Further solving related adjustments

[joschmitt]

I learned that nullspace(::RingElement) returns a "rational nullspace" EXCEPT for ZZMatrix.
The output of Solve.kernel is still not exactly the same as of kernel; I will probably adjust some more.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (cd38e14) 87.04% compared to head (4b6c5e5) 87.02%.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1598      +/-   ##
==========================================
- Coverage   87.04%   87.02%   -0.03%     
==========================================
  Files         115      115              
  Lines       29593    29603      +10     
==========================================
+ Hits        25760    25762       +2     
- Misses       3833     3841       +8     

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[joschmitt]

Regarding the structure of the kernel, which do you prefer? M is a ZZMatrix.

julia> kernel(M, side = :right)
[ 1    0    0    0]
[ 0    1    0    0]
[ 0    0    1    0]
[ 0    0    0    1]
[-2   -1    0   -2]
[ 1    0   -1    1]

julia> kernel(change_base_ring(QQ, M), side = :right)
[ 1   -1    0    1]
[-2    0   -1   -2]
[ 1    0    0    0]
[ 0    1    0    0]
[ 0    0    1    0]
[ 0    0    0    1]

julia> kernel(change_base_ring(GF(17), M), side = :right)
[ 1   16    0    1]
[15    0   16   15]
[ 1    0    0    0]
[ 0    1    0    0]
[ 0    0    1    0]
[ 0    0    0    1]

julia> kernel(M, side = :left)
[1   0   0   0   -2    1]
[0   1   0   0   -1    0]
[0   0   1   0    0   -1]
[0   0   0   1   -2    1]

julia> kernel(change_base_ring(QQ, M), side = :left)
[ 1   -2   1   0   0   0]
[-1    0   0   1   0   0]
[ 0   -1   0   0   1   0]
[ 1   -2   0   0   0   1]

julia> kernel(change_base_ring(GF(17), M), side = :left)
[ 1   15   1   0   0   0]
[16    0   0   1   0   0]
[ 0   16   0   0   1   0]
[ 1   15   0   0   0   1]

Notice that the functions over QQ and GF(17) call flint, so it would be somewhat easier to adjust the ZZ functionality. Then again, there are some instances in Hecke relying particularly on the shape in the ZZ case (although I haven't checked yet whether the 180° flip to get from the ZZ shape to the QQ shape would create a problem there).
EDIT: Apparently QQMatrix is the only type for which we compute kernels in C? (And only for those only since last week.)

[joschmitt]

This is good to go for now. Would be good if we could make a minor release after it is merged because I need it for Hecke. (I can already adjust the version number here, if you want.)

[thofma]

Yes, please bump the version number.

[joschmitt]

This should only be breaking because it tests with Oscar on top of my Hecke js/solve branch (I didn't no CI is this "clever"!). So the changes in Hecke are breaking, but not the ones here.