基于特征列与Groebner基内蕴关系的消去理论和方法

多项式消去理论研究如何通过消元、整序、分解将任意多元多项式系统化为具有特定代数结构的等价系统。它是计算交换代数与代数几何的基础，是符号计算的核心研究方向，在几何推理、密码学等科学与工程领域有广泛应用。目前流行的消去法主要基于特征列、三角分解和Groebner基。..申请人的最新研究表明，Ritt特征列与字典序Groebner基之间有着非常深刻的内在关联。本项目拟通过挖掘特征列与Groebner基的内蕴关系，研究并揭示两种不同理论和方法的本质差异，从而发展、建立融合两者优势、性能良好的多项式消去理论和方法。具体研究内容包括特征列与Groebner基的内蕴关系、字典序Groebner基的结构性质、非正规Ritt特征列的构造以及多项式系统的特征分解。申请人希望本项目的研究工作对特征列与Groebner基的融合产生深远影响。

Polynomial elimination theory studies how to reduce arbitrary systems of multivariate polynomials to equivalent systems with desired algebraic structures by means of variable elimination, triangularization, and decomposition. It is a fundamental tool for computational commutative algebra and algebraic geometry, is a core direction of research in symbolic computation, and has diverse applications in geometric reasoning, cryptography, and many other areas of science and engineering. Most popular elimination methods are based on characteristic sets, triangular decomposition, and Groebner bases...Recent research of the proposer shows that there are very deep and inherent connections between Ritt characteristic sets and lexicographic Groebner bases. This project is devoted to developing new and powerful theories and methods of polynomial elimination that combine the advantages of both characteristic sets and Groebner bases by exploring inherent relationships and substantial differences between characteristic sets and Groebner bases. Specific problems of research include inherent relationships between characteristic sets and Groebner bases, structural properties of lexicographic Groebner bases, construction of abnormal Ritt characteristic sets, and characteristic decomposition of polynomial systems. It is hoped that the outcome of research in the frame of this project will have a profound impact on the combination of characteristic sets and Groebner bases.