Domain of attraction of normal law and zeros of random polynomials

‎Let‎ ‎$ P_{n}(x)= \sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraic‎ ‎polynomial‎, ‎where $A_{0},A_{1}‎, ‎\ldots $ is a sequence of independent‎ ‎random variables belong to the domain of attraction of the normal law‎. ‎Thus‎ ‎$A_j$ʹs for $j=0,1,\ldots $‎ ‎possess the characteristic functions $\exp‎ \{ ‎-\frac{1}{2}t^{2}H_{j}(t)\}$,‎ ‎where $H_j(t)$ʹs are complex slowly‎ ‎varying functions‎. ‎Under the assumption that there exist a real positive slowly varying function $H(\cdot)$ and positive constants $t_{0}$‎, ‎$ C_{\ast}$ and‎ ‎$C^{\ast}$ that $C_{\ast} H(t) \leq \mbox{Re}[H_{j}(t)] \leq C^{\ast} H(t),\;t\leq‎ ‎t_{0},\;j=1,\ldots,n$‎, ‎we find that while the variance of coefficients are bounded‎, ‎real zeros are concentrated around $\pm 1$‎, ‎and‎ ‎the expected number of real zeros of $P_n(x)$ round the origin at a distance $(\log n)^{-s}$ of $\pm 1$ are at most of order‎ ‎$O\left( (\log n)^s \log (\log n)\right)$‎.