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.