Igor Schein on Thu, 3 Apr 2003 18:50:23 -0500

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Re: polredabs() again

On Fri, Apr 04, 2003 at 01:36:01AM +0200, Karim BELABAS wrote:
> On Fri, 4 Apr 2003, Bill Allombert wrote:
> > On Fri, Apr 04, 2003 at 12:18:34AM +0200, Karim BELABAS wrote:
> > > On the other hand, the current specification of polredabs is quite useless.
> > > There's no application whatsoever for a "polynomial of absolute smallest
> > > T2-norm". It's not even guaranteed to have minimal discriminant, or to
> > > yield smallest coefficients. The only one I can see is to give a
> > > pseudo-canonical representative for the field (this helps table builders,
> > > less isomorphism tests...)
> >
> > I do not fully agree. Having a canonical defining polynomial is quite
> > useful when you are generating lots of (small) isomorphic files (try
> > galoisubfields on a large non abelian Galois groups).
> I said pseudo-canonical, it is not at all canonical. It depends in a
> complicated way on the available stack space (which affects the cache
> algorithm described in my previous post), the quality of the LLL-reduced
> basis (which depends on the original polynomial). There could be many
> polynomials with the same T2-bound (hundreds of them).

Yes, canonicalization of number fields is a shady subject.  At
different times, I found different canonical criteria being more
useful than others:

- index
- L2 norm
- coefficient size
- automorphism size ( for Galois polynomials )
- sparseness ( number of zero coefficients )

Latter is my favorite, as you might have figured out.