Abstract
Let Q(n)(x) = Sigma(n)(i = 0) A(i)x(i) be a random algebraic polynomial where the coefficients A(0), A(1), ... form a sequence of centered Gaussian random variables. Moreover, we assume that the increments Delta(j) = A(j) -A(j-1) = 0, 1, 2, ..., n are independent normal random variables with mean zero, finite variances sigma(2)(j) and the conventional notation of A(-1) = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. Assuming the symmetric property of A(j)'s, we investigate some new and considerable results about the distribution of zeros. We prove that the expected number of real zeros of Q(n)(x) is asymptotically of order logn.