Built-in Mathematica Symbol

GroebnerBasis

GroebnerBasis[{poly₁, poly₂, ...}, {x₁, x₂, ...}]
gives a list of polynomials that form a Gröbner basis for the set of polynomials poly_i.

GroebnerBasis[{poly₁, poly₂, ...}, {x₁, x₂, ...}, {y₁, y₂, ...}]
finds a Gröbner basis in which the y_i have been eliminated.

The set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials.

For linear functions in any number of variables, GroebnerBasis is equivalent to Gaussian elimination.

The Gröbner basis in general depends on the ordering assigned to monomials. This ordering is affected by the ordering of the x_i.

MonomialOrder	Lexicographic	the criterion used for ordering monomials
CoefficientDomain	Automatic	the type of objects assumed to be coefficients
Method	Automatic	the method to use
Modulus	0	the modulus for numerical coefficients

Possible settings for MonomialOrder are Lexicographic, DegreeLexicographic, DegreeReverseLexicographic or an explicit weight matrix. Monomials are specified for the purpose of MonomialOrder by lists of the exponents with which the x_i appear in them.