On the zeros of polynomials
type: Item
instance of: scholarly article
title: On the zeros of polynomials
zbMATH Open document ID: 0868.30007
DOI: 10.1007/BF02837165
author: A. Aziz
author: B. A. Zargar
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1997-08-19T00:00:00Z
review text: The classical Cauchy result [Exercises de mathématiques, IV Année de Bure Fréses, 1829] said that if \[ P(z)= z^n+ a_{n-1}z^{n-1}+\cdots+ a_1z+a_0 \] than all the zeros of \(P(z)\) lie in a circle \(|z|\leq 1+M\) where \(M=\max|a_j|\), \(j=0,1,2,\dots,n-1\). In this paper, the authors extend the above Cauchy result on the class of lacunary type polynomials: Theorem. If \[ P(z)= z^n+ a_pz^p+\cdots+ a_1z+a_0,\quad 0\leq p\leq n-1 \] then all the zeros of \(P(z)\) lie in \(|z|<K\), where \(K\) is a unique positive root of the trinomial \[ X^{n-p}- X^{n-p-1}-M=0 \] for \(M=\max|a_j|\), \(j=0,1,\dots,p\). For \(p=n-1\) this gives the Cauchy result. The other results giving some ring shaped region for the zeros of lacunary polynomials are also present.
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 26C10
zbMATH DE Number: 955215
reviewed by: Jan Stankiewicz
MaRDI profile type: Publication
cites work: On the location of the zeros of a polynomial
cites work: Bounds for the Zeros of Polynomials
cites work: On the Location of Zeros of Polynomials
cites work: Q5546070
cites work: Analytic methods in the study of zeros of polynomials
full work available at URL: https://doi.org/10.1007/bf02837165
OpenAlex ID: W2024029614
description: scientific article; zbMATH DE number 955215