The bimodal distribution in the derivative of unitary polynomials

TL;DR

This paper investigates the bimodal distribution of zeros of derivatives of unitary polynomials, identifying the source of the second mode through theoretical analysis and numerical experiments with random matrices.

Contribution

It reveals the origin of the bimodal distribution of derivative zeros and supports findings with numerical evidence from random unitary matrices.

Findings

Zeros of derivatives tend to cluster near two distinct circles.

The second mode in the distribution is explained by the polynomial's structure.

Numerical simulations confirm the theoretical predictions.

Abstract

The derivative of a polynomial with all zeros on the unit circle has the zeros of its derivative on or inside the unit circle. It has been observed that in many cases the zeros of the derivative have a bimodal distribution: there are two smaller circles near which it is more likely to find those zeros. We identify the likely source of the second mode. This idea is supported with numerical examples involving the characteristic polynomials of random unitary matrices.