Count of many zeros of a class of random algebraic polynomials (Q2731015)

Let \(F_n(x,\omega)= \sum^n_{k=0} Y_kx^k\), where \(Y_k\), \(k= 0,1,\dots, n\), is a sequence of independent and identically distributed random variables defined on a probability space \((\Omega,{\mathcal A},\text{Pr})\). There are many estimates concerning the number of real zeros of \(F_n(x,\omega)\) for different assumptions for the distribution of the coefficients \(Y_k\). These estimates were initiated by \textit{J. E. Littlewood} and \textit{A. C. Offord} [Proc. Camb. Philos. Soc. 35, 133-148 (1939; Zbl 0021.03702)] and reviewed by \textit{A. T. Bharucha-Reid} and \textit{M. Sambandham} [``Random polynomials'' (1986; Zbl 0615.60058)] and the reviewer [``Topics in random polynomials'' (1998; Zbl 0949.60010)]. The present paper considers a case when the coefficients belong to the domain of attraction of normal law. Their theorem states that there exists a positive integer \(n_0\) such that, for \(n> n_0\), the number of real zeros of \(F_n(x,\omega)\) is at least \(\mu(\log\log n)^2\log n\), except for a set of measure at most \(\mu'(\log n_0-\log\log\log n_0)^{-1-\varepsilon}\), \(0< \varepsilon< 1\), where \(\mu\) and \(\mu'\) are constants. The value of \(\varepsilon\) is not defined and this reviewer has failed to establish its origin. If the theorem had considered the case of domain of attraction of the stable law instead, a similar parameter as the authors' \(\varepsilon\) would have been required.