Bill Allombert on Fri, 4 Apr 2003 12:19:30 +0200

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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).

I have yet to meet this problem. I have no example of isomorphic
polredabs'ed polynomials.

> I am not convinced that you save that many isomorphism tests [ also for
> quadratic and cubic fields, one has much better methods ]

[... which are not implemented ]. Just try some examples and get
convinced. The problem is that doing all the isomorphism tests is
tedious to do manually, not that they are costly (when they fails).  

Anyway that do not address the fact that polred() output could be
made more useful so that we can stop using polredabs when we do not
need proven T2 minimality.