Hole radii for the Kac polynomials and derivatives

Abstract

The Kac polynomial fn(x)=∑i=0nξixi with independent coefficients of variance 1 is one of the most studied models of random polynomials.It is well-known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. In a beautiful work (Michelen, 2020), Michelen showed that the holes at ±1 are of order 1/n. We show that in fact, all the hole radii are of the same order. The same phenomenon is established for the derivatives of the Kac polynomial as well.