Reduce a polynomial p with respect to a list of polynomials qq:

p is a linear combination of polynomials qq and the remainder polynomial r:

Reduce a polynomial over an algebra with commutative and scalar variables:

p is a linear combination of polynomials qq and the remainder polynomial r:

Reduce a polynomial over an algebra with symbolic property names:

p is a linear combination of polynomials qq and the remainder polynomial r:

The result of reduction depends on the monomial order:

Reduce a polynomial modulo a list of polynomials that is not a Gröbner basis:

The remainder is not zero, even though p belongs to the ideal generated by qq:

When p belongs to the ideal generated by qq, the remainder modulo gb must be zero:

Simplify an expression involving matrices and :

Find the inverses in the expression:

Generate the relations satisfied by the inverses:

Pick a variable order that puts the most complicated variables first:

Compute the Gröbner basis of the ideal generated by the relations over the algebra of matrices:

Reducing the expression modulo the Gröbner basis shows that it is equal to the identity matrix:

Use ArraySimplify to do the above simplification steps automatically:

Prove the Woodbury matrix identity:

In the standard formulation, is an matrix, is a matrix, with , is a matrix, and is a matrix. However, by replacing , and with the block matrices , and , one may assume that all matrices belong to the algebra of matrices. It will be shown that the difference of the sides of the identity reduces to zero modulo the Gröbner basis of the ideal generated by relations implied by the properties of the matrix inverse.

Compute the difference of the sides of the Woodbury matrix identity:

Find the inverses in :

Generate the relations satisfied by the inverses:

Pick a variable order that puts the most complicated variables first:

Compute the Gröbner basis of the ideal generated by the relations over the algebra of matrices:

Reduce modulo the Gröbner basis. The result is a zero matrix, which proves the identity:

Use ArraySimplify to do the above simplification steps automatically:

Construct a finitely presented group. The dicyclic group is given by generators and relations . The group algebra of is given by four generators:

The generators satisfy the following relations:

Compute the Gröbner basis of the ideal generated by the relations:

Note that reducing an arbitrary monomial modulo the Gröbner basis gives a monomial that does not contain and . This shows that reducing an arbitrary monomial in and of total degree yields a monomial of a total degree at most :

Hence, all elements of the group can be represented as reduced monomials of degree at most :

It has been proven that is a finite group of order :

Compute in the Clifford algebra . The algebra has six generators:

The generators satisfy the following relations:

The set of relations is a Gröbner basis:

Compute the canonical representation of :

Show that is idempotent:

Compute the canonical representation of :

Show that is nilpotent:

Compute in the Weyl algebra associated to the differential ring of bivariate polynomials. The algebra has four generators:

The generators satisfy the following relations:

The set of relations is a Gröbner basis:

Compute the canonical representation of and :

Compute the canonical representation of :

Elements of the Weyl algebra correspond to differential operators with product representing composition:

Verify that the operator corresponding to is the composition of the operators corresponding to and :

Reduce a polynomial p with respect to a list of polynomials qq:

p is a linear combination of polynomials qq and the remainder polynomial r:

A polynomial belongs to the ideal generated by a Gröbner basis iff it reduces to zero:

This shows that p is in the ideal generated by gb:

Use PolynomialReduce to a commutative polynomial with respect to a list of polynomials:

p is a linear combination of polynomials qq and the remainder polynomial r: