Geodesic
On the variance of the number of real roots of random polynomials
Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 1, pp. 36-51 
N. B. Maslova. On the variance of the number of real roots of random polynomials. Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 1, pp. 36-51. http://geodesic.mathdoc.fr/item/TVP_1974_19_1_a3/
@article{TVP_1974_19_1_a3,
     author = {N. B. Maslova},
     title = {On the variance of the number of real roots of random polynomials},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {36--51},
     year = {1974},
     volume = {19},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1974_19_1_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\xi_0,\xi_1,\dots,\xi_n,\dots$ be a sequence of independent identically distributed random variables, $N_n$ be the number of real roots of the polynomial $\sum_{j=0}^n\xi_jx^j$. The main result is Theorem 1. {\em If $\mathbf P\{\xi_j=0\}=0$, $\mathbf E\xi_j=0$, $\mathbf E|\xi_j|^{2+s}<\infty$ for some positive number $s$, then} $$ \mathbf DN_n\sim4\biggl(\frac1\pi-\frac2{\pi^2}\biggr)\ln n\quad(n\to\infty). $$