Polynomials on the Cauchy circle
type: Item
instance of: scholarly article
title: Polynomials on the Cauchy circle
zbMATH Open document ID: 0821.65009
author: Stephan Ruscheweyh
author: Luis Salinas
author: Diego Alejandro Mejía
published in: Numerische Mathematik
publication date: 1995-09-10T00:00:00Z
review text: Let \({\mathcal P}_ n^ c\) consist of the polynomials \[ P(z) = 1 + \sum_{k=1}^ n a_ kz^ k, \quad \sum_{k=1}^ n | a_ k | = 1. \] For \(\varepsilon_ n \geq 0\) the order of magnitude of the number \[ N = N(\varepsilon_ n,n) : = \min \Bigl\{ \nu \in \mathbb{N} : \forall P \in {\mathcal P}_ n^ c \quad \min_{1 \leq k \leq n} \bigl | P(e^{2 \pi ik/ \nu}) \bigr | \leq 1 - \varepsilon_ n \Bigr\} \] is discussed. For \(\varepsilon_ n = O(1/n)\) previous estimates of \(N = O(n^{3/2})\) are improved to \(N = O(n \log n)\). For \(n\leq 199\) the best upper bounds for \(N({1 \over 16n}, n)\) are presented. Moreover, an upper bound for the modulus of the zero of least modulus of \(P \in {\mathcal P}_ n^ c\) is established.
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 12Y05
zbMATH DE Number: 739405
zbMATH Keywords: polynomials
zbMATH Keywords: Cauchy circle
zbMATH Keywords: Cauchy radius
zbMATH Keywords: methods of descent
zbMATH Keywords: best upper bounds
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
full work available at URL: s002110050103
OpenAlex ID: W2001774243
DOI: 10.1007/S002110050103
description: scientific article; zbMATH DE number 739405