### Computing μ-Bases of Univariate Polynomial Matrices Using Polynomial Matrix Factorization

HUANG Bingru · CHEN Falai

1. Department of Mathematics, University of Science and Technology of China, Hefei 230000, China.
Email: hbr666@ustc.edu.cn; chenfl@ustc.edu.cn.
• Online:2021-06-25 Published:2021-03-11

HUANG Bingru · CHEN Falai. Computing μ-Bases of Univariate Polynomial Matrices Using Polynomial Matrix Factorization[J]. Journal of Systems Science and Complexity, 2021, 34(3): 1189-1206.

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 O(dn4+d2n3), 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.

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