The Bimodal Distribution in the Derivative of Unitary Polynomials

This is the authors' abstract. We don't add key points for this paper.

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 the derivative 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.