Alternating Roots of Polynomials over Cayley-Dickson Algebras

TL;DR

This paper introduces the concept of alternating roots for polynomials over Cayley-Dickson algebras, establishing a link between roots of original and alternating polynomials, and provides computational tools for quaternions.

Contribution

It defines alternating roots over Cayley-Dickson algebras, proves their relation to polynomial roots, and offers practical Octave code for quaternion computations.

Findings

Established a connection between roots of polynomials and their alternating counterparts

Provided an algorithmic approach for computing alternating roots over quaternions

Extended polynomial root theory to non-associative Cayley-Dickson algebras

Abstract

We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding alternating polynomial over the Cayley-Dickson doubling of the algebra. We also include a detailed Octave code for the computation of alternating roots over Hamilton's quaternions.