Extending the GVW Algorithm to Local Ring

Abstract

A new algorithm, which combines the GVW algorithm with the Mora normal form algorithm, is presented to compute the standard bases of ideals in a local ring. Since term orders in local ring are not well-orderings, there may not be a minimal signature in an infinite set, and we can not extend the GVW algorithm from a polynomial ring to a local ring directly. Nevertheless, when given an anti-graded order in R and a term-over-position order in Rm that are compatible, we can construct a special set such that it has a minimal signature, where R , Rm are a local ring and a R -module, respectively. That is, for any given polynomial v0 ın R, the set consisting of signatures of pairs (u,v)ın Rm x R has a minimal element, where the leading power products of v and v0 are equal. In this case, we prove a cover theorem in R , and use three criteria (syzygy criterion, signature criterion and rewrite criterion) to discard useless J-pairs without any reductions. Mora normal form algorithm is also extended to do regular top-reductions in Rm x R, and the correctness and termination of the algorithm are proved. The proposed algorithm has been implemented in the computer algebra system Maple, and experiment results show that most of J-pairs can be discarded by three criteria in the examples.