Re: Solve an non-homogeneous system of equations mod Z.

[Hongyi Zhao <hongyi.zhao@gmail.com>]

On Sun, Jan 8, 2023 at 3:35 AM Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
>
> On Sat, Jan 07, 2023 at 06:49:01PM +0100, Bill Allombert wrote:
> > On Sat, Jan 07, 2023 at 09:29:55PM +0800, Hongyi Zhao wrote:
> > > On Sat, Jan 7, 2023 at 5:31 PM Bill Allombert
> > > > One possible solution
> > > > Take w = [0,0,1/4]~
> > >
> > > This solution is not given in advance. We need to find such things
> > > first. How did you find such a solution?
> >
> > You should start by replacing mat1 by matriqz(mat1,-2) as I suggested.
> > Then everything is be easier.  For example
> >
> > mat1 = [ -210, -210, -220; -221, -222, -232; 410, 411, 430 ];
>
> I think I understand why we are struggling with your example:
>
> the set of x in Q^3 such that
>
> mat1 * x - vec1 is in Z^3
>
> is neither a affine lattice nor an affine space but a mix:
>
> It is
>
> x1 + B1*Z^3 + ker mat1
>
> where x1 is a vector, B1*Z^3 is a lattice and ker mat1 is a vector space.

Thank you again for your in-depth analysis and comments.

> Cheers,
> Bill.

Best,
Zhao