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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.
  • The set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials.
  • 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:
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 MonomialOrder are Lexicographic, DegreeLexicographic, DegreeReverseLexicographic 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".
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