Gröbner环及其环上特殊线性群的性质与算法研究

Gröbner基理论与计算是代数与符号计算研究领域的核心内容之一，在工程与数学计算中有很好的科学意义和应用前景。本项目主要研究Krull维数不超过1的环上多项式理想Gröbner基及其特殊线性群的性质与算法，具体内容如下：(1) 围绕Gröbner环猜想，研究一些环的Gröbner性质，寻找一类具有Gröbner性质的新环，并给出计算其多项式理想Gröbner基算法；(2) 运用动态方法研究Gröbner环上多项式理想的动态Gröbner基与合冲模的动态Gröbner基及其算法；(3) 利用Gröbner基理论研究维数不超过1的环上多项式环中特殊线性群的结构及其分解算法。这些研究有利于寻找更多环上多项式理想的Gröbner基算法及其矩阵分解算法。

Problem of theory and conputation for Gröbner bases is one of key problems in the research area of algebra and symbolic computation, which has a vast range of application background in mathematical theory and engineering computation. The main goal of this project is to study properties and algorithms of ideals Gröbner bases and special linear groups in multivariate polynomial ring over a ring with Krull dimension no more than 1, the specific content is as follows: (1) Environing the conjecture of Gröbner ring, we study Gröbner properties of some rings and find a new kind of rings which is Gröbner ring, furthermore obtain algorithm for computing Gröbner bases of polynomial ideals over this ring; (2) Using the method of “dynamical proof”, we study dynamical Gröbner bases agorithm for computing polynomial ideals Gröbner bases and syzygy modules on Gröbner ring; (3) Using the theory of Gröbner basis , we investigate properties and algorithms of special linear groups in polynomial ring over a ring with Krull dimension no more than 1. All of these researches are beneficial to seek more algorithms for computing ideals Gröbner bases and polynomial matrices factorization in polynomial ring.