Bounds for the Zeros of Quaternionic Polynomials and Regular Functions Using Matrix Techniques

TL;DR

This paper explores bounds for the zeros of quaternionic polynomials and regular functions by employing advanced matrix techniques, extending previous work on the relationship between polynomial zeros and eigenvalues.

Contribution

It introduces new matrix-based methods to establish bounds for quaternionic polynomial zeros, broadening the understanding of their properties and eigenvalue relationships.

Findings

Established bounds for zeros of quaternionic polynomials.

Extended the relationship between polynomial zeros and eigenvalues.

Enhanced understanding of quaternionic polynomial properties.

Abstract

We investigate the problem of determining the zeros of quaternionic polynomials using matrix method. In a recent paper, Dar et al. \cite{RD} proved that the zeros of a quaternionic polynomial and the left eigenvalues of the corresponding companion matrix are identical. Building on this, we employ various newly developed matrix techniques to establish several results concerning the location of the zeros of regular polynomials of a quaternionic variable with quaternionic coefficients. These findings significantly enhance the understanding of quaternionic polynomials and their eigenvalues, offering a broader perspective on their mathematical properties.