Random polynomials having few or no real zeros

Abstract

Consider a polynomial of large degree n 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 k real zeros with probability n − b + o ( 1 ) n^{-b+o(1)} as n → ∞ n \rightarrow \infty through integers of the same parity as the fixed integer k ≥ 0 k \ge 0 . In particular, the probability that a random polynomial of large even degree n n has no real zeros is n − b + o ( 1 ) n^{-b+o(1)} . The finite, positive constant b b is characterized via the centered, stationary Gaussian process of correlation function s e c h ( t / 2 ) {\mathrm {sech}} (t/2) . The value of b b depends neither on k 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 ) n^{-b+o(1)} one may specify also the approximate locations of the k k zeros on the real line. The constant b b is replaced by b / 2 b/2 in case the i.i.d. coefficients have a nonzero mean.