Zero-free neighborhoods around the unit circle for Kac polynomials

In this paper, we study how the roots of the Kac polynomials concentrate around the unit circle when the coefficients are independent and identically distributed non-degenerate real random variables. It is well-known that the roots of a Kac polynomial concentrate around the unit circle as the dimension growths if and only if some log-moment is finite. Under the finiteness of the second moment, we show that there exists an annulus of small width around the unit circle which is free of roots with high probability. The proof relies on small ball probability inequalities and the least common denominator.