Concentration inequalities for the number of real zeros of Kac polynomials

We study concentration inequalities for the number of real roots of the classical Kac polynomials f_n (x) = ∑_i=0^n ξ_i x^i where ξ_i are independent random variables with mean 0, variance 1, and uniformly bounded (2+_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.