On the number of real roots of random polynomials

By Hoi H. Nguyen, Oanh Nguyen, Van H. Vu. 2014

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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 Erdős–Offord, showed that the expectation of the number of real roots is 2 πlog n + o(log n). In this paper, we determine the true nature of the error term by showing that the expectation equals 2 πlog n + O(1). Prior to this paper, the error term O(1) has been known only for polynomials with Gaussian coefficients.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/55e4c0a1188b7703870a5b30ff73dfc64503bd89