Random polynomials having few or no real zeros

Consider a polynomial of large degreennwhose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactlykkreal zeros with probabilityn−b+o(1)n⁻ᵇ⁺ᵒ⁽¹⁾asn→∞n → integers of the same parity as the fixed integerk≥0k ≥ 0. In particular, the probability that a random polynomial of large even degreennhas no real zeros isn−b+o(1)n⁻ᵇ⁺ᵒ⁽¹⁾. The finite, positive constantbbis characterized via the centered, stationary Gaussian process of correlation functionsech(t/2){sech} (t/2). The value ofbbdepends neither onkknor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probabilityn−b+o(1)n⁻ᵇ⁺ᵒ⁽¹⁾one may specify also the approximate locations of thekkzeros on the real line. The constantbbis replaced byb/2b/2in case the i.i.d. coefficients have a nonzero mean.