Abstract
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.
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This research was supported by the National Natural Science Foundation of China under Grant No. 61972368.
This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.
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Huang, B., Chen, F. Computing μ-Bases of Univariate Polynomial Matrices Using Polynomial Matrix Factorization. J Syst Sci Complex 34, 1189–1206 (2021). https://doi.org/10.1007/s11424-020-9314-6
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DOI: https://doi.org/10.1007/s11424-020-9314-6