The theory of Groebner bases is the underlying theoretical base of the computational tools being developed in commutative and noncommutative ring theory. Important results have been achieved in commutative algebra and algebraic geometry, including algorithms for polynomial factorization and computation of the greatest common divisor of polynomials. There are algorithms for the primary decomposition of an ideal, the construction of projective resolutions and syzygies, the codimension of an ideal, the Poincare and Hilbert series, and the determination of ideal membership. For noncommutative rings, Groebner bases are becoming increasingly relevant. There are complications here since free algebras and path algebras are not Noetherian in general. Recently, algorithms have been designed for finding syzygies, ideal membership, and computation of the Cartan determinant. The theory itself has led to interesting questions about algorithms and their implementations as programs. Theoretical questions on termination as well as effectiveness are being studied. This project will support a Conference on Computational Algebra to be held in May 1992 at George Mason University. The conference will focus on some theoretical and practical applications of computational algebra, including topics in commutative and noncommutative ring theory which utilize Groebner bases, as well as applications to other fields.
