It is shown how to compute quotients efficiently in non-commutative univariate polynomial rings. This expands on earlier work where generic efficient quotients were introduced with a primary focus on commutative domains. Asymptotically fast algorithms are given for left and right quotients when the polynomial variable commutes with coefficients. These algorithms are based on the concept of the ``whole shifted inverse'', which is a specialized quotient where the dividend is a power of the polynomial variable. When the variable does not commute with coefficients, that is for skew polynomials, left and right whole shifted inverses are defined and the left whole shifted inverse may be used to compute the right quotient, although not with asymptotically fast complexity. Examples are shown of polynomials with matrix coefficients and differential operators and a proof-of-concept Maple implementation is given.

翻译：本文展示了如何在非交换一元多项式环中高效计算商。该研究扩展了先前以交换域为主要关注对象的通用高效商运算工作。当多项式变量与系数可交换时，给出了左商和右商的渐近快速算法。这些算法基于"整体移位逆"概念——这是一种特殊商，其中被除数为多项式变量的幂。当变量与系数不可交换（即扭多项式）时，定义了左、右整体移位逆，并可使用左整体移位逆计算右商，但无法实现渐近快速复杂度。文中展示了矩阵系数多项式与微分算子的示例，并给出了概念验证的Maple实现。