Journal of Applied Mathematics and Stochastic Analysis, 16:3 (2003), 249-255. Printed in the USA c©2003 by North Atlantic Science Publishing Company REAL ZEROS OF CLASSES OF RANDOM ALGEBRAIC POLYNOMIALS

• Main Authors: K. Farahmand, M. Sambandham
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Published: 2003
• Subjects: Random Algebraic Polynomials; Kac-Rice Formula; Random Variables; Binomial Coefficients; North Atlantic
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.530.1189; http://emis.maths.adelaide.edu.au/journals/HOA/JAMSA/Volume16_3/255.pdf

There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0 + a1x+ a2x2 + · · ·+ an−1xn−1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients {aj}n−1j=0 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 (n j 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.