McLaughlin, James on Mon, 01 Dec 2003 23:21:15 +0100

 RE: Computing a system of fundamental units

• To: "pari-users list" <pari-users@list.cr.yp.to>
• Subject: RE: Computing a system of fundamental units
• From: "McLaughlin, James" <James.McLaughlin@trincoll.edu>
• Date: Mon, 1 Dec 2003 17:04:47 -0500
• Delivery-date: Mon, 01 Dec 2003 23:21:16 +0100
• Mailing-list: contact pari-users-help@list.cr.yp.to; run by ezmlm
• Thread-topic: Computing a system of fundamental units

```I meant this response to go to the list also.

Here is a related question (at least related to what I am trying to do):

Given an irreducible polynomial in Z{x], is there any simple way of calculating the degree of the associated splitting field over Q?
All I need is the degree of the extension, and not any of the other invariants of the splitting field.

Thanks for the response. I have been able to get the set of fundamental units for my degree 31 polynomial (in fact it did not take that long really - a few hours) and am presently waiting to see if it can do the same for a degree 37 polynomial.

I really wanted to check if I was doing this in the most efficient way, since I would like to push this as high as possible. I do not have prior information about the regulator. (I am working on a  problem that necessitates finding solutions to certain Thue equations).

Do you happen to know the record for pari/gp in this regard - the highest degree irreducible polynomial that pari/gp has been able to compute a set of fundamental units for the corresponding number field?

I will investigate the CVS version.

Jimmy Mc Laughlin.

-----Original Message-----
From: Karim BELABAS [mailto:Karim.Belabas@math.u-psud.fr]
Sent: Sunday, November 30, 2003 3:09 PM
To: pari-users list
Subject: Re: Computing a system of fundamental units

On Sun, 30 Nov 2003, McLaughlin, James wrote:
> I have an irreducible polynomial in Z[x], P say, of degree 31 or higher
> and I want to compute a system of fundamental units for the number filed K
> defined by this polynomial over Q.
>
> I know that
>
>   bnfinit(P,1)[8][5]
>
> will output a set of fundamental units

bnfinit(P, 1).fu looks nicer.

And will not break when the representation changes [ as it should since some
of the data inside bnf structure is obsolete and nowadays unused ]

> but I am wondering if there is a more efficient/less time-consuming way to
> do it, since I do not need any of the other information that bnfinit
>
> Can anyone suggest some more efficient code?

If you happen to know the regulator in advance, then shortcuts are available.
Esp. if the fundamental units have small height.

Is it the case, or do you require a general purpose algorithm ? (in the
latter case, this looks hopeless).

Note that the CVS version is much more efficient for these computations than
any of the released versions. (Especially if the units are huge.)

In case of trouble with the CVS version, you can send me your polynomial of
degree 31.

Cheers,

Karim.
--
Karim Belabas                     Tel: (+33) (0)1 69 15 57 48
Dép. de Mathématiques, Bât. 425   Fax: (+33) (0)1 69 15 60 19
Université Paris-Sud              http://www.math.u-psud.fr/~belabas/
F-91405 Orsay (France)            http://pari.math.u-bordeaux.fr/  [PARI/GP]

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