Gröbner Basis

Abstract

A Gröbner basis is a set of multivariate polynomi-als that has desirable algorithmic properties. Every set of polynomials can be transformed into a Gröb-ner basis. This process generalizes three familiar techniques: Gaussian elimination for solving linear systems of equations, the Euclidean algorithm for computing the greatest common divisor of two univariate polynomials, and the Simplex Algorithm for linear programming; see [3]. For example, the input for Gaussian elimination is a collection of linear forms such as F = 2x + 3y + 4z − 5, 3x + 4y + 5z − 2 , and the algorithm transforms F into the Gröbner basis G = x − z + 14, y + 2z − 11 .