Re: Find an invertible integer matrix that satisfies given conditions.

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

On Wed, Apr 12, 2023 at 09:00:00PM +0800, Hongyi Zhao wrote:
> > The difficult part is obviously 3).
> > For n=2, the Diophantine equation is a binary quadratic form, so can be solved by PARI.
> > Hofmann shows that solving the equation can be reduced to solve a set of
> > norm equations in orders of number fields.
> 
> I am very grateful to Bill Allombert and Aurel Page for providing me
> with valuable information and algorithm implementations.
> 
> Based on the clues here, I checked the works done by Tommy Hofmann
> [1], and it seems that there are no publically available
> implementations of his method, such as in open source codes, like GAP
> or PARI/GP. So, I'm still not sure if it's possible to implement his
> algorithms based on open source tools solely, and then use them as a
> starting point to do the further study.

Well, in some case, this can be solved using the GP functions bnfisintnorm
or bnfunits.

Cheers,
Bill