Let Qn(x) =Sni=0 Aixi be a random algebraic polynomial where the coefficients A0,A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments Dj = Aj - Aj−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(nk), for some k > 0.