Domain of attraction of normal law and zeros of random polynomials

Let P n (x)=sum n i=0 A i x i be a random algebraic polynomial, where A 0, A 1, cdots is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus A j 's for j=0,1cdots possesses the characteristic functions exp−frac12t 2 H j (t), where H j (t) 's 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)leqmboxRe[H j (t)]leqC ast H(t),;tleqt 0, ;j=1,cdots,n, we find that while the variance of coefficients are bounded, real zeros are concentrated around pm1, and the expected number of real zeros of P n (x) round the origin at a distance (logn) (−s) of pm1 are at most of order Oleft((logn) s log(logn)right).