Polynomial squares of the form \(aX^m+b(1-X)^n+c\)
type: Item
instance of: scholarly article
title: Polynomial squares of the form \(aX^m+b(1-X)^n+c\)
zbMATH Open document ID: 1167.11304
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2007-02-02T00:00:00Z
full work available at URL: https://eudml.org/doc/108645
review text: Summary: It is an open diophantine problem to prove the finiteness of the integral solutions for equations like \(y^2=3^m+2^n+1\). This may be shown to correspond to one of the simplest unknown cases of a conjecture by Lang and Vojta and is probably very difficult; actually also the most basic analogous questions over function fields seem not to follow easily from known principles. With this in mind, in the present note we consider the equation \(y(X)^2=aX^m + b(1-2X)^n+c\), to be solved in the unknowns \(a,b,c\in \mathbb{C}^*\), \(m,n\in\mathbb{N}\) and polynomials \(y\in\mathbb{C}[X]\); we show that only pairs \((m , n)\) with \(m,n\leq 8\) may give rise to solutions. Our arguments are somewhat ad hoc and in a way surprising; the problem is left to find a more natural and general approach.
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 5121476
author: Umberto Zannier
MaRDI profile type: Publication
cites work: Vanishing sums in function fields
cites work: On the diophantine equation $f(a^m,y)=b^n$
cites work: Number theory III: Diophantine geometry
cites work: Q4488096
cites work: Equations in roots of unity
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description: scientific article; zbMATH DE number 5121476