Random polynomials having few or no real zeros

Abstract:

Consider a polynomial of large degree $n$ whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly $k$ real zeros with probability $n^{-b+o(1)}$ as $n \rightarrow \infty$ through integers of the same parity as the fixed integer $k \ge 0$. In particular, the probability that a random polynomial of large even degree $n$ has no real zeros is $n^{-b+o(1)}$. The finite, positive constant $b$ is characterized via the centered, stationary Gaussian process of correlation function ${\mathrm {sech}} (t/2)$. The value of $b$ depends neither on $k$ nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability $n^{-b+o(1)}$ one may specify also the approximate locations of the $k$ zeros on the real line. The constant $b$ is replaced by $b/2$ in case the i.i.d. coefficients have a nonzero mean.