Algorithm for computing μ-bases of univariate polynomials
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We present a new algorithm for computing a μ-basis of the syzygy module of n polynomials in one variable over an arbitrary field K. The algorithm is conceptually different from the previously-developed algorithms by Cox, Sederberg, Chen, Zheng, and Wang for n=3, and by Song and Goldman for an arbitrary n. It involves computing a "partial" reduced row-echelon form of a (2d+1)× n(d+1) matrix over K, where d is the maximum degree of the input polynomials. The proof of the algorithm is based on standard linear algebra and is completely self-contained. It includes a proof of the existence of the μ-basis and as a consequence provides an alternative proof of the freeness of the syzygy module. The theoretical (worst case asymptotic) computational complexity of the algorithm is O(d^2n+d^3+n^2). We have implemented this algorithm (HHK) and the one developed by Song and Goldman (SG). Experiments on random inputs indicate that SG gets faster than HHK when d gets sufficiently large for a fixed n, and that HHK gets faster than SG when n gets sufficiently large for a fixed d.
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