Computer and Computational Algebra

这是一个跨学科的研究项目，旨在开发、实现和应用计算机和计算代数中的算法。将在四个领域进行研究：计算代数和代数几何；实闭域和代数单元分解；积分和求根技术；以及在流和动力系统中的应用。在计算代数和代数几何领域，将研究寻找计算初等分解的改进方法、求逆变形、理想逆系统的计算和计算Hilbert函数的问题，以及开发寻找Groebner基的并行算法、将子代数隶属度确定应用于寻找多项式函数分解和改进利用有理函数域的子域计算的子代数算法。我们将研究多元结式在一般交换环上的推广。计算机代数系统Macaulay将扩展到处理样条、初等分解、理想根、非齐次Groebner基、多项式因式分解、有理算术和浮点数。Macaulay将在求根、Hilbert函数计算、表示单项理想的数据结构以及减少虚拟内存抖动方面进行改进。在实数域、代数闭域和代数元胞分解方面，将改进和实现Ben-or、Kozen、Reif判定过程，并利用多项式求根技术给出计算代数元胞分解完全邻接关系的改进算法。在积分和求根技术方面，将研究将正特征积分提升回特征零的问题，并将实现逼近实系数多项式的所有实根的Ben-or、Feig、Kozen、Tiwari算法。最后，在流和动力系统领域，Groebner基技术将应用于计算多项式流的极限环个数的界，并将开发和实现几种相关摄动方法的通用实用程序。

This is an interdisciplinary research project for development, implementation, and application of algorithms in computer and computational algebra. Research will be done in four areas: Computational Algebra and Algebraic Geometry; Real Closed Fields and Algebraic Cell Decomposition; Integration and Root Finding Techniques; and Applications in Flows and Dynamical Systems. In the area of computational algebra and algebraic geometry, such problems as finding improved methods for computing primary decompositions, finding versal deformations, computing with inverse systems of ideals, and calculating Hilbert functions will be investigated as well as developing parallel algorithms for finding Groebner bases, applying subalgebra membership determination to finding polynomial functional decomposition and improving subalgebra algorithms for computing with subfields of rational function fields. A generalization of multivariate resultants to general commutative rings will be investigated. The computer algebra system Macaulay will be extended to handle splines, primary decompositions, ideal radicals, inhomogeneous Groebner bases, polynomial factorization, rational arithmetic and floats. Macaulay will be improved with regard to root finding, Hilbert function calculation, data structures representing monomial ideals, and reducing virtual memory thrashing. In the area of real and algebraically closed fields and algebraic cell decomposition, the Ben-Or, Kozen, Reif decision procedure will be improved and implemented and polynomial root finding techniques will be used to give an improved algorithm for computing full adjacency relations in algebraic cell decomposition. In the area of integration and root finding techniques, the problem of lifting positive characteristic integration back to characteristic zero will be investigated and the Ben-Or, Feig, Kozen, Tiwari algorithm for approximating all real roots of a polynomial with real coefficients will be implemented. Finally, in the area of flows and dynamical systems, Groebner basis techniques will be applied to the problem of calculating a bound on the number of limit cycles of a polynomial flow and general utilities for the implementation of several related perturbation methods will be developed and implemented.