多元多项式环的Hermite性质与多项式矩阵的分解

Algorithmic algebra and symbolic computation have important scientific significance and a vast application background in mathematical theory and engineering computation. The Hermite property of multivariate polynomial ring and multivariate polynomial matrix is an important research direction which is formed by the cross penetration of algorithmic algebra and symbolic computations. The main researches of our project are Hermite property of ring and factorization of multivariate polynomial matrix and relationship between them, the specific content are as follows: (1) Discussing the conjecture of Hermite ring, study a new kind of ring R such that R[X] is Hermite ring. (2) Based on Hermite property of R, discuss the Hermite property of multivariate polynomial ring over R. (3) Study the matrix on the Hermite ring R[X], focusing on studying a new kind of ring R so that unimodular matrix on R[X] can be factorized into the product of some elementary matrices, and factorization algorithm. (4) Study and seek an algorithm in which a ZLP matrix on a multivariate polynomial ring over R can be embedded in the unimodular matrix. The significance of these study are that: On the one hand, we figure out the relationship between the polynomial ring and matrix to discover a new class of Hermite ring; on the other hand, we find new algorithms about the factorization and embedding of polynomial matrix.

算法代数与符号计算在数学理论与工程计算中有很好的科学意义与应用背景，多元多项式环的Hermite性质与多元多项式矩阵是算法代数与符号计算交叉渗透而形成的重要研究方向。本项目主要研究环的Hermite性质与多元多项式矩阵的分解以及它们之间的联系，具体内容如下：(1) 围绕Hermite环猜想，研究一类新的环R使得R[X]是Hermite环。(2) 在R是Hermite环的基础上，研究R上的多元多项式环的Hermite性质。(3) 研究Hermite环R[X]上的矩阵，重点研究一类新的环R使得R[X]上的幺模矩阵分解成一些初等矩阵的乘积与分解的算法。(4) 研究并寻找R上的多元多项式环上ZLP矩阵嵌入到幺模矩阵中的算法。这些研究的意义在于一方面可以利用多项式环与矩阵的关系发现一类新的Hermite环，另一方面发现多项式矩阵分解与嵌入的新算法。

算法代数与符号计算在数学理论与工程计算中有很好的科学意义与应用背景，多元多项式环的Hermite性质与多元多项式矩阵是算法代数与符号计算交叉渗透而形成的重要研究方向。本项目主要研究环的Hermite性质与多元多项式矩阵的分解以及它们之间的联系，具体内容如下：（1）围绕Hermite环猜想，研究一类新的环（赋值环）R使得R[X]是Hermite环。（2）在R是Hermite环的基础上，研究R上的多元多项式环的Hermite性质。（3）研究Hermite环R[X]上的矩阵，重点研究一类新的环R使得R[X]上的幺模矩阵分解成一些初等矩阵的乘积与分解的算法（4）研究并寻找R上的多元多项式环上ZLP矩阵嵌入到幺模矩阵中的算法。最后，我们研究与Hermite环和矩阵相关的4个猜想以及 Gröbner基的判定准则。