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.
Citation
K. Farahmand. Jianliang Gao. "Algebraic polynomials with symmetric random coefficients." Rocky Mountain J. Math. 44 (2) 521 - 529, 2014. https://doi.org/10.1216/RMJ-2014-44-2-521
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