New features on real zeros of random polynomials Original Research Article

Let View the MathML sourceQn(x)=∑k=0nAkxk be a random algebraic polynomial in which the coefficients A0,A1,A2,…A0,A1,A2,… form a sequence of independent normally distributed random variables. In this work we study the behavior of the expected density of real zeros of Qn(x)Qn(x) for the case that the variances of the middle coefficients are substantially large, say Var(Ak)=ρ(k−n/2)2Var(Ak)=ρ(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 ρρ. We also consider the case where the variances of the coefficients are decreasing as View the MathML sourceVar(Ak)=e−k2/2n7/4, and we show that the asymptotic behavior of the expected number of real zeros of Qn(x)Qn(x) is of order n3/8n3/8.