gives a list of polynomials that form a Gröbner basis for the set of polynomials polyi.
finds a Gröbner basis in which the yi have been eliminated.
Details and Options
- 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 xi.
- 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 MonomialOrder are Lexicographic, DegreeLexicographic, DegreeReverseLexicographic, EliminationOrder, or an explicit weight matrix. Monomials are specified for the purpose of MonomialOrder by lists of the exponents with which the xi appear in them.
- The ordering of the xi and the setting for MonomialOrder can substantially affect the efficiency of GroebnerBasis.
- Possible settings for CoefficientDomain are InexactNumbers, Rationals, RationalFunctions, and Polynomials[x].
- Possible settings for the Method option include "Buchberger" and "GroebnerWalk".
Examplesopen allclose all
The Automatic method setting uses "GroebnerWalk" for lexicographic bases over the rationals:
By default, GroebnerBasis uses the Lexicographic monomial order:
By default, GroebnerBasis is not allowed to reorder the variables:
Properties & Relations (6)
Use PolynomialReduce to show that p1 is in the ideal generated by g1 and g2:
Use FindInstance to find a solution instance:
GroebnerBasis of linear polynomials is equivalent to a Gaussian elimination process:
GroebnerBasis is used to solve systems of polynomial equations:
Use Reduce to directly solve the system:
Solve gives solutions in terms of replacement rules:
Eliminate a variable using Resolve:
Eliminate a variable using Eliminate:
Eliminate a variable using Resultant:
Wolfram Research (1991), GroebnerBasis, Wolfram Language function, https://reference.wolfram.com/language/ref/GroebnerBasis.html (updated 2007).
Wolfram Language. 1991. "GroebnerBasis." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/GroebnerBasis.html.
Wolfram Language. (1991). GroebnerBasis. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GroebnerBasis.html