On the number of real roots of random polynomials

Roots of random polynomials have been studied intensively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the expectation of the number of real roots is 2/π log n + o(log n). In this paper, we determine the true nature of the error term by showing that the expectation equals 2/π log n + O(1). Prior to this paper, the error term O(1) has been known only for polynomials with Gaussian coefficients.