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:

This was not correct, sorry! I forgot that the solutions were
projective.

Let
sym2sq(M)=my([a,b]=M[1,],[c,d]=M[2,]);[a^2,2*b*a,b^2;c*a,d*a+c*b,d*b;c^2,2*d*c,d^2]
the symmetric square function

The transfer matrix is 
P=[1,-62;5,13]
We have
M2*sym2sq(P)==-M

So indeed M2 and M are equivalent as far as qfparam is concerned.

Cheers,
Bill.