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Bill Allombert on Sat, 07 Jan 2023 20:36:53 +0100
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Re: Solve an non-homogeneous system of equations mod Z.
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- To: pari-users@pari.math.u-bordeaux.fr
- Subject: Re: Solve an non-homogeneous system of equations mod Z.
- From: Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>
- Date: Sat, 7 Jan 2023 20:35:44 +0100
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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.