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.

我们考虑四元数的偏斜域上的二元多项式，其中不确定项与所有系数并相互交换。我们分析单变量的存在分解，即具有单变量线性因子的分解。单变量分解存在的必要条件是将范数多项式分解为单变量多项式的乘积。然而，这个条件是不够的。我们的中心结果表明，只要满足必要的分解条件，就可以在与合适的单变量实数多项式相乘后进行单变量分解。我们提出了一种用于计算这个实数多项式和相应的单变量分解的算法。如果存在原始多项式的单变量分解，算法的合适输入会产生一个恒定的乘法因子，从而给出后验存在单变量分解的条件。以这种方式获得的一些因式分解在机制科学中很有趣。我们展示了一个具有八个旋转关节的奇怪闭环机构的示例。