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Real zeros of classes of random algebraic polynomials. (English) Zbl 1051.60057
Summary: There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial \(a_0+a_1x+ a_2x^2+\cdots+a_{n-1}x^{n-1}\) with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients \(\{a_j\}^{n-1}_{j=0}\) 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 \({n\choose j}\) the expected number of zeros of the polynomial increases to \(O(\sqrt n)\). 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.

60G99 Stochastic processes
60H99 Stochastic analysis
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