Support for polytopes with (real) QQBar coefficients?

[fingolfin]

Our implementation of the algebraic closure of the rationals, QQBar / CalciumQQBar, is quite nice and it comes with an "automatic" real or complex embedding of its elements.

It would be quite handy if one could directly take a bunch of vectors over this field, and create polyhedra/polytopes from this. E.g. construct a convex hull and visualize it. I guess this is related to currently ongoing work for doing this for embedded number fields.

Example:

julia> e = one(CalciumQQBar)
Root 1.00000 of x - 1

julia> s, c = sinpi(2*e/5), cospi(2*e/5)
(Root 0.951057 of 16x^4 - 20x^2 + 5, Root 0.309017 of 4x^2 + 2x - 1)

julia> pts = [ [e, e], [s,c], [c,s] ]
3-element Vector{Vector{qqbar}}:
 [Root 1.00000 of x - 1, Root 1.00000 of x - 1]
 [Root 0.951057 of 16x^4 - 20x^2 + 5, Root 0.309017 of 4x^2 + 2x - 1]
 [Root 0.309017 of 4x^2 + 2x - 1, Root 0.951057 of 16x^4 - 20x^2 + 5]

julia> P = convex_hull(pts)
ERROR: MethodError: Cannot `convert` an object of type Int64 to an object of type qqbar

This initial error is easy to fix, just something missing in Nemo (see Nemocas/Nemo.jl#1650).

But after adding the missing conversion method, there are more errors inside PolyhedralGeometry specific code I am not familiar with. So in particular I don't know hard it would be to do in general.

But in this case, I managed to proceed quite nicely after converting to floating point:

julia> Base.convert(::Type{Float64}, x::qqbar) = Float64(ArbField(64)(x))  # TODO: add to Nemo

julia> Vector{Vector{Float64}}(pts)
3-element Vector{Vector{Float64}}:
 [1.0, 1.0]
 [0.9510565162951535, 0.30901699437494745]
 [0.30901699437494745, 0.9510565162951535]

julia> P = convex_hull(Vector{Vector{Float64}}(pts))
Polyhedron in ambient dimension 2

julia> visualize(P)

which gave me

[benlorenz]

I will try to add this in during the refactoring. To make it work nicely it will need to be added to a few type collections.

But the manual approach currently also fails due to some type / constructor issues, I will collect these and create an issue:

julia> pts = [ e e e; e s c; e c s]
3×3 Matrix{qqbar}:
 Root 1.00000 of x - 1  Root 1.00000 of x - 1               Root 1.00000 of x - 1
 Root 1.00000 of x - 1  Root 0.951057 of 16x^4 - 20x^2 + 5  Root 0.309017 of 4x^2 + 2x - 1
 Root 1.00000 of x - 1  Root 0.309017 of 4x^2 + 2x - 1      Root 0.951057 of 16x^4 - 20x^2 + 5

julia> ppts = Polymake.Matrix{Polymake.OscarNumber}(pts)
WARNING: cfunction: return type of cmp does not match
pm::Matrix<common::OscarNumber>
(Root 1.00000 of x - 1) (Root 1.00000 of x - 1) (Root 1.00000 of x - 1)
(Root 1.00000 of x - 1) (Root 0.951057 of 16x^4 - 20x^2 + 5) (Root 0.309017 of 4x^2 + 2x - 1)
(Root 1.00000 of x - 1) (Root 0.309017 of 4x^2 + 2x - 1) (Root 0.951057 of 16x^4 - 20x^2 + 5)


julia> pp = Polymake.polytope.Polytope{Polymake.OscarNumber}(POINTS=ppts)
ERROR: TypeError: in cfunction, expected Int64, got a value of type Int32

Here the issue is that Base.cmp for qqbar returns an Int32 instead of the expected Int64, which could be fixed by a cast there or a wrapper function in the OscarNumber code.

I also needed something like this to allow the embedding to work with the OscarNumber:

(::Nemo.CalciumQQBarField)(r::Rational{BigInt}) = qqbar(fmpq(r))

(i.e. we need a constructor from Rational{BigInt}.)

There will probably be more small things missing to get this working.

[fingolfin]

@benlorenz great! And yeah, we need Rational->QQBar conversion, already requested on Nemocas/Nemo.jl#1650 -- it shouldn't be hard either, but right now I have other things to finish (and perhaps we can convince someone else to work on that ;-)