Re: size of the coefficients returned by bnfisnorm()

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

Le Sun, Oct 31, 2021 at 09:40:49AM +0100, Bill Allombert a écrit :
> Le Sat, Oct 30, 2021 at 10:12:07PM -0400, Max Alekseyev a écrit :
> > Dear Bill,
> > 
> > I did not have a chance to thank you for your suggestion on reducing
> > coefficients of bnfisnorm() based on qfparam(), but now I have a similar
> > question about qfparam() itself.
> > Consider an example:
> > 
> > ? G = matdiagonal([650, -104329, -104329]);
> > ? M = qfparam(G, qfsolve(G))
> > %1 =
> > [-33698267 -161709950 -194002198]
> > [  -521645   -2487100   -2964370]
> > [ -2608225  -12519480  -15023350]
> > 
> > I claim that the following matrix works equally well (i.e. it could have
> > been returned by qfparam), but it has much smaller entries:
> > 
> > ? M2 = [323, 0, 323; 5, 50, -5; 25,- 10, - 25]
> 
> As I understand, your solution is not a full parametrization since it
> does not reach all the rational solutions:

Sorry, now I am not sure my explanation was correct...

In any case, there is a flag to qfparam to reduce the quadratic forms,
which gives a smaller result:
? qfparam(G, [323,-5,25]~,1)
%158 = [-8398,-1938,-1296199;-130,16120,21925;650,3380,-99935]

Cheers,
Bill