动态Gröbner 基与GVW算法

Gröbner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Gröbner基是利用局部实现原理将动态证明方法应用到Gröbner基的理论与计算中。本项目主要研究计算多项式理想Gröbner基的新算法—GVW算法与动态Gröbner基及其关系：根据诺特赋值环上已有计算多项式理想Gröbner基的Buchberger算法，研究寻找该环上多项式理想Gröbner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Gröbner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Gröbner基的GVW算法。

Theory and application of Gröbner bases is one of key problems in the research area of algorithmic algebra and symbolic computation. Dynamical Gröbner bases is an approach that is based on gluing local realizability appeals to use of dynamical proof methods to theory and computation of Gröbner bases. In this project, we mainly study the new algorithm for computing ideals Gröbner bases -GVW algorithm and dynamical Gröbner bases and relationship between them. Based on Buchberger algorithm for computing polynomial ideals Gröbner bases on Noetherian valuation rings, we study GVW algorithm for computing ideals Gröbner bases on this kind of ring. We discuss properties of Dedekind ring localization and study dynamical Gröbner bases algorithm for computing Gröbner 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 Gröbner bases algorithm of this class of ring.