Wow, I'm really surprised at the lukewarm reaction so far.
To me, Groebner Bases over ZZ are essential!
I guess it is a matter of research area. I'm interested
in arithmetic geometry, which basically means that you
pretend that ZZ is a curve and study the geometry of
"varieties" that are finite-type extensions of this curve.
The closest analogy is that ZZ is very much like FF_p[t].
Both rings are Dedekind domains with finite residue fields,
etc. So in algebraic geometry you might study the affine
line over with coordinate ring R = FF_p[t] and then consider
a relative genus-1 curve (i.e., an elliptic surface) defined
by, say, S = R[x,y]/(-y^2 - y + x^3 - t*x + 1). The equivalent thing
over ZZ is to consider, say, T = ZZ[x,y]/(-y^2 - y + x^3 + 7*x + 1).
You wouldn't want study the geometry of S by saying: "oh, you
have to work on the curve FF_p(t)[x,y]/(-y^2 - y + x^3 - t*x + 1)
over the field FF_p(t)". That would loose all the structure of
having a surface -- you would only see the generic behavior,
not the fine detail of how it behaves at, say, the fiber t = 0.
For similar reasons, I don't want to make do with
QQ[x,y]/(-y^2 - y + x^3 + 7*x + 1) instead of T.
An example in Magma:
> A<x,y> := PolynomialRing(Integers(), 2);
> B<X,Y> := PolynomialRing(Rationals(),2);
> f := -y^2 - y + x^3 + 7*x + 1;
> fx := Derivative(f,x);
> fy := Derivative(f,y);
>
> // Suppose I'm only allowed to do Groebner Bases over fields.
> // OK, I can find the generic singularities of f by coercing
> // my polynomials to B = QQ[X,Y] and doing the following
> GroebnerBasis([B!f, B!fx, B!fy]);
[
1
]
>
> // Since the output is 1, I know that there are no generic
> // singularities.
>
> // To look at the singularities of the arithmetic surface,
> // I need to do the corresponding computation over ZZ:
> I := GroebnerBasis([f, fx, fy]);
> I;
[
x + 20607,
y + 11314,
22627
]
>
> Factorization(I[3]);
[
<11, 3>,
<17, 1>
]
This Groebner Basis gives me a lot of information. First, the only
fibers (over ZZ) that are not smooth are at 11 = 0, and 17 = 0.
Examining the Groebner Basis, we see that we have a simple node
in both the fiber at 11 and at 17. From the factorization, we
see that the node at 17 is regular on the surface (an I_1 node),
but the node at 11 is not. After blowing up this non-regular point,
we find that it is an I_3 node.
Granted, much of this information could be obtained from some resultant
computations and equation manipulations, but a Groebner Basis is much
easier to interpret.
In summary: Yes, there are mathematicians who use Groebner Bases
over ZZ (and rings of integers in number fields, and localizations
of the above, etc.) on a regular basis. I would like to see it
in SAGE.
Joe
William Stein wrote:
> Can anybody who cares about Groebner Basis over ZZ pipe up!
>
> ------- Forwarded message -------
> From: "Michael Abshoff" <Michael.Abshoff@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
> To: wstein@xxxxxxxxx
> Cc:
> Subject: Gröbner Basis computation over ZZ
> Date: Thu, 08 Feb 2007 16:18:33 -0700
>
> Dear William,
>
> I hope you got my last EMail - I am sure you are quite busy at the moment.
> We had some discussion about Gröbner Bases over ZZ here:
> http://cocoa.mathematik.uni-dortmund.de/forum/viewtopic.php?t=624
>
> John's time estimate is probably realistic since we (at least the 'we' in
> Dortmund :)) couldn't find an application for that sort of computation, so
> could you please enlighten us why one would need to compute GBases over
> rings.
> I don't think it is particularly difficult to implement, but without an
> application it is rather low on the list of priorities.
>
> Cheers,
>
> Michael
>
>
>
> >
>
>
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