Efficient Quotients of Non-Commutative Polynomials

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.