In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real ("motion polynomials"), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition is also sufficient for existence of factorizations. We present an algorithm to compute factorizations of these polynomials and use it for new constructions of mechanisms which cannot be obtained by existing factorization algorithms for motion polynomials. While they produce mechanisms with rotational or translational joints, our approach yields mechanisms consisting of "vertical Darboux joints". They exhibit mechanical deficiencies so that we explore ways to replace them by cylindrical joints while keeping the overall mechanism sufficiently constrained.

中文翻译：

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无研究条件的对偶四元数多项式的因式分解

在本文中，我们研究了对偶四元数环上的多项式分解为线性因子。虽然较早的结果假设范数多项式是实数（“运动多项式”），但我们只需要原始部分中不存在实数多项式因子，以及对偶数上的范数多项式可分解为单次二次因子。这个明显的必要条件也是因式分解存在的充分条件。我们提出了一种算法来计算这些多项式的因式分解，并将其用于新的机制构造，而现有的运动多项式因式分解算法无法获得这些新构造。虽然它们产生具有旋转或平移关节的机制，但我们的方法产生了由“垂直 Darboux 关节”组成的机制。