A new algorithm for computing $\mu$-bases of the univariate polynomial vector

TL;DR

This paper introduces a new, efficient algorithm for computing μ-bases of univariate polynomial vectors by leveraging the relationship between Groebner bases and u-bases, with proven complexity and practical implementation.

Contribution

The paper presents a novel algorithm for μ-base computation that improves efficiency and is based on a new characterization linking Groebner bases and u-bases.

Findings

The MinGb algorithm outperforms the SG algorithm for large n and d.

The MinGb algorithm has a theoretical complexity of O(n^3d^2).

Experimental results show competitive performance with existing methods.

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.