Real zeros of classes of random algebraic polynomials

K. Farahmand and M. Sambandham

International Journal of Stochastic Analysis, 2003, vol. 16, 1-7

Abstract:

There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a 0 + a 1 x + a 2 x 2 + ⋯ + a n − 1 x n − 1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients { a j } j = 0 n − 1 it is shown that the above expected number is asymptotic to O ( log n ) . 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 j th 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.

Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:328104

DOI: 10.1155/S1048953303000194

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