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GroebnerBasis

FilledSmallSquare GroebnerBasis[, , ... , , , ... ] gives a list of polynomials that form a Gröbner basis for the set of polynomials .

FilledSmallSquare GroebnerBasis[, , ... , , , ... , , , ... ] finds a Gröbner basis in which the have been eliminated.

FilledSmallSquare The set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials.

FilledSmallSquare For polynomials in one variable, GroebnerBasis reduces to PolynomialGCD.

FilledSmallSquare For linear functions in any number of variables, GroebnerBasis is equivalent to Gaussian elimination.

FilledSmallSquare The Gröbner basis in general depends on the ordering assigned to monomials. This ordering is affected by the ordering of the .

FilledSmallSquare The following options can be given:

FilledSmallSquare 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 appear in them.

FilledSmallSquare The ordering of the and the setting for MonomialOrder can substantially affect the efficiency of GroebnerBasis.

FilledSmallSquare Possible settings for CoefficientDomain are InexactNumbers, Rationals, RationalFunctions and Polynomials[x].

FilledSmallSquare See Section 3.3.4.

FilledSmallSquare Implementation Notes: see Section A.9.5.

FilledSmallSquare See also: PolynomialReduce, PolynomialGCD, Reduce, Solve, RowReduce, Eliminate, FindInstance, CylindricalDecomposition.

FilledSmallSquare New in Version 2; modified in 3.

Further Examples