Buchberger's Algorithm

An algorithm that, given a finite set of multivariate polynomials in a polynomial ring over a field and a monomial ordering, produces a Gröbner basis for the ideal they generate; the output is a finite generating set whose leading terms generate the ideal of leading terms.

An iterative procedure that constructs a Gröbner basis from a given finite set of polynomials by repeatedly forming S-polynomials of pairs, reducing them modulo the current basis, and adjoining nonzero remainders until all S-polynomials reduce to zero.