Abstract
This paper provides an asymptotic estimate for the expected number
of real zeros of a random algebraic polynomial a0+a1x+a2x2+…+an−1xn−1. The coefficients aj(j=0,1,2,…,n−1)
are assumed to be independent normal random
variables with mean zero. For integers m
and k=O(logn)2
the
variances of the coefficients are assumed to have nonidentical
value var(aj)=(k−1j−ik), where n=k⋅m
and i=0,1,2,…,m−1. Previous results are mainly for identically
distributed coefficients or when
var(aj)=(nj). We
show that the latter is a special case of our general theorem.