Hole radii for the Kac polynomials and derivatives

TL;DR

This paper investigates the size of root-free regions (holes) in Kac polynomials and their derivatives, revealing that all such holes are of order 1/n, extending previous results at specific points.

Contribution

The paper proves that all holes in the roots of Kac polynomials and their derivatives are of order 1/n, generalizing prior findings at specific points.

Findings

All holes are of order 1/n in radius.

The phenomenon extends from roots to derivatives.

Confirms uniformity of hole sizes across the unit disk.

Abstract

The Kac polynomial $$f_n(x) = \sum_{i=0}^{n} \xi_i x^i$$ 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 \cite{michelen2020real}, Michelen showed that the holes at $\pm 1$ 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.