Signature-based standard basis algorithm under the framework of GVW algorithm

IF 0.6 4区 数学 Q4 COMPUTER SCIENCE, THEORY & METHODS
Dong Lu , Dingkang Wang , Fanghui Xiao , Xiaopeng Zheng
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引用次数: 0

Abstract

The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm.

GVW 算法框架下基于签名的标准基础算法
GVW 算法是最重要的所谓基于签名的算法之一,旨在消除布赫伯格算法中大量无用的多项式还原。覆盖定理是 GVW 算法的理论基础,到目前为止,它只适用于某一类单项式阶,即全局阶和一类特殊的局部阶。在本文中,我们将这一定理扩展到任何半群阶,它既可以是全局阶,也可以是局部阶,甚至可以是混合阶。在莫拉正则表达式算法这一开创性思想的基础上,我们提出了一个更全面、更通用的覆盖定理证明,同时绕过了所有现有证明中需要从无限单项式集合中选择一个最小元素的问题。因此,我们在 GVW 算法的框架下,针对任何半群阶提出了基于签名的标准基算法,并举例说明了该算法。
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来源期刊
Journal of Symbolic Computation
Journal of Symbolic Computation 工程技术-计算机:理论方法
CiteScore
2.10
自引率
14.30%
发文量
75
审稿时长
142 days
期刊介绍: An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.
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