Topics on Computational Algebra

计算Groebner基和求多项式理想的准素分解是计算代数几何中两个密切相关的基本课题。Groebner基为代数计算提供了一个重要的工具，特别是在求解多元多项式系统时。准素分解定理是由已故国际象棋大师伊曼纽尔·拉斯克（英语：Emanuel Lasker）在1905年证明的多项式环，以及埃米·诺特（英语：Emmy Noether）在20世纪20年代早期证明的一般诺特环。初等分解是代数几何中计算格式的关键步骤，然而对于合理大小的多项式组提供有效的算法仍然是一个很大的挑战。主要的瓶颈是在计算过程中出现的多项式系统的Groebner基地的计算小学分解。该项目的主要目标是开发新的有效算法来计算Groebner基和寻找初等分解。求解多项式系统在科学和工程中无处不在。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，在这个项目中的工作将有利于主要的计算机代数系统及其用户在教育和工业。它还直接应用于从互联网商业到军事战斗以及从手机到外层空间探索的可靠和安全通信。

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.