## Rocky Mountain Journal of Mathematics

### Algebraic polynomials with symmetric random coefficients

#### 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

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

#### 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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