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

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

On Sun, Jan 8, 2023 at 1:49 AM Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> 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.

There is a typo in the command "matriqz" mentioned above, which should
be written as "matrixqz".

IIRC, as far as this problem is concerned, you haven't mentioned
matrixqz in the whole discussion process.

> Then everything is be easier.  For example
>
> mat1 = [ -210, -210, -220; -221, -222, -232; 410, 411, 430 ];
> vec1 = [ -27, -28, 105/2 ]~;
> M1 = matrixqz(mat1,-2)
> B1 = matinverseimage(mat1,M1)
> x1 = B1*matsolvemod(M1,2,2*vec1)/2
> \\%64 = [3/7, 1/42, 0]~
> mat1*x1-vec1
> \\%65 = [-68, -72, 133]~

Thank you for your complete example. It can solve the problem.

Another puzzle for me is that I actually hope to combine PARI/GP and
GAP to carry out relevant work. But I'm not sure if there is a way to
completely embed the above code in the GAP script. Although there is a
related GAP package alnuth [1], its features seem to be very limited.

> So you get a solution.

However, based on our previous discussions and the general
characteristics of this problem, I still have the following questions:

1. For a problem like this, if there are solutions, then there should
be infinitely many solutions, how do I find and represent all of them
in a reasonable form?
2. How to quickly determine if there is no solution?

[1] https://www.gap-system.org/Packages/alnuth.html

> Cheers,
> Bill.

Best,
Zhao