for example in [@husty09:_algebraic_geometry_kinematics].

Now we make this group homomorphism parametric. Denote by ${{\mathbb{D}}{\mathbb{H}}}[t]$ the skew ring of polynomials over ${{\mathbb{D}}{\mathbb{H}}}$ with indeterminate $t$. We define multiplication in this ring by the convention that $t$ commutes with all coefficients. This is a natural convention because $t$ will later act as a real motion parameter and ${\mathbb{R}}$ is in the center of ${{\mathbb{D}}{\mathbb{H}}}$. Some notions that have already been defined for ${{\mathbb{D}}{\mathbb{H}}}$ can be transferred to ${{\mathbb{D}}{\mathbb{H}}}[t]$. For $C \in {{\mathbb{D}}{\mathbb{H}}}[t]$ the *conjugate polynomial* ${\overline{C}}$ is obtained by conjugating all coefficients of $C$. If $C = P + {\varepsilon}Q$ with $P, Q \in {\mathbb{H}}[t]$, then $P$ and $Q$ are called *primal* and *dual part,* respectively. The norm polynomial is ${N(C)} \coloneqq C{\overline{C}} = P{\overline{P}} + {\varepsilon}(P{\overline{Q}} + Q{\overline{P}})$. Its coefficients are dual numbers. If $C = \sum_{i=0}^n c_it^i$, the value of $C$ at $h \in {{\mathbb{D}}{\mathbb{H}}}$ is defined as $C(h) \coloneqq \sum_{i=0}^n c_ih^i$. With these definitions, evaluation of polynomials at a fixed value $h \in {{\mathbb{D}}{\mathbb{H}}}$ is not a ring homomorphism. Still, the dual quaternion zeros of polynomials over ${{\mathbb{D}}{\mathbb{H}}}$ have a meaning in our algorithms.

The polynomial $C \in {{\mathbb{D}}{\mathbb{H}}}[t]$ is called a *motion polynomial,* if $C{\overline{C}} \in {\mathbb{R}}[t] \setminus 