Abstract
Let Q(n)(x) = Σ(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, assume that the increments Δ(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 u-sharp crossings, u > 0, of polynomial Q(n)(x). We refer to u-sharp crossings as those zero up-crossings with slope greater than u, or those down-crossings with slope smaller than - u. We consider the cases where u is unbounded and increasing with n, say u = o(n (5/4)), and u = o(n (3/2)).