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PolynomialReduce

Q: I have a polynomial
[Graphics:../Images/index_gr_84.gif]
and a list of three other polynomials
[Graphics:../Images/index_gr_85.gif]
p is a linear combination of the polynomials from the list,
[Graphics:../Images/index_gr_86.gif]
[Graphics:../Images/index_gr_87.gif]
but PolynomialReduce does not seem to notice.
[Graphics:../Images/index_gr_88.gif]
[Graphics:../Images/index_gr_89.gif]
The documentation says that the last element in PolynomialReduce is "minimal," but it does not elaborate. Wouldn't it be reasonable to check for linear dependence?
A: Daniel Lichtblau (danl@wolfram.com) answers: PolynomialReduce[[Graphics:../Images/index_gr_90.gif],{[Graphics:../Images/index_gr_91.gif],[Graphics:../Images/index_gr_92.gif],[Graphics:../Images/index_gr_93.gif]},{[Graphics:../Images/index_gr_94.gif],[Graphics:../Images/index_gr_95.gif],[Graphics:../Images/index_gr_96.gif]}] performs what is called generalized division with respect to a given set of polynomials, [Graphics:../Images/index_gr_97.gif], and a given term ordering. This latter is lexicographic by default and moreover depends on the ordering of the variables. This in turn is determined by the order in which they are given in the optional "variables" argument, [Graphics:../Images/index_gr_98.gif], to PolynomialReduce or by the order in which they get put internally if they are not explicitly given as an argument. See also Section 3.3.4 of The Mathematica Book.
If we reduce with respect to a Groebner basis, then we are guaranteed to get a reduction to zero if [Graphics:../Images/index_gr_99.gif] is in the ideal generated by [Graphics:../Images/index_gr_100.gif] (as it is in this case). This is a bit subtle, though. A set may be a Groebner basis with respect to one ordering but not another; the ordering that matters is the one by which we reduce, not the one by which the basis was created (if they are different orderings).
We will use the variable order
[Graphics:../Images/index_gr_101.gif]
to create a Groebner basis
[Graphics:../Images/index_gr_102.gif]
Now we reduce with respect to this basis, but the reduction term order may be different because we do not specify a variable order.
[Graphics:../Images/index_gr_103.gif]
[Graphics:../Images/index_gr_104.gif]
We did not get zero. This time we reduce with respect to the same order we used to create the Groebner basis.
[Graphics:../Images/index_gr_105.gif]
[Graphics:../Images/index_gr_106.gif]


Converted by Mathematica      September 29, 1999

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