On the number of roots of self-inversive polynomials on the complex unit circle

TL;DR

This paper establishes a sufficient condition for self-inversive polynomials to have a fixed number of simple roots on the complex unit circle, extending previous results to partial root configurations.

Contribution

It introduces a new sufficient condition ensuring a specific number of roots on the unit circle and proves their simplicity under this condition.

Findings

Identifies a condition for fixed roots on the unit circle

Proves roots are simple when the condition is met

Generalizes previous full-root-on-circle results

Abstract

We present a sufficient condition for a self-inversive polynomial to have a fixed number of roots on the complex unit circle. We also prove that these roots are simple when that condition is satisfied. This generalizes the condition found by Lakatos and Losonczi for all the roots of a self-inversive polynomial to lie on the complex unit circle.