Built-in Mathematica Symbol

PolynomialReduce

PolynomialReduce[poly, {poly₁, poly₂, ...}, {x₁, x₂, ...}]
yields a list representing a reduction of poly in terms of the poly_i. The list has the form {{a₁, a₂, ...}, b}, where b is minimal and a₁ poly₁+a₂ poly₂+... +b is exactly poly.

MORE INFORMATION

The polynomial b has the property that none of its terms are divisible by leading terms of any of the poly_i.

If the poly_i 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 type 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:

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

In[1]:=

In[2]:=

Out[2]=

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

In[3]:=

Out[3]=

Scope (1)

Reduce a polynomial modulo a list of polynomials which 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)

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:

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

By default, PolynomialReduce uses the Lexicographic monomial order:

Any MonomialOrder allowed by GroebnerBasis can be used:

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: