Karim Belabas on Fri, 25 Jun 2010 00:50:50 +0200

 Re: Fwd: bnf unit normalization A12-140

* John Cremona [2010-06-24 23:46]:
> I was wondering what normalization, if any, is done to the system of
> fundamental units as computed by bnfinit?
>
> I think it makes sense to talk of a unit basis being LLL-reduced, in
> the sense that the lattice which is the image of the units under the
> logarithmic embedding has many bases and these may (or may not) be
> LLL-reduced.
>
> So the question is:  is the unit basis LLL-reduced in this sense?  If
> not, could it be?

It is. :-)  [ cf getfu() ]

Actually, starting from an arbitrary LLL-reduced basis (u_1,...,u_r) in
your sense, we then normalize so that each u \in {u_1,..., u_r} is
replaced by the "nicest" one among { u, -u, 1/u, -1/u }, for some silly
ad hoc notion of "nice" usually translating to "using fewer characters
when fully printed out as a t_POLMOD".

The result is still LLL-reduced of course.

Cheers,

K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite Bordeaux 1          Fax: (+33) (0)5 40 00 69 50
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~belabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]

`