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GroebnerBasis—Wolfram Language Documentation

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• See Also
  • PolynomialReduce
  • PolynomialGCD
  • Reduce
  • Solve
  • RowReduce
  • Eliminate
  • FindInstance
  • CylindricalDecomposition
  • MonomialList
  • CoefficientRules
• Related Guides
  • Polynomial Systems
  • Polynomial Division
  • Theorem Proving
  • Computational Geometry
  • Polynomial Algebra
• Tech Notes
  • Algebraic Operations on Polynomials
  • Polynomials Modulo Primes
  • Complex Polynomial Systems
  • Implementation notes: Algebra and Calculus

GroebnerBasis[{poly₁,poly₂,…},{x₁,x₂,…}]

gives a list of polynomials that form a Gröbner basis for the set of polynomials poly_i.

GroebnerBasis[{poly₁,poly₂,…},{x₁,x₂,…},{y₁,y₂,…}]

finds a Gröbner basis in which the y_i have been eliminated.

• 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 x_i.
• 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 x_i appear in them.
• The ordering of the x_i 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".

Compute a Gröbner basis:

Prove that polynomials have no common roots:

Polynomials with a finite number of common roots:

Polynomials with an infinite number of common roots:

Polynomials with no common roots:

Eliminate a variable:

A lexicographic Gröbner basis:

A degree reverse lexicographic Gröbner basis:

Polynomial equations can be given instead of polynomials:

By default, Gröbner bases are computed over the field of rational numbers:

This computes the strong Gröbner basis over the ring of integers:

This computes the Gröbner basis over the field of rational functions (a):

This uses approximate arithmetic:

The Automatic method setting uses "GroebnerWalk" for lexicographic bases over the rationals:

In this case the "Buchberger" method is much slower than "GroebnerWalk":

These polynomials are "close" to the lexicographic Gröbner basis:

The "GroebnerWalk" method computes the degree reverse lexicographic basis first:

Computing the lexicographic basis directly with the "Buchberger" method is faster here:

This computes the Gröbner basis over the field of integers modulo 7:

By default, GroebnerBasis uses the Lexicographic monomial order:

This gives the Gröbner basis in the DegreeReverseLexicographic monomial order:

A monomial order may be specified by giving a full rank square rational weight matrix:

For the order to be well-founded the first nonzero entry in each column must be positive:

Eliminate z and return a degree reverse lexicographic basis with respect to {x,y}:

Parameters are ordered lexicographically after all other variables:

This is an equivalent input:

By default, GroebnerBasis is not allowed to reorder the variables:

Reordering the variables may make computations faster; the Gröbner basis may be different:

Find an approximate GCD of a pair of univariate polynomials:

The polynomials are close to polynomials with integer coefficients:

With the default setting Tolerance->0, the approximate GCD has a too low degree:

With a higher setting of Tolerance, GroebnerBasis gives a "better" approximate GCD:

Solve a system of polynomial equations:

A Gröbner basis has the same set of roots as the input polynomials:

Solve the first polynomial of the Gröbner basis for its only variable x:

Solve the second polynomial of the Gröbner basis for the other variable y:

This method finds all common roots of polys:

Reduce and Solve use Gröbner bases to solve systems of equations:

Get a fuzzy solution to a system of overdetermined equations:

This gives a low-precision approximate solution to this overdetermined set of polynomials:

A Gröbner basis generates the same ideal as the input polynomials:

Use PolynomialReduce to show that p1 is in the ideal generated by g1 and g2:

By Hilbert's Nullstellensatz, if the ideal is then the polynomials have no common zero:

Reduce or Solve proves that there is no common solution:

Conversely, if the ideal is not , then there is at least one common zero:

Use FindInstance to find a solution instance:

GroebnerBasis of univariate polynomials is equivalent to computing PolynomialGCD:

GroebnerBasis of linear polynomials is equivalent to a Gaussian elimination process:

GroebnerBasis is used to solve systems of polynomial equations:

Use Reduce to directly solve the system:

Solve gives solutions in terms of replacement rules:

Eliminate a variable from a system of polynomial equations:

Eliminate a variable using Resolve:

Eliminate a variable using Eliminate:

Eliminate a variable using Resultant:

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