A strong law of large numbers for real roots of random polynomials

We consider random polynomials pₙ(x)=ξ₀+ξ₁++ξₙ xⁿ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded (2+ε)ᵗʰ moment (for some ε>0), also known as the Kac polynomials. Let Nₙ be the number of real roots of pₙ. In this paper, answering a question from Igor Pritsker, we prove that almost surely the following convergence holds: {eqnarray*} limn→∞ N_n/log n &=& 2 π. {eqnarray*} This convergence can be viewed as a strong law of large numbers for the real roots of random Kac polynomials.