Parallel algorithms for Gröbner-basis construction

Abstract

The problem of the Grobner-basis construction is important both from the theoretical and applied points of view. As examples of applications of Grobner bases, one can mention the consistency problem for systems of nonlinear algebraic equations and the determination of the number of solutions to a system of nonlinear algebraic equations. The Grobner bases are actively used in the constructive theory of polynomial ideals and at the preliminary stage of numerical solution of systems of nonlinear algebraic equations. Unfortunately, many real examples cannot be processed due to the high computational complexity of known algorithms for computing the Grobner bases. However, the efficiency of the standard basis construction can be significantly increased in practice. In this paper, we analyze the known algorithms for constructing the standard bases and consider some methods for increasing their efficiency. We describe a technique for estimating the efficiency of paralleling the algorithms and present some estimates.