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:
  • MonomialOrderLexicographicthe criterion used for ordering monomials
    CoefficientDomainAutomaticthe type of objects assumed to be coefficients
    MethodAutomaticthe method to use
    Modulus0the 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 .
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