Concentration inequalities for the number of real zeros of Kac polynomials

Abstract

We study concentration inequalities for the number of real roots of the classical Kac polynomials $$f_{n} (x) = \sum_{i=0}^n \xi_i x^i$$ where $\xi_i$ are independent random variables with mean 0, variance 1, and uniformly bounded $(2+\ep_0)$-moments. We establish polynomial tail bounds, which are optimal, for the bulk of roots. For the whole real line, we establish sub-optimal tail bounds.