Factorization of Multivariate Quaternionic Polynomials

Quaternion polynomials and their factorizations have attracted a lot of attention in recent years. This interest stems from the relationships between linear factors of such polynomials and the geometry of a mechanical linkage capable of realising the motion it parametrises. This led ultimately to an extension of Kempe's universality theorem to rational, spatial curves. Generalising these results to the multivariate case is of great interest to roboticists, as it could give a purely algebraic design method for parallel manipulators. In this article, we describe a factorization algorithm for general multivariate quaternion polynomials, as well as characterise some classes of polynomials admitting multiple non-equivalent factorizations. One of these classes-one we called "aromatic"- is particularity interesting from the perspective of theoretical kinematics, as it is factorable into polynomials parametrising rotations about constant axes. (c) 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.