##
**A multiplication technique for the factorization of bivariate quaternionic polynomials.**
*(English)*
Zbl 1497.16027

The authors study the problem of factorizing bivariate polynomials (in two indeterminates \(t\) and \(s\)) over the skew field of quaternions, denoted \(\mathbb{H}\). It is assumed that the variables commute with each other and with the quaternionic coefficients, so that the only non-commutativity comes from the multiplication in \(\mathbb{H}\).

Quaternionic polynomials play an important role in linkage design, as they are a means to represent algebraic motions. Multiplication of such polynomials then corresponds to the composition of their underlying motions. The base cases, i.e., polynomials of degree \(1\), realize motions that are performed by (revolute or translational) joints. Hence, the factorization of a quaternionic polynomial indicates how the corresponding motion can be decomposed into simpler motions, and in case of a factorization into linear factors, allows to realize the motion by a linkage.

For univariate polynomials in \(\mathbb{H}[t]\) the factorization problem is well understood, but much less is known about factoring in \(\mathbb{H}[t,s]\). For \(Q\in\mathbb{H}[t,s]\), denote by \(Q^*\) its conjugate polynomial, obtained by conjugating all coefficients of \(Q\). The one can show that the norm polynomial \(QQ^*\) is a real polynomial in \(\mathbb{R}[t,s]\). In contrast to the univariate situation, the norm polynomial need not factorize, but still plays an important role in the algorithm. For example, the authors can show that a “univariate” factorization \(Q=h_0\cdot(u_1-h_1)\cdots(u_k-h_k)\) with \(h_i\in\mathbb{H}\) and \(u_i\in\{t,s\}\), i.e., a factorization into linear univariate factors, can only occur if \(QQ^*=PR\) for some \(P\in\mathbb{R}[t]\) and some \(R\in\mathbb{R}[s]\). Unfortunately, the converse does not hold, as the authors demonstrate with Beauregard’s example. However, they are able to show the following: for every polynomial \(Q\in\mathbb{H}[t,s]\) whose norm polynomial is the product of two univariate real polynomials, there exist a real univariate polynomial \(K\in\mathbb{R}[t]\) or \(K\in\mathbb{R}[s]\) such that \(KQ\) admits a univariate factorization. Moreover, the authors can find all different factorizations of this type (there may be up to \(m!+n!\) many where \((m,n)\) denotes the bi-degree of \(Q\)).

As an application, by extending their algorithm to the dual quaternions, the authors are able to construct a curious spatial closed-loop linkage with eight revolute joints. Since it is constructed from a bivariate polynomial, it has two degrees of freedom, but the remarkable feature is that locking any of its joints automatically locks the other three joints that are parametrized by the same variable, but leaving one degree of freedom to the remaining four joints (that are parametrized by the other variable).

Quaternionic polynomials play an important role in linkage design, as they are a means to represent algebraic motions. Multiplication of such polynomials then corresponds to the composition of their underlying motions. The base cases, i.e., polynomials of degree \(1\), realize motions that are performed by (revolute or translational) joints. Hence, the factorization of a quaternionic polynomial indicates how the corresponding motion can be decomposed into simpler motions, and in case of a factorization into linear factors, allows to realize the motion by a linkage.

For univariate polynomials in \(\mathbb{H}[t]\) the factorization problem is well understood, but much less is known about factoring in \(\mathbb{H}[t,s]\). For \(Q\in\mathbb{H}[t,s]\), denote by \(Q^*\) its conjugate polynomial, obtained by conjugating all coefficients of \(Q\). The one can show that the norm polynomial \(QQ^*\) is a real polynomial in \(\mathbb{R}[t,s]\). In contrast to the univariate situation, the norm polynomial need not factorize, but still plays an important role in the algorithm. For example, the authors can show that a “univariate” factorization \(Q=h_0\cdot(u_1-h_1)\cdots(u_k-h_k)\) with \(h_i\in\mathbb{H}\) and \(u_i\in\{t,s\}\), i.e., a factorization into linear univariate factors, can only occur if \(QQ^*=PR\) for some \(P\in\mathbb{R}[t]\) and some \(R\in\mathbb{R}[s]\). Unfortunately, the converse does not hold, as the authors demonstrate with Beauregard’s example. However, they are able to show the following: for every polynomial \(Q\in\mathbb{H}[t,s]\) whose norm polynomial is the product of two univariate real polynomials, there exist a real univariate polynomial \(K\in\mathbb{R}[t]\) or \(K\in\mathbb{R}[s]\) such that \(KQ\) admits a univariate factorization. Moreover, the authors can find all different factorizations of this type (there may be up to \(m!+n!\) many where \((m,n)\) denotes the bi-degree of \(Q\)).

As an application, by extending their algorithm to the dual quaternions, the authors are able to construct a curious spatial closed-loop linkage with eight revolute joints. Since it is constructed from a bivariate polynomial, it has two degrees of freedom, but the remarkable feature is that locking any of its joints automatically locks the other three joints that are parametrized by the same variable, but leaving one degree of freedom to the remaining four joints (that are parametrized by the other variable).

Reviewer: Christoph Koutschan (Linz)

### MSC:

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

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

70B10 | Kinematics of a rigid body |

### Keywords:

multiplication technique; bivariate factorization; necessary factorization condition; mechanism science; quaternionic polynomial; motion polynomial### Software:

Maple### References:

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