The Grobner package in Maple and computer algebra system for solving multivariate polynomial equations / Shamsatun Nahar Ahmad and Nor'aini Aris

This paper is a preliminary survey on the developments in the techniques of solving multivariate polynomial equations. Currently two main approaches originating from algebraic geometry have been used to compute the roots of a zero dimensional polynomial system. The first approach involves Grobner bases computations. This method involved computing common roots by eliminating a set of variables from a system of polynomial equations and thereby reducing the problem to a sequence of univariate polynomials. The other approach is based on resultant formulations, which can eliminate many variables simultaneously and can also be performed in floating point arithmetic. The resultant techniques can also be viewed from linear algebra to reduce the root computations to a nonsingular eigenvector problem and to find approximate values of the solutions. In this paper, we present an overview of the stages and development in the Grobner basis techniques and to discuss some basic implementations of the Grobner package in Maple and the computer algebra system related to solving multivariate polynomial equations.