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

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

On Sat, Jan 7, 2023 at 5:31 PM Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
>
> On Sat, Jan 07, 2023 at 09:14:04AM +0800, Hongyi Zhao wrote:
> >
> > I tried the following method, but failed:
> >
> > ? mat1 = [ -210, -210, -220; -221, -222, -232; 410, 411, 430 ]
> > %1 =
> > [-210 -210 -220]
> >
> > [-221 -222 -232]
> >
> > [ 410  411  430]
> >
> > ? vec1 = [ -27, -28, 105/2 ]~
> > %2 = [-27, -28, 105/2]~
> > ? matsolvemod(mat1,[1,1,1]~, 2 * vec1~,1)
> >   ***   at top-level: matsolvemod(mat1,[1,1,1]~,2*vec1~,1)
> >   ***                 ^------------------------------------
> >   *** matsolvemod: incorrect type in matsolvemod (Y) (t_VEC).
>
> There are three issues:
> 1/ The error is that you applied ~ to vec1.

But based on my testing with your code snippet below, you still used
vec1 like this:

vec1 = [ -27, -28, 105/2 ]~

> 2/ You need to multiply everything by 2 to compensate, not just vec1.
> 3/ matsolvemod looks for integral solutions. Obviously there are none,
> so you will get the empty set.
>
> 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?

> then  mat1*w-vec1 = [-28,-30,55]~ is integral
>
> ? matsolvemod(mat1,1,mat1*w-vec1,1)
> %20 = [[0,0,0]~,[1,0,0;0,1,0;0,0,1]]

Here is the complete test using your above code example:

? mat1 = [ -210, -210, -220; -221, -222, -232; 410, 411, 430 ]
%9 =
[-210 -210 -220]

[-221 -222 -232]

[ 410  411  430]

? vec1 = [ -27, -28, 105/2 ]~
%10 = [-27, -28, 105/2]~
? w = [0,0,1/4]~
%11 = [0, 0, 1/4]~
? mat1*w-vec1
%12 = [-28, -30, 55]~
? matsolvemod(mat1,1,mat1*w-vec1,1)
%13 = [[0, 0, 0]~, [1, 0, 0; 0, 1, 0; 0, 0, 1]]


> so the set of solution is w + Z^3
>
> Cheers,
> Bill

Best,
Zhao