Solving explicitly \(F(x,y)=G(x,y)\) over function fields
type: Item
instance of: scholarly article
title: Solving explicitly \(F(x,y)=G(x,y)\) over function fields
zbMATH Open document ID: 1239.11149
author: István Gaál
author: Michael E. Pohst
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2011-11-03T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1317045233
review text: The authors consider the Diophantine equation \[ w F(x,y)=zG(x,y), \] where \(F\) and \(G\) are binary forms with coefficients in \(\mathbb Q[t]\) and moreover \(F\) is an irreducible form of degree \(\geq 3\). Further \(x,y,w,z \in \mathbb Q[t]\), \(\mathrm{gcd}(x,y)=\mathrm{gcd}(w,z)=1\) and \(w,z \in U_S\), where \(U_S\) denotes the group of \(S\)-units in \(\mathbb Q(t)\) for a finite set \(S\) of places. The authors prove the finiteness of solutions \((x,y,z,w)\) to such Diophantine equations by proving effective upper bounds for the heights of these solutions. Thus they also provide an algorithm to solve such Diophantine equations explicitly. Their proof is based on a result on \(S\)-unit equations of the form \(x+y=1\) over function fields established in an earlier paper of the authors [J. Number Theory 119, No. 1, 49--65 (2006; Zbl 1157.11011)]. The reviewer wants to emphasize that in most cases Diophantine equations over function fields are considered for function fields with algebraic closed constant field or global function fields.
reviewed by: Volker Ziegler
Mathematics Subject Classification ID: 11Y50
Mathematics Subject Classification ID: 11D57
zbMATH DE Number: 5967689
zbMATH Keywords: Diophantine equations
zbMATH Keywords: Thue equations
zbMATH Keywords: function fields
describes a project that uses: KANT/KASH
MaRDI profile type: Publication
OpenAlex ID: W2061481073
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DOI: 10.7169/FACM/1317045233
description: scientific article; zbMATH DE number 5967689