Linear solving for p-adic residue rings

[joschmitt]

Use the Howell form for linear solving over the new LocalFieldValuationRingResidueRings.

The kernel method is copied from the zzModMatrix version in Nemo. I have no idea whether this is correct; it seems to work.

[codecov]

Codecov Report

Attention: Patch coverage is 98.73418% with 2 lines in your changes missing coverage. Please review.

Project coverage is 75.58%. Comparing base (7a5bac9) to head (1d6498d).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1533      +/-   ##
==========================================
+ Coverage   75.50%   75.58%   +0.08%     
==========================================
  Files         355      355              
  Lines      113184   113320     +136     
==========================================
+ Hits        85464    85658     +194     
+ Misses      27720    27662      -58     

Files	Coverage Δ
src/LocalField/ResidueRing.jl	99.44% <100.00%> (+4.80%)	⬆️
src/LocalField/Ring.jl	75.59% <100.00%> (+2.57%)	⬆️
src/NumFieldOrd/NfOrd/StrongEchelonForm.jl	55.61% <98.55%> (+12.79%)	⬆️

... and 33 files with indirect coverage changes

[joschmitt]

@simonbrandhorst does this help you in #1510?

[simonbrandhorst]

Thanks for contributing this! When I tried it I got the following error message:

ERROR: MethodError: no method matching ^(::Hecke.LocalFieldValuationRingResidueRingElem{Hecke.LocalField{…}, Hecke.LocalFieldElem{…}}, ::QQFieldElem)

Closest candidates are:
  ^(::RealFieldElem, ::QQFieldElem)
   @ Nemo ~/.julia/packages/Nemo/my7xb/src/arb/Real.jl:788
  ^(::RealFieldElem, ::QQFieldElem, ::Int64)
   @ Nemo ~/.julia/packages/Nemo/my7xb/src/arb/Real.jl:788
  ^(::ComplexFieldElem, ::QQFieldElem)
   @ Nemo ~/.julia/packages/Nemo/my7xb/src/arb/Complex.jl:315
  ...

Stacktrace:
  [1] annihilator(a::Hecke.LocalFieldValuationRingResidueRingElem{Hecke.LocalField{…}, Hecke.LocalFieldElem{…}})
    @ Hecke ~/.julia/dev/Hecke/src/LocalField/ResidueRing.jl:302
  [2] strong_echelon_form_naive!(A::AbstractAlgebra.Solve.LazyTransposeMatElem{…})
    @ Hecke ~/.julia/dev/Hecke/src/NumFieldOrd/NfOrd/StrongEchelonForm.jl:158
  [3] howell_form!(A::AbstractAlgebra.Solve.LazyTransposeMatElem{Hecke.LocalFieldValuationRingResidueRingElem{…}, AbstractAlgebra.Generic.MatSpaceElem{…}})
    @ Hecke ~/.julia/dev/Hecke/src/NumFieldOrd/NfOrd/StrongEchelonForm.jl:210
  [4] _can_solve_internal_no_check(A::AbstractAlgebra.Generic.MatSpaceElem{…}, b::AbstractAlgebra.Generic.MatSpaceElem{…}, task::Symbol; side::Symbol)
    @ Hecke ~/.julia/dev/Hecke/src/NumFieldOrd/NfOrd/StrongEchelonForm.jl:263
  [5] _can_solve_internal_no_check
    @ ~/.julia/dev/Hecke/src/NumFieldOrd/NfOrd/StrongEchelonForm.jl:246 [inlined]
  [6] _can_solve_internal(A::AbstractAlgebra.Generic.MatSpaceElem{…}, b::AbstractAlgebra.Generic.MatSpaceElem{…}, task::Symbol; side::Symbol)
    @ AbstractAlgebra.Solve ~/.julia/packages/AbstractAlgebra/dLVCE/src/Solve.jl:644
  [7] _can_solve_internal
    @ ~/.julia/packages/AbstractAlgebra/dLVCE/src/Solve.jl:636 [inlined]
  [8] #can_solve_with_solution#11
    @ ~/.julia/packages/AbstractAlgebra/dLVCE/src/Solve.jl:456 [inlined]
  [9] can_solve_with_solution
    @ ~/.julia/packages/AbstractAlgebra/dLVCE/src/Solve.jl:455 [inlined]
 [10] #solve#9
    @ ~/.julia/packages/AbstractAlgebra/dLVCE/src/Solve.jl:417 [inlined]
 [11] solve
    @ ~/.julia/packages/AbstractAlgebra/dLVCE/src/Solve.jl:416 [inlined]

The bug lies in ResidueRing.jl

function annihilator(a::LocalFieldValuationRingResidueRingElem)
  if is_zero(a)
    return one(parent(a))
  end
  pi = uniformizer(_valuation_ring(parent(a)))
  return parent(a)(pi)^(_exponent(parent(a)) - (valuation(data(a))))
end

the valuation seems to be a QQFieldElem

[joschmitt]

This is fixed now. I should have tested with more obscure extensions, thanks for catching it!

[joschmitt]

From my side, this is good to go. I'm even convinced for the moment that the algorithm for the kernel is correct.