Computing H-bases via minimal bases for syzygy modules

摘要

The main objective of this paper is to present an improved algorithm to compute H-bases by relying on techniques from linear algebra. An H-basis is a specific generating set for a polynomial ideal whose computation depends on the prior knowledge of a generating set for a syzygy module. The well-known method to calculate a set of generators of a syzygy module is based on a Grobner basis computation which may computationally complicate the problem. To deal with this issue, we propose a linear-algebra-based method to compute a minimal generating set for the syzygy module of a given homogeneous ideal. Our main idea depends on constructing syzygies degree by degree and removing the redundant generators in each degree. The effectiveness of employing our proposed method to compute a minimal basis for a syzygy module in the H-bases structure is demonstrated by means of a diverse set of benchmark polynomials in terms of running time and required memory compared to the original algorithm. We have implemented all the proposed algorithms in MAPLE.