A new method of finding all Roots of simple quaternionic polynomials

TL;DR

This paper introduces a more efficient and simpler method for finding all roots of simple quaternionic polynomials, improving upon previous techniques and providing new insights into their solutions.

Contribution

The paper presents a novel, more efficient method for solving simple quaternionic polynomials, along with new results and conditions for the nature of their solutions.

Findings

Method is more efficient and simpler than previous approaches.

Provides necessary and sufficient conditions for finite solutions.

Recovers and extends known results in quaternionic polynomial equations.

Abstract

In this paper, we provide a new method to find all zeros of polynomials with quaternionic coefficients located on only one side of the powers of the variable (these polynomials are called simple polynomials). This method is much more efficient and much simpler than the known one in [D. Janovska and G. Opfer, A note on the computation of all zeros of simple quaternionic polynomials, SIAM J. Numer. Anal., 48(1)(2010), pp. 244-256]. We recover several known results, and deduce several interesting consequences concerning solving equations with all real coefficients or complex coefficients which do not seem to be deduced easily from the results in [D. Janovska and G. Opfer, A note on the computation of all zeros of simple quaternionic polynomials, SIAM J. Numer. Anal., 48(1)(2010), pp. 244-256]. We also give a necessary and sufficient condition for a simple quaternionic polynomials to have finitely many solutions (only isolated solutions).