gives a list of polynomials that form a Gröbner basis for the set of polynomials .
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:
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 are , , , , or an explicit weight matrix. Monomials are specified for the purpose of by lists of the exponents with which the appear in them.
- The ordering of the and the setting for can substantially affect the efficiency of GroebnerBasis.
- Possible settings for are , Rationals, , and .
- Possible settings for the Method option include and .
Introduced in 1991
(2.0)| Updated in 2007