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.
Euclidean algorithm for polynomial greatest common divisor computation and Gaussian elimination of linear systems are special cases of Buchberger's algorithm when the number of variables or the degrees of the polynomials are respectively equal to one.
For other Gröbner basis algorithms, see Gröbner basis § Algorithms and implementations.