Complex Roots of Quaternion Polynomials

TL;DR

This paper investigates the complex roots of quaternion polynomials, providing conditions for different types of roots and establishing bounds on their size using Bézout matrices.

Contribution

It introduces necessary and sufficient conditions for quaternion polynomials to have complex, spherical, or isolated roots, utilizing Bézout matrices for analysis.

Findings

Conditions for the existence of complex roots

Criteria for spherical roots

Bounds on root sizes

Abstract

The polynomials with quaternion coefficients have two kind of roots: isolated and spherical. A spherical root generates a class of roots which contains only one complex number $z$ and its conjugate $\bar{z}$, and this class can be determined by $z$. In this paper, we deal with the complex roots of quaternion polynomials. More precisely, using B\'{e}zout matrices, we give necessary and sufficient conditions, for a quaternion polynomial to have a complex root, a spherical root, and a complex isolated root. Moreover, we compute a bound for the size of the roots of a quaternion polynomial.