Computing Mu-Bases Of Univariate Polynomial Matrices Using Polynomial Matrix Factorization

This paper extends the notion of mu-bases to arbitrary univariate polynomial matrices and present an efficient algorithm to compute a mu-basis for a univariate polynomial matrix based on polynomial matrix factorization. Particularly, when applied to polynomial vectors, the algorithm computes a mu-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 O(dn(4)+d(2)n(3)), 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 mu-bases of planar rational curves, the algorithm is among the two fastest algorithms.