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 --> infinity through integers of the same parity as the fixed integer k >= 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 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.