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

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

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.

Cheers,
Bill.