Real zeros of classes of random algebraic polynomials

[摘要] There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomiala0+a1x+a2x2+⋯+an−1xn−1with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients{aj}j=0n−1it is shown that the above expected number is asymptotic toO(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 thejth term is(nj)the expected number of zeros of the polynomial increases toO(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.

[学科分类] 应用数学