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