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

f is a linear combination of polynomials p and a remainder term r:

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

The remainder is not zero, even though f belongs to the ideal generated by polys:

When f belongs to the ideal generated by polys, the remainder modulo gb must be zero:

Test whether polynomials belong to the ideal generated by a set of polynomials:

The remainder is zero, hence f belongs to the ideal generated by polys:

The remainder is not zero, hence g does not belong to the ideal generated by polys:

Replace variables in a polynomial using equations relating old and new variables:

The remainder gives a representation of poly in terms of a and b:

This proves correctness of the representation:

Compute the representation of a polynomial in an algebra :

Introduce tag variables and order them last in the monomial ordering:

Since the remainder is in , this shows that :

Check the result:

Reduce a polynomial with respect to a list of polynomials:

f is equal to the linear combination of polys with coefficients qs plus the remainder r:

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

This shows that p1 is in the ideal {g1,g2}:

Univariate PolynomialReduce is equivalent to PolynomialQuotientRemainder:

Reduce a noncommutative polynomial with respect to a list of noncommutative polynomials:

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

PolynomialReduce can give different results for the same inputs but with variables ordered differently:

Now change the ordering of the variables: