Rocky Mountain Journal of Mathematics

Algebraic polynomials with symmetric random coefficients

K. Farahmand and Jianliang Gao

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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

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

First available in Project Euclid: 4 August 2014

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Zentralblatt MATH identifier

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

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


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.

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