Re: Groebner basis question.
- From: Marc Olschok <nobody@xxxxxxxxxxxxxxx>
- Date: Wed, 2 Sep 2009 14:56:02 +0000 (UTC)
Ray Vickson <RGVickson@xxxxxxx> wrote:
I am confused about the case where the number of elements in a
Groebner basis exceeds the original number of polynomials. For
example, a set of three polynomials f(x,y,z), g(x,y,z), h(x,y,z) may
have a Groebner basis consisting of six polynomials k_i(x,y,z),
i=1,...,6 (at least according to Maple, in a specific example). Isn't
the set of roots of the system {f=0,g=0,h=0} the same as that of the
system {k_i = 0 for all i}?
Well, yes, actually more is true: {f, g, h} and { k_i |i=1,..6}
in your example generate the same ideal. Beware that having the
same set of roots does not in general give you that the ideals
are the same (take e.g. the principal ideals (x) and (x^2)).
I am bothered by the fact that the
Groebner system has six equations while the original system has only
three. I would not be bothered if the Groebner system had fewer than
three equations, because that would be a great help in solving for the
roots of the original system.
Could it be that you confuse the generating polynomials with
their zero-sets? The point of computing a Groebner basis is, that
you can then decide effectively if some given polynomial is in
the ideal.
You can view each generating polynomials as rewrite rule, where
the highest monomial (w.r.t. some ordering) is replaced by
the negative of the remaining monomials. So you can hope
to decide membership in the ideal by just applying these
rules to a given polynomial as long as possible and check if
you end up with 0 in doing so.
But without a Groebner Basis this might not work.
Take a small example with two variables which is manageable by hand:
let I = ( xy-x , x^2-y )
as rewrite rules
xy -> x
x^2 -> y
Now the polynomial
y^2 - y = y^2 - x^2y + x^2y - x^2 + x^2 - y
= y(y-x^2) + x(xy-x) + x^2 - y
lies in I. But neither xy or x^2 appear in it, so the above rewrite
rules do not lead anywhere. Although y^2 - y is in I we could not
detect this.
Also the result of rewriting depends on which rules you choose:
take x^2y = x(xy)
If you start with the first rule you obtain
x(xy) -> xx -> y
If you start with the second rule you obtain
x^2y -> y^2
So it is not so unexepected that you may need a third rule
y^2 -> y
to repair these deficiencies and end up with
I = ( xy-x , x^2-y, y^2-y )
--
Marc
.
- References:
- Groebner basis question.
- From: Ray Vickson
- Groebner basis question.
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