# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# GroebnerBasis

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

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

## Details and OptionsDetails 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".

## ExamplesExamplesopen allclose all

### Basic Examples  (1)Basic Examples  (1)

Compute a Gröbner basis:

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Prove that polynomials have no common roots:

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