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, . . ., with A(-1) = 0, are independent. The coefficients can be considered as n consecutive observations of a Brownian motion. We study the asymptotic behaviour of the expected number of local maxima of Qn( x) below level u = O(n(k)), for some k > 0.