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. Igor