On the number of real roots of random polynomials

Abstract

Roots of random polynomials have been studied exclusively 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 $\frac{2}{\pi }\log n+o(\log n)$. In this paper, we determine the true nature of the error term by showing that the expectation equals $\frac{2}{\pi }\log n+O(1)$. Prior to this paper, such estimate has been known only in the gaussian case, thanks to works of Edelman and Kostlan.

Cited by in corpus (4)

• Real eigenvalues of non-Gaussian random matrices and their products
• Real zeros of random Dirichlet series
• Salem-Zygmund Inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices
• Real roots of random polynomials: asymptotics of the variance