A new algorithm for computing $\mu$-bases of the univariate polynomial vector
Abstract
In this paper, we characterized the relationship between Groebner bases and u-bases: any minimal Groebner basis of the syzygy module for n univariate polynomials with respect to the term-over-position monomial order is its u-basis. Moreover, based on the gcd computation, we construct a free basis of the syzygy module by the recursive way. According to this relationship and the constructed free basis, a new algorithm for computing u-bases of the syzygy module is presented. The theoretical complexity of the algorithm is O(n^3d^2) under a reasonable assumption, where d is the maximum degree of the input n polynomials. We have implemented this algorithm (MinGb) in Maple. Experimental data and performance comparison with the existing algorithms developed by Song and Goldman (2009) (SG algorithm) and Hong et al. (2017) (HHK algorithm) show that MinGb algorithm is more efficient than SG algorithm when n and d are sufficiently large, while MinGb algorithm and HHK algorithm both have their own advantages.
Cite
@article{arxiv.2011.10924,
title = {A new algorithm for computing $\mu$-bases of the univariate polynomial vector},
author = {Dingkang Wang and Hesong Wang and Fanghui Xiao},
journal= {arXiv preprint arXiv:2011.10924},
year = {2021}
}
Comments
Through a researcher's reminder and after reading related references, we found that the main result (Theorem 10 and 15) in the preprint have been mentioned in a dissertation. Although we presented a new algorithm for computing u-bases, analyzed theoretical complexity of the algorithm, and gave performance comparison with existing algorithms, the contribution of this article is not enough