Algorithm for computing μ-bases of univariate polynomials. (May 2017) - Hong, Hoon; Hough, Zachary; Kogan, Irina A.
Abstract: 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 . The algorithm involves computing a "partial" reduced row-echelon form of a ( 2 d + 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. The proof 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 2 n + 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 is faster than HHK when d is sufficiently large for a fixed n, and that HHK is faster than SG when n is sufficiently large for a fixed d .