We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials.

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分裂四元数多项式的因式分解算法

我们提出了一种算法来计算所有分解为分裂四元数上的单变量多项式的线性因子，前提是存在这样的分解。该算法的失败相当于不可分解性，我们还根据对不可逆分裂四元数的二次曲线的规则给出了几何解释。然而，分裂四元数多项式的合适的实多项式倍数仍然可以被分解，我们将描述如何找到这些实多项式。分裂四元数多项式描述双曲平面中的有理运动。使用线性因子的分解对应于将有理运动分解为双曲线旋转。由于与实数多项式相乘不会改变运动，因此这种分解总是可能的。我们的一些想法可以转移到运动多项式的分解理论中。这些是具有实范数多项式的对偶四元数的多项式，它们描述了欧几里德运动学中的有理运动。我们转移为分裂四元数开发的技术来计算某些对偶四元数多项式的新因式分解。