Rocky Mountain Journal of Mathematics

Algebraic polynomials with symmetric random coefficients

K. Farahmand and Jianliang Gao

Full-text: Open access

Abstract

This paper provides an asymptotic estimate for the expected number of real zeros of algebraic polynomials $P_n (x)=a_0 + a_1 x + a_2 x^2 + \cdots + a_{n-1}x^{n-1} $, where $a_j$'s ($j=0,1,2,\cdots,n-1$) are a sequence of normal standard independent random variables with the symmetric property $a_j \equiv a_{n-1-j}$. It is shown that the expected number of real zeros in this case still remains asymptotic to $(2/\pi)\log n$. In the previous study, it was shown for the case of random trigonometric polynomials this expected number of real zeros is halved when we assume the above symmetric properties.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 2 (2014), 521-529.

Dates
First available in Project Euclid: 4 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1407154912

Digital Object Identifier
doi:10.1216/RMJ-2014-44-2-521

Mathematical Reviews number (MathSciNet)
MR3240512

Zentralblatt MATH identifier
1296.60137

Subjects
Primary: 60G99: None of the above, but in this section
Secondary: 60H99: None of the above, but in this section

Keywords
Number of real zeros real roots random algebraic p olynomials Kac-Rice formula random trigonometric p olynomial

Citation

Farahmand, K.; Gao, Jianliang. Algebraic polynomials with symmetric random coefficients. Rocky Mountain J. Math. 44 (2014), no. 2, 521--529. doi:10.1216/RMJ-2014-44-2-521. https://projecteuclid.org/euclid.rmjm/1407154912


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