Re: Strange results in lower bound for regulator in bnfcertify()

[Georgi Guninski <guninski@guninski.com>]

KASH3 computes better lower bound for regulator,
but appears much slower.

On Tue, May 06, 2014 at 09:34:19AM +0200, Karim Belabas wrote:
> * Georgi Guninski [2014-05-06 08:33]:
> > 2.5.5 on linux x86_64.
> > 
> > With debug info the lower bound for the regulator
> > in bnfcertify() is greater than the regulator.
> > 
> > Why so?
> 
> This is due to an incorrect use of fudge factors in minimforunits(),
> which returns
>   m := min_{u unit, not a root of 1} T_2(u).
> 
> This is done using a variant of qfminim() which (since T_2 can in general
> only be approximated and not computed exactly) actually computes, for 
> a know unit u_0,
> 
>   min_{T_2(u) < T_2(u_0) * (1 + epsilon), u unit} T_2(u)
> 
> i.e. possibly
> 
>   m * (1 + epsilon).
> 
> i.e. an upper bound for m (but very close to m). This was harmless in
> our application since this is only used to approximate a rational
> integer via rounding, and the epsilon cannot change that value.
> 
> I just changed that code so that the routine returns an acctual lower
> bound (but very close to m).
> 
> Cheers,
> 
>     K.B.
> 
> P.S. See also
> 
>   http://pari.math.u-bordeaux1.fr/archives/pari-dev-1207/msg00013.html
> 
> This lower bound implementation should be rewritten, since the original
> author no longer supports it, we have no written reference for it, and
> the Magma implementation (for which there is a good published reference)
> is quite a bit faster. Not enthusiastic about doing it myself: been
> aware of the issue since 1995, postponed fixing it since then. I'll open
> an issue in the BTS shortly.
> 
> --
> 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-bordeaux1.fr/~kbelabas/
> F-33405 Talence (France)       http://pari.math.u-bordeaux1.fr/  [PARI/GP]
> `