# zbMATH — the first resource for mathematics

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.

##### MSC:
 60G99 Stochastic processes 60H99 Stochastic analysis
Full Text: