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.
