the proof, Puiseux expansions of algebraic functions are used.
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 1088553
zbMATH Keywords: polynomial diophantine equations
zbMATH Keywords: polynomial bounds for the height of integer points
zbMATH Keywords: Puiseux expansions
zbMATH Keywords: algebraic curves
reviewed by: István Gaál
MaRDI profile type: Publication
full work available at URL: jnth.1997.2153
OpenAlex ID: W2029310755
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DOI: 10.1006/JNTH.1997.2153
description: scientific article; zbMATH DE number 1088553