Abstract
Let Q(n)(x) = Σ(n)(k=0) A(k)x(k) be a random algebraic polynomial in which the coefficients A 0; A 1; A(2), ... form a sequence of independent normally distributed random variables. In this work we study the behavior of the expected density of real zeros of Q(n)(x) for the case that the variances of the middle coefficients are substantially large, say Var(A(k)) = rho((k-n/2)2). We find some new and interesting features about the distribution and the expected number of real zeros of such a polynomial for different values of rho. We also consider the case where the variances of the coefficients are decreasing as Var(A(k)) = e (k2/2n7/4), and we show that the asymptotic behavior of the expected number of real zeros of Q(n)(x) is of order n(3/8).