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PolynomialReduce
PolynomialReduce[
poly
,
,
, ... 
,
,
, ... 
] yields a list representing a reduction of poly in terms of the  .
The list has the form 
,
, ... 
,
b
 , where b is minimal and 
+
+ ... +
b is exactly poly.
The polynomial b has the property that none of its terms are divisible by leading terms of any of the  .
If the  form a Gröbner basis then this property uniquely determines the remainder obtained from PolynomialReduce.
The following options can be given, as for GroebnerBasis:
See the Mathematica book: Section 3.3.4.
See also: GroebnerBasis, PolynomialRemainder, PolynomialMod.
Related package: Algebra`SymmetricPolynomials`.
Further Examples
In the polynomial every term is divisible by either a or x, so reducing by a and x, in either order, gives  . We check that we recover the original polynomial.
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This reduces  assuming that the polynomials in the list are zero.
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We form a Groebner basis from a list of polynomials. When we reduce one of the polynomials (here the last one) in that Groebner basis by a different Groebner basis for the same list, we get zero.
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See the Further Examples for GroebnerBasis.
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