On the distribution of polynomials with bounded roots. I: Polynomials with real coefficients
type: Item
instance of: scholarly article
title: On the distribution of polynomials with bounded roots. I: Polynomials with real coefficients
zbMATH Open document ID: 1305.33006
author: Shigeki Akiyama
author: Attila Pethő
published in: Journal of the Mathematical Society of Japan
publication date: 2014-09-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jmsj/1406206977
review text: Let \(d\) be a positive integer, and denote by \(E_d\) the set of all \(d\)-dimensional polynomials whose roots lie within the ball of radius 1 centered at the origin. This set was first studied by \textit{I. Schur} in 1918, who found conditions, which imply that the boundary of \(E_d\) is the union of finitely many algebraic surfaces. In [IEEE Trans. Autom. Control 23, 454--458 (1978; Zbl 0377.93021)] \textit{A. T. Fam} and \textit{J. S. Meditch} improved this result by showing that the boundary of \(E_d\) is the union of two hyperplanes and one hypersurfaces. The authors of the paper under review prove a generalization of the above results for the boundary of the set \(v^{(s)}_d\), which is the set of polynomials having signature \(s\). They prove that the \(d\)-dimensional Lebesgue measure of this last set can be computed by a certain multiple integral (which is too complicated to be stated here), related to the well-known Selberg integral and its generalization.
reviewed by: József Sándor
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 26B15
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 33B20
zbMATH DE Number: 6350045
zbMATH Keywords: Selberg integral
zbMATH Keywords: polynomials with bounded roots
MaRDI profile type: Publication
OpenAlex ID: W1992095407
cites work: A Short Proof of Selberg’s Generalized Beta Formula
cites work: Q4222737
cites work: Jacobi Polynomials Associated with Selberg Integrals
cites work: A canonical parameter space for linear systems design
cites work: The importance of the Selberg integral
cites work: Q4377092
cites work: Finite and periodic orbits of shift radix systems
cites work: Q5845916
DOI: 10.2969/JMSJ/06630927
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description: scientific article; zbMATH DE number 6350045