Real zeros of random algebraic polynomials with binomial elements

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