Buchberger's algorithm

Summary

Buchberger's algorithm turns a finite set of generators of an ideal of a polynomial ring into a Gröbner basis of the same ideal. It repeatedly forms S-polynomials of pairs of elements, divides them by the current set with the multivariate division algorithm, and adds the nonzero remainders. It was introduced by Bruno Buchberger, who also named Gröbner bases after his advisor Wolfgang Gröbner. It plays the role for polynomial systems that Gaussian elimination plays for linear systems and the Euclidean algorithm plays for polynomials in one variable.