# Wolfram Language & System 10.4 (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 .

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

## ExamplesExamplesopen allclose all

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

Compute a Gröbner basis:

 In[1]:=
 Out[1]=

Prove that polynomials have no common roots:

 In[2]:=
 Out[2]=