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

for lexicographic bases over the rationals:

In this case the

method is much slower than

:

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

The

method computes the degree reverse lexicographic basis first:

Computing the lexicographic basis directly with the

method is faster here:

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

By default,

GroebnerBasis uses the

monomial order:

This gives the Gröbner basis in the

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

:

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, the approximate GCD has a too low degree:

With a higher setting of

Tolerance,

GroebnerBasis gives a "better" approximate GCD: