Add tropical prevarieties generated by intersecting tropical hypersurfaces

[danteluber]

Want to introduce methods to intersect tropical hypersurfaces and produce topical prevarieties from finite collections of polynomials.

Follows discussion with @lkastner

@YueRen Interested in your thoughts as well

[YueRen]

Hi @danteluber,

Thanks for the pr draft! Would it be possible to do the following changes:

1. Wrap Polymake.fan.common_refinement for OSCAR, so that users can use common_refinement(...) for arbitrary polyhedral complexes.
2. Make the tropical_prevariety function rely on it.

(Number 1. can be its own separate pull-request, if you want to tackle it step-by-step)

[YueRen]

Oh, so there was a common_refinement already? That is great to know.

As for the remaining changes, you can construct tropical hypersurfaces from polynomials using tropical_hypersurface(f), so there is no need to define your own functions unless there is a problem with the existing ones.

[YueRen]

Hi Dante, thanks again for your effort! I just have two final requests (see enclosed).

[YueRen]

Suggested change

	tropical_prevariety(A::Vector{<:MPolyRingElem}) = tropical_prevariety(min, A)
	function tropical_prevariety(F::Vector{<:MPolyRingElem}, nu::TropicalSemiringMap=tropical_semiring_map(coefficient_ring(first(F))))
	return tropical_prevariety(tropical_polynomial.(F,Ref(nu))
	end

I suggest adding a second optional TropicalSemiringMap that encodes a valuation + convention (default: trivial valuation, min convention). This allows users working over fields to create tropical hypersurfaces with respect to non-trivial valuations.

[YueRen]

Suggested change

	function tropical_prevariety(convention::minmax, A::Vector{<:MPolyRingElem}) where {minmax<:Union{typeof(min),typeof(max)}}
	length(A) > 0 || error("Empty array.")
	nu = tropical_semiring_map(coefficient_ring(A[1]), convention)
	ftrops = [tropical_polynomial(f, nu) for f in A]
	HTA = [tropical_hypersurface(f) for f in ftrops]
	return reduce(common_refinement, [h.polyhedralComplex for h in HTA])
	end

I would suggest removing this, as it is superseded by the suggested function above.

[lkastner]

Looks good to me.

[benlorenz]

Please add the function to some documentation page, i.e. an .md file in docs/src.

[benlorenz]

Suggested change

#Input array of tropical polynomials

This line needs to be removed for the docstring to work.

The doctest is also not run currently.

[benlorenz]

Suggested change

	#Give input as tropical polynomials
	Dr25 = tropical_prevariety(tropical_polynomial.([f for f in gens(Gr25)]))
	Polyhedral complex in ambient dimension 10
	julia> Dr25 = tropical_prevariety(tropical_polynomial.([f for f in gens(Gr25)])) # Give input as tropical polynomials
	Polyhedral complex in ambient dimension 10

There is a julia> missing, also I don't think comments like this before the julia> lines will work, we do have it like this in other cases so I added this as a suggestion:

julia> some + code   # some explanation

[benlorenz]

Suggested change

	#Compute with respect to max convention
	julia> nu = tropical_semiring_map(QQ,max)
	julia> nu = tropical_semiring_map(QQ,max) #Compute with respect to max convention

[benlorenz]

Suggested change

	#Compute Dressian without specified tropicalization map.
	julia> Dr25 = tropical_prevariety(gens(Gr25))
	julia> Dr25 = tropical_prevariety(gens(Gr25)) #Compute Dressian without specified tropicalization map.

[benlorenz]

Suggested change

	tropical_prevariety(F::Vector{MPolyRingElem},nu::TropicalSemiringMap)
	tropical_prevariety(F::Vector{MPolyRingElem},nu::TropicalSemiringMap)

This needs to be indented by 4 spaces.