Factorization Results for Left Polynomials in Some Associative Real Algebras: State of the Art, Applications, and Open Questions

TL;DR

This paper reviews and advances the understanding of polynomial factorizations in associative real algebras, especially quaternions and Clifford algebras, with implications for kinematics and mechanism science.

Contribution

It presents new results on polynomial factorizations over various associative algebras and discusses algorithms and open questions in the field.

Findings

Existence of factorizations for generic polynomials over division rings.

Counterexamples illustrating limitations of factorization.

Application relevance to kinematics and mechanism science.

Abstract

We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put particular emphasis on factorization results for quaternion, dual quaternion and split quaternion polynomials. A general algorithm ensures existence of a factorization for generic polynomials over division rings but we also consider factorizations for non-division rings. We explain the current state of the art, present some new results and provide examples and counter examples.