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

We consider random polynomials pₙ (x) =₀+₁++ₙ xⁿ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded (2+) ^th moment (for some >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* ₍ Nₙ n &=& 2. eqnarray* This convergence can be viewed as a strong law of large numbers for the real roots of random Kac polynomials.