A note on the complex roots of complex random polynomials
type: Item
instance of: scholarly article
title: A note on the complex roots of complex random polynomials
zbMATH Open document ID: 0963.60049
DOI: 10.1016/S0167-7152(99)00007-3
published in: Statistics \& Probability Letters
publication date: 2001-06-21T00:00:00Z
review text: Let \(a_k(\omega)\) and \(b_k(\omega)\), \(\omega\in\Omega\), \(k=0,1, \dots,n\), be sequences of independent, normally distributed random variables defined on a probability space \((\Omega,A,\text{Pr})\) with mean zero and variance one. Denote \(P(z,\omega)\equiv P(z)= \sum^n_{k=0} c_k(\omega)z^k\) where \(c_k(\omega) \equiv c_k=a_k (\omega)+i b_k(\omega)\). For the case of \(b_k\equiv 0\), \(k=0,1,\dots,n\), it is well known that one can find the density function \(h_n (x)\) such that the expected number of real zeros of \(P(x)\) denoted by \(v_n (R)\), \(R\in\mathbb{R}\), satisfies \(v_n(R)= \int_Rh_n(x)dx\), \(R\in\mathbb{R}\). This formula then could be used to obtain an asymptotic formula for the expected number of real zeros. However, in comparison, very little is known about the complex roots of \(P(z)\). It is obvious, see for example \textit{J. E. A. Dunnage} [Proc. Lond. Math. Soc., III Ser. 18, 439-460 (1968; Zbl 0164.19002)], that there could be no analogue of the above asymptotic formula for the expected number of real zeros for \(P(z)\) with complex coefficients. Also the problems concerning the complex roots are involved with analysis of the different levels of difficulties. This can be seen in the classical work of \textit{J. M. Hammersley} [in: Proc. 3rd Berkeley Sympos. Math. Statist. Probab. 2, 89-111 (1956; Zbl 0074.34302)] who obtained \(h_n(z)\) where the coefficients of \(P(z)\) are multivariate normal random vector with mean \(\mu_c\) and a nondegenerate covariance matrix. However, recently \textit{I. Ibragimov} and \textit{O. Zeitouni} [Trans. Am. Math. Soc. 349, No. 6, 2427-2441 (1997; Zbl 0872.30002)] used an idea of \textit{R. J. Adler} [``The geometry of random fields'' (1981; Zbl 0478.60059)] and significantly simplified the derivation of the density of complex roots. A complete and comprehensive background of up-to-date results can be found in a book by this reviewer [``Topics on random polynomials'' (1998; Zbl 0949.60010)], which als…
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 1395148
author: Alessandro Ramponi
reviewed by: Kambiz Farahmand
MaRDI profile type: Publication
cites work: Q3935962
cites work: Q4722950
cites work: The Number of Real Zeros of a Class of Random Algebraic Polynomials
cites work: Complex roots of a random algebraic polynomial
cites work: Q3239646
cites work: On roots of random polynomials
cites work: A correction to “On the average number of real roots of a random algebraic equation”
cites work: The Complex Zeros of Random Polynomials
description: scientific article; zbMATH DE number 1395148