Topics on Computational Algebra

Clemson University, Clemson SC

Investigators

Abstract

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.