# PolynomialReduce

PolynomialReduce[poly,{poly1,poly2,},{x1,x2,}]

yields a list representing a reduction of poly in terms of the polyi. The list has the form {{a1,a2,},b}, where b is minimal and a1 poly1+a2 poly2++b is exactly poly.

# Details and Options • The polynomial b has the property that none of its terms are divisible by leading terms of any of the polyi.
• If the polyi form a Gröbner basis, then this property uniquely determines the remainder obtained from PolynomialReduce.
• The following options can be given, as for GroebnerBasis:
•  MonomialOrder Lexicographic the criterion used for ordering monomials CoefficientDomain Rationals the types of objects assumed to be coefficients Modulus 0 the modulus for numerical coefficients

# Examples

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## Basic Examples(1)

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:

## Scope(1)

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:

## Options(4)

### CoefficientDomain(1)

By default, PolynomialReduce works over the field of rational functions of parameters:

Compute the Gröbner basis of polys over the field of rational functions :

Reduce poly modulo gb1 over the field of rational functions :

Compute the Gröbner basis and reduce poly over the integers:

Compute the Gröbner basis and reduce poly over the rationals:

Compute the Gröbner basis and reduce poly using approximate arithmetic:

The precision used is chosen automatically, based on the precision of the Gröbner basis:

### Modulus(1)

Compute a Gröbner basis and reduce a polynomial over the integers modulo 7:

### MonomialOrder(1)

By default, PolynomialReduce uses the Lexicographic monomial order:

Any MonomialOrder allowed by GroebnerBasis can be used:

### Tolerance(1)

Compute approximate quotients:

With the default zero tolerance, d does not divide p:

Increase the tolerance to obtain an approximate quotient and a zero remainder:

## Applications(3)

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:

## Properties & Relations(3)

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: