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. In this article, 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. For skew polynomials, the computation of whole shifted inverses is not asymptotically fast, but once obtained, quotients may be computed with one multiplication. 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单变量多项式环中高效计算商。这扩展了先前以交换域为核心引入通用高效商的研究工作。本文针对多项式变量与系数可交换的情形，给出了左商与右商的快速算法。这些算法基于“完备移位逆”概念——即一种被除数为多项式变量幂次的特化商。当变量与系数不可交换（即斜多项式）时，定义了左、右完备移位逆，且可利用左完备移位逆计算右商。对于斜多项式，完备移位逆的计算不具备渐近快速性，但一旦获得，可通过一次乘法完成商的计算。文中展示了矩阵系数多项式与微分算子的示例，并给出了概念验证性的Maple实现。