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.

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