动态Gr？bner 基与GVW算法

项目名称： 动态Gr？bner 基与GVW算法

项目编号： No.11426101

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 李冬梅

作者单位： 湖南科技大学

项目金额： 3万元

中文摘要： Gr？bner基理论与计算是算法代数与符号计算研究领域的核心问题之一，动态Gr？bner基是利用局部实现原理将动态证明方法应用到Gr？bner基的理论与计算中。本项目主要研究计算多项式理想Gr？bner基的新算法—GVW算法与动态Gr？bner基及其关系：根据诺特赋值环上已有计算多项式理想Gr？bner基的Buchberger算法，研究寻找该环上多项式理想Gr？bner基的GVW算法；研究Dedekind环局部化的性质及该环上多项式理想基于GVW算法的动态Gr？bner基算法；找到一类新的算术环局部化后是诺特赋值环，并给出该环上多项式理想动态Gr？bner基的GVW算法。

中文关键词： Gr？bner基；GVW算法；动态Gr？bner 基；多项式环；矩阵分解

英文摘要： Theory and application of Gr？bner bases is one of key problems in the research area of algorithmic algebra and symbolic computation. Dynamical Gr？bner bases is an approach that is based on gluing local realizability appeals to use of dynamical proof methods to theory and computation of Gr？bner bases. In this project, we mainly study the new algorithm for computing ideals Gr？bner bases -GVW algorithm and dynamical Gr？bner bases and relationship between them. Based on Buchberger algorithm for computing polynomial ideals Gr？bner bases on Noetherian valuation rings, we study GVW algorithm for computing ideals Gr？bner bases on this kind of ring. We discuss properties of Dedekind ring localization and study dynamical Gr？bner bases algorithm for computing Gr？bner bases of polynomial ideals on this kind of ring, which is based on GVW agorithm. We wish to find a new kind of arithmetic ring whose localization are Noetherian valuation rings, then we can present dynamical Gr？bner bases algorithm of this class of ring.

英文关键词： Gr？bner basis；GVW algorithm；Dynamical Gr？bner bases；Polynomial ring；Matrix factorization