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Journal of Applied Mathematics and Stochastic Analysis
Volume 16 (2003), Issue 3, Pages 249-255

Real zeros of classes of random algebraic polynomials

1University of Ulster, Department of Mathematics, Jordanstown Co., Antrim BT37 0QB, UK
2Morehouse College, Department of Mathematics, Atlanta, GA 30114, USA

Received 1 October 2002; Revised 1 March 2003

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0+a1x+a2x2++an1xn1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients {aj}j=0n1 it is shown that the above expected number is asymptotic to O(logn). This order for the expected number of zeros remains valid for the case when the coefficients are grouped into two, each group with a different variance. However, it was recently shown that if the coefficients are non-identically distributed such that the variance of the jth term is (nj) the expected number of zeros of the polynomial increases to O(n). The present paper provides the value for this asymptotic formula for the polynomials with the latter variances when they are grouped into three with different patterns for their variances.