A Multiplication Technique for the Factorization of Bivariate Quaternionic Polynomials

TL;DR

This paper introduces a multiplication technique for bivariate quaternionic polynomials that enables univariate factorization by multiplying with a suitable real polynomial, with applications in mechanism science.

Contribution

It provides a novel algorithm to determine and compute the necessary real polynomial for univariate factorization of quaternionic polynomials.

Findings

Univariate factorizations exist after multiplication with a suitable real polynomial.

The algorithm can identify when the original polynomial admits a univariate factorization.

Applications include the analysis of mechanisms with revolute joints.

Abstract

We consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate linear factors. A necessary condition for existence of univariate factorizations is factorization of the norm polynomial into a product of univariate polynomials. This condition is, however, not sufficient. Our central result states that univariate factorizations exist after multiplication with a suitable univariate real polynomial as long as the necessary factorization condition is fulfilled. We present an algorithm for computing this real polynomial and a corresponding univariate factorization. If a univariate factorization of the original polynomial exists, a suitable input of the algorithm produces a constant multiplication factor, thus giving an a posteriori condition for existence of univariate factorizations. Some factorizations obtained in this way are of interest in mechanism science. We present an example of a curious closed-loop mechanism with eight revolute joints.