# On monotonicity of zeros of paraorthogonal polynomials on the unit circle

## TL;DR

This paper establishes necessary and sufficient conditions for the monotonicity of zeros of paraorthogonal polynomials on the unit circle, using tridiagonal matrix coefficients, with practical conditions and an example.

## Contribution

It provides a new characterization of zero monotonicity for paraorthogonal polynomials based on primary coefficients in the tridiagonal framework.

## Key findings

- Necessary and sufficient conditions for zero monotonicity.
- Tractable sufficient conditions provided.
- Application example demonstrating the theory.

## Abstract

The purpose of this note is to establish, in terms of the primary coefficients in the framework of the tridiagonal theory developed by Delsarte and Genin in the environment of nonnegative definite Toeplitz matrices, necessary and sufficient conditions for the monotonicity with respect to a real parameter of zeros of paraorthogonal polynomials on the unit circle. It is also provided tractable sufficient conditions and an application example. These polynomials can be regarded as the characteristic polynomials of any matrix similar to an unitary upper Hessenberg matrix with positive subdiagonal elements.