Abstract
We study a random algebraic polynomial Q(n)(x)= Σ(n)(i=0) A(i)x(i), where the coefficients A(0), A(1), ... form a sequence of centred Gaussian random variables. Moreover, we assume that the increments Λ(j) = A(j) - A(j-1), j = 0, 1, 2,... are independent, assuming A(-1) = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. We obtain the asymptotic behaviour of the expected number of times that such a random polynomial assumes the real value K, where K is any non-zero real constant. It is shown that the results are valid even for K -> infinity, as long as K = o(n(1/4)).