算法代数与符号计算在数学理论与工程计算中有重要的科学意义与应用背景，Serre猜想与多维矩阵的Serre约化是算法代数与符号计算交叉融合形成的重要方向。本项目主要研究内容：（1）围绕Serre猜想，研究一般赋值环R是Hermite环这一开问题，进一步证明SLn(R[x])=En(R[x])。（2）研究多维矩阵的递归Serre约化，寻找实现分级Serre约化的原理与方法。(3)研究多维矩阵可Serre约化到它的Smith型的充要条件，并给出条件的精细刻画。（4）运用理想和模的Gröbner基理论与方法，寻找Serre约化新的充要条件，并设计检验条件的程序算法。这些研究将极大地丰富算法代数与符号计算的内容、思想与方法。

Algorithmic Algebra and Symbolic Computation have important scientific signficance and wide application in mathematical theory and engineering calculation. Serre conjecture and Serre reduction of multidimensional matrices are important directrions for the cross-fusion of algorithm algebra and symbolic computation .The main researches of this project are as follows: (1) Surround the Serre's conjecture, the paper studies and proves the open problem that valuation ring R is Hermite, Furthermore we will prove that SLn(R[x])=En(R[x]). (2) The recursive Serre reduction of multidimensional matrices is studied, and search for the principle and method of realizing hierarchical Serre reduction. (3) The necessary and sufficient conditions for a multidimensional matrix to Serre reduce to its Smith-type are studied, and the fine characterization of the conditions is given. (4) Using the theory and method of Gröbner for ideals and modules,we find some new necessary and sufficient conditions for Serre reduction,and design program algorithms for checking conditions. These researches will greatly enrich the contents ideas and methods of algebraic and symbolic computation.