This paper extends the notion of μ-bases to arbitrary univariate polynomial matrices and present an efficient algorithm to compute a μ-basis for a univariate polynomial matrix based on polynomial matrix factorization. Particularly, when applied to polynomial vectors, the algorithm computes a μ-basis of a rational space curve in arbitrary dimension. The authors perform theoretical complexity analysis in this situation and show that the computational complexity of the algorithm is \({\cal O}\left( {d{n^4} + {d^2}{n^3}} \right)\), where n is the dimension of the polynomial vector and d is the maximum degree of the polynomials in the vector. In general, the algorithm is n times faster than Song and Goldman’s method, and is more efficient than Hoon Hong’s method when d is relatively large with respect to n. Especially, for computing μ-bases of planar rational curves, the algorithm is among the two fastest algorithms.

本文将μ基的概念扩展到任意一元多项式矩阵，并提出了一种基于多项式矩阵分解的有效算法来计算一元多项式矩阵的μ基。特别地，当应用于多项式矢量时，该算法计算任意维度上的有理空间曲线的μ基。作者在这种情况下进行了理论复杂度分析，并表明该算法的计算复杂度为\（{\ cal O} \ left（{d {n ^ 4} + {d ^ 2} {n ^ 3}} \ right ）\），其中n是多项式向量的维数，d是向量中多项式的最大次数。在一般情况下，该算法是ñ当d相对于n较大时，它的速度比Song和Goldman的方法快两倍，并且比Hoon Hong的方法更高效。特别地，为了计算平面有理曲线的μ-基，该算法是两个最快的算法之一。