GroebnerBasis[, , ... , , , ... ] gives a list of polynomials that form a Gröbner basis for the set of polynomials .
GroebnerBasis[, , ... , , , ... , , , ... ] finds a Gröbner basis in which the have been eliminated.
The set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials.
For polynomials in one variable, GroebnerBasis reduces to PolynomialGCD.
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 .
The following options can be given:
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 appear in them.
The ordering of the and the setting for MonomialOrder can substantially affect the efficiency of GroebnerBasis.
Possible settings for CoefficientDomain are InexactNumbers, Rationals, RationalFunctions and Polynomials[x].
See The Mathematica Book: Section 3.3.4.
Implementation Notes: see section A.9.5.
See also: PolynomialReduce, PolynomialGCD, Solve, RowReduce, Eliminate.