Factorization results for left polynomials in some associative real algebras: state of the art, applications, and open questions.

*(English)*Zbl 1425.12001This paper is suited for graduate students and researchers of hypercomplex algebra applied to advanced mechanical problems. The origins of this work date back to the middle of the last (20th) century. It gradually evolved from motion problems described by quaternion (isomorphic to the even subalgebras of \(\mathrm{Cl}(3,0)\) or \(\mathrm{Cl}(0,3)\)) polynomials to the use of dual quaternions (with an extra commutative unit squaring to zero, isomorphic to the even subalgebra of \(\mathrm{Cl}(3,0,1)\)), split quaternions (isomorphic to the even subalgebra of \(\mathrm{Cl}(1,2)\)), and Clifford algebras in general.

Section 2 reviews the subject of polynomial factorizations of (non-commutative) rings, including an algorithm (Alg. 1), that allows the computation of linear right factors from zeros. Section 3 is on the existence (Th. 3) of polynomial factorizations over quaternions (or more general finite-dimensional associative real involutive algebras), key roles being played by the so-called maximal real polynomial factor (being equal to one) and the involution. Based on this theorem, the second algorithm (Alg. 2) produces factorizations.

Additionally, Section 4 provides examples of polynomial factorizations by first introducing Clifford algebras and spin groups based on these algebras. Of particular interest are Clifford algebras that allow isomorphisms to (non-)Euclidean space transformation groups, because factorization there can decompose rational motions into elementary motion products. In particular, here we find brief characterizations of quaternions, split quaternions and dual quaternions, and a range of characteristic examples (some of which conclude that no factorizations exist) for polynomial factorizations involving these algebras.

Section 5 is on applications of factorization in mechanism science, including concrete examples of such mechanisms (anti-parallelogram mechanism in Euclidean and hyperbolic geometry, parallelogram linkage and a scissor linkage). Finally, Section 6 goes beyond the previous theory showing more results and examples regarding the wider applicability of Alg. 2 (with example), the factorization of quadratic split quaternion polynomials, factorization of non-generic motion polynomials, of unbounded motion polynomials, and by using a projection method (for non-motion polynomials in dual quaternions, with a generalization to polynomials in Clifford algebras \(\mathrm{Cl}(p,q,1)\)).

Section 2 reviews the subject of polynomial factorizations of (non-commutative) rings, including an algorithm (Alg. 1), that allows the computation of linear right factors from zeros. Section 3 is on the existence (Th. 3) of polynomial factorizations over quaternions (or more general finite-dimensional associative real involutive algebras), key roles being played by the so-called maximal real polynomial factor (being equal to one) and the involution. Based on this theorem, the second algorithm (Alg. 2) produces factorizations.

Additionally, Section 4 provides examples of polynomial factorizations by first introducing Clifford algebras and spin groups based on these algebras. Of particular interest are Clifford algebras that allow isomorphisms to (non-)Euclidean space transformation groups, because factorization there can decompose rational motions into elementary motion products. In particular, here we find brief characterizations of quaternions, split quaternions and dual quaternions, and a range of characteristic examples (some of which conclude that no factorizations exist) for polynomial factorizations involving these algebras.

Section 5 is on applications of factorization in mechanism science, including concrete examples of such mechanisms (anti-parallelogram mechanism in Euclidean and hyperbolic geometry, parallelogram linkage and a scissor linkage). Finally, Section 6 goes beyond the previous theory showing more results and examples regarding the wider applicability of Alg. 2 (with example), the factorization of quadratic split quaternion polynomials, factorization of non-generic motion polynomials, of unbounded motion polynomials, and by using a projection method (for non-motion polynomials in dual quaternions, with a generalization to polynomials in Clifford algebras \(\mathrm{Cl}(p,q,1)\)).

Reviewer: Eckhard Hitzer (Tokyo)

##### MSC:

12D05 | Polynomials in real and complex fields: factorization |

15A66 | Clifford algebras, spinors |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |

70B15 | Kinematics of mechanisms and robots |

##### Keywords:

split quaternion; Clifford algebras; kinematics; quaternion; mechanism science; dual quaternion; polynomial division
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\textit{Z. Li} et al., J. Comput. Appl. Math. 349, 508--522 (2019; Zbl 1425.12001)

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