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

Yen Q. Do

Published 2024-03-11Version 1

We consider random polynomials $p_n(x)=\xi_0+\xi_1+\dots+\xi_n x^n$ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded $(2+\epsilon)^{th}$ moment (for some $\epsilon>0$), also known as the Kac polynomials. Let $N_n$ be the number of real roots of $p_n$. In this paper, answering a question from Igor Pritsker, we prove that almost surely the following convergence holds: \begin{eqnarray*} \lim_{n\to\infty} \frac{N_n}{\log n} &=& \frac 2 \pi. \end{eqnarray*} This convergence can be viewed as a strong law of large numbers for the real roots of random Kac polynomials.