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The bimodal distribution in the derivative of unitary polynomials

David W. Farmer

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.

The bimodal distribution in the derivative of unitary polynomials

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.

Paper Structure

This paper contains 9 sections, 1 theorem, 4 equations.

Table of Contents

  1. Random characteristic polynomials
  2. Some experiments to gain intuition
  3. The local arrangement is most important, when close to the unit circle
  4. Relative frequency of neighbor spacings
  5. Curvature effects
  6. The source of the second bump
  7. A toy model
  8. Excised roots of unity
  9. A final remark

Key Result

Proposition 3.1

The derivative of $f_n$ has a zero within $O(1/n^2)$ of $1 - b_0/n$, where $b_0 = 2.3565\ldots$ is the unique real root of $4 \pi^2 +2 b + b^2 - 2 b e^b$.

Theorems & Definitions (3)

  • Claim 2.1
  • Proposition 3.1
  • proof