Buchberger's algorithm

In the theory of multivariate polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials that have the same common zeros and are more convenient for extracting information on these common zeros. It was introduced by Bruno Buchberger simultaneously with the definition of Gröbner bases.

The Euclidean algorithm for computing the polynomial greatest common divisor is a special case of Buchberger's algorithm restricted to polynomials of a single variable. Gaussian elimination of a system of linear equations is another special case where the degree of all polynomials equals one.

For other Gröbner basis algorithms, see Gröbner basis § Algorithms and implementations.