Real zeros of algebraic polynomials with nonidentical dependent random coefficients

Abstract

The expected number of real zeros of a random algebraic polynomial a 0 + a 1 x + a 2 x 2 + a 3 x 3 + .... + a n−1 x n−1 depends on the types of random coefficients, with large n. In all works, the coefficients are either independent or dependent but varience of coefficients a i is one. In these cases the exepected number of real zeros is found out to be asymptotic to 2 π logn. In this article, we have considered the negatively correlated dependent random coefficients {a i } n−1 i=0 with varience σ 2i , for σ > 1 and coefficient of correlation ρ ij = −ρ |i−j| , where 0 < ρ < 1 3 . The expected number of real zeros is found to be 2 πσ logn, which is depended on σ.