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, ther...
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ISBN:
(纸本)9781450355506
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. R, the set consisting of signatures of pairs (u, v). 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.
gvw algorithm was given by Gao, Wang, and Volny in computing a Grobuer bases for ideal in a polynomial ring, which is much faster and more simple than F5. In this paper, the authors generalize gvw algorithm and presen...
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gvw algorithm was given by Gao, Wang, and Volny in computing a Grobuer bases for ideal in a polynomial ring, which is much faster and more simple than F5. In this paper, the authors generalize gvw algorithm and present an algorithm to compute a Grobner bases for ideal when the coefficient ring is a principal ideal domain. K
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