On the distribution of the number of real roots of random polynomials

Let $\xi_0,\xi_1,\dots$ be a sequence of independent identically distributed random variables and $N$ be the number of real roots of the polynomial
$$ Q(x)=\sum_{j=0}^n\xi_jx^j. $$

The main result is the following
Theorem. {\it If $\mathbf P\{\xi_j=0\}=0$, $\mathbf E\xi_j=0$ and $\mathbf E|\xi_j|^{2+s}<\infty$ for some positive number $s$, then, for any real} $t$,
$$ \mathbf E\exp\{it(N-\mathbf EN)(\mathbf DN)^{-1/2}\}\underset{n\to\infty}\longrightarrow е^{-t^2/2}. $$