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

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 $${\\cal O}\\left( {d{n^4} + {d^2}{n^3}} \\right)$$ O ( d n 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 μ -bases of planar rational curves, the algorithm is among the two fastest algorithms.