Journal of Applied Mathematics and Stochastic Analysis 
Volume 2006 (2006), Article ID 13980, 6 pages
doi:10.1155/JAMSA/2006/13980

Real zeros of random algebraic polynomials with binomial elements

A. Nezakati1 and K. Farahmand2

 1Faculty of Mathematics, Shahrood University of Technology, P.O. Box 316-36155, Shahrood, Iran
2Department of Mathematics, University of Ulster, Jordanstown Campus, County Antrim BT37 0QB, United Kingdom
Received 26 August 2005; Revised 26 September 2005; Accepted 26 September 2005

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(log⁡n)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.