A generalization of Mason's theorem for four polynomials
type: Item
instance of: scholarly article
title: A generalization of Mason's theorem for four polynomials
zbMATH Open document ID: 1079.12002
author: Yanyan Li
published in: Elemente der Mathematik
publication date: 2004-09-22T00:00:00Z
review text: Let \(f\), \(g\), \(h\), \(k \in {\mathbb C}[t]\) be non-constant relatively prime polynomials satisfying \(f+g+h=k\), and write \(n_0(F)\) for the number of distinct roots of a polynomial \(F\). The author proves that the maximum of the degrees of \(f\), \(g\), \(h\), and \(k\) is bounded above by \(2n_0(fghk)-3\). This is a generalization of Mason's Theorem for three polynomials, in which the maximal degree of \(f\), \(g\), and \(h\), where \(f+g=h\) and \((f,g,h)=1\), is bounded above by \(n_0(fgh)-1\). The proof makes use of the logarithmic derivative, some elementary linear algebra, and divisibility properties. As an application, the author deduces that if \(n_1, \dots, n_4\) are integers with \(n_i \geq 8\) for each \(i\), then there is no solution of the generalized Fermat equation \(a(t)^{n_1} + b(t)^{n_2} + c(t)^{n_3} = d(t)^{n_4}\) with \(a\), \(b\), \(c\), and \(d\) non-constant relatively prime polynomials over \({\mathbb C}\).
reviewed by: Scott T. Parsell
Mathematics Subject Classification ID: 12D05
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 2102275
zbMATH Keywords: roots of polynomials
zbMATH Keywords: Fermat's equation
MaRDI profile type: Publication
full work available at URL: s00017-001-0203-2
OpenAlex ID: W1971197552
cites work: Q4257157
cites work: Q3315578
cites work: An alternate proof of Mason's theorem
cites work: POLYNOMIAL IDENTITIES AND HAUPTMODULN
DOI: 10.1007/S00017-001-0203-2
description: scientific article; zbMATH DE number 2102275