Topics on Computational Algebra

计算Groebner基和寻找多项式理想的初等分解是计算代数几何中两个密切相关的基本问题。Groebner基为代数中的计算，特别是在求解多元多项式的系统中提供了一个重要的工具。已故国际象棋大师伊曼纽尔·拉斯克于1905年证明了多项式环的初等分解定理，20世纪20年代初由Emmy Noether证明了一般Notherian环的初等分解定理。初等分解是代数几何中计算机化的关键步骤，但如何为合理大小的多项式系统提供有效的算法仍然是一个巨大的挑战。主要的瓶颈是在计算初等分解过程中出现的多项式系统的Groebner基的计算。该项目的主要目标是开发新的高效算法来计算Groebner基和寻找初等分解。求解多项式系统在科学和工程中普遍存在。它的应用包括但不限于计算机视觉、计算机辅助设计、编码理论、密码学、机器人运动学、计算生物学等。该项目的工作将使教育和工业中的主要计算机代数系统及其用户受益。它还直接应用于可靠和安全的通信，从互联网、商业到军事战斗，从手机到外层空间探索。

Computing Groebner bases and finding primary decomposition of polynomial ideals are two closely related topics that are fundamental in computational algebraic geometry. Groebner bases provide an essential tool for computation in algebra, especially in solving systems of multivariate polynomials. The primary decomposition theorem was proved by the late chess Master Emanuel Lasker in 1905 for polynomial rings and Emmy Noether in early 1920s for general Noetherian rings. Primary decomposition is a crucial step in computerizing schemes in algebraic geometry, yet it is still a big challenge to provide efficient algorithms for reasonable sized systems of polynomials. The major bottleneck is in computing Groebner bases for systems of polynomials that appear in the process of computing primary decomposition. The main goal of the project is to develop new efficient algorithms for computing Groebner bases and for finding primary decomposition.Solving polynomial systems is ubiquitous in sciences and engineerings. Its applications include, but not limited to, computer vision, computer-aided designs, coding theory, cryptography, robot kinematics, computational biology, etc. Work in this project would benefit major computer algebra systems and their users in education and industry. It also bears direct applications in reliable and secure communications from Internet commercial to military combats and from cell phones to outer space explorations.