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# 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.
• 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.
 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
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:
 Out[2]=
f is a linear combination of polynomials p and a remainder term r:
 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:
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:
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