环的Gröbner性质及多元多项式矩阵的分解与等价问题研究

环的Gröbner性质及多元多项式矩阵分解与等价是计算机代数中的重要研究方向，它在代数结构与符号计算的研究中发挥重要作用。本项目主要研究内容为:（1）围绕Gröbner环猜想，研究一些环的Gröbner性质，证明阿基米德环是Gröbner环，寻求一类具有Gröbner性质的算术环，并给出这些Gröbner环上计算多项式理想的动态Gröbner基算法；（2）运用Gröbner基理论与方法研究多元多项式矩阵及其Smith型等价问题，寻求判别它们等价的充分、必要条件和可执行算法；（3）研究多元多项式矩阵的Serre约化问题，寻求实现一些多元多项式矩阵具有Serre约化的有效判别条件及算法；（4）研究多项式环中可逆矩阵的结构及其分解成第三类初等矩阵乘积的算法。这些研究不仅可以发现一些新的环上计算多项式理想Gröbner基的算法，也可以得到多项式矩阵分解与等价的新算法。

The property of Gröbner rings and multilvariate polynomial matrix factorization and equivalence is one of important research direction in computer algebra. It plays a crucial part in the study of algebraic structure and symbolic computation. The main researches of this project are as follows: (1) Based on the Gröbner ring conjecture, we study the Gröbner properties of some rings, and prove that Archimedean rings are Gröbner rings, and sought a kind of arithmetic ring with Gröbner property. Furthermore, we obtain the dynamical Gröbner basis algorithm of polynomial ideal in Gröbner rings. (2) Using the Gröbner basis theory and method , we study the equivalence problems of the multivariate polynomial matrices and their Smith-type, and find the sufficient conditions, necessary conditions and executable algorithms for their equivalence. (3) The problem of Serre's reduction for the multivariate polynomial matrices is studied, and some effective discriminant conditions and algorithms of which are sought. (4) We research on the structure of invertible matrices in polynomial ring and the algorithm of an invertible matrix is decomposed into product of the third elementary matrices. These researches not only present some new algorithms for calculating the Gröbner bases of the polynomial ideal in the ring, but also obtain new algorithms of polynomial matrix factorization and equivalence.