TropicalGeometry: new positive tropicalizations

[YueRen]

This is the final pull request from the Leipzig workshop, adding the capability of computing positive tropicalizations for linear and binomial ideals (required for example for Telek-Rose).

It is a draft for now, as it is contingent on #4061 being merged. (I don't think it makes a lot of sense if tropical_variety returns a list of TropicalVariety while positive_tropical_variety returns a single TropicalVariety)

[codecov]

Codecov Report

Attention: Patch coverage is 84.21053% with 3 lines in your changes missing coverage. Please review.

Project coverage is 84.43%. Comparing base (3e86180) to head (588fccd).
Report is 162 commits behind head on master.

Files with missing lines	Patch %	Lines
src/TropicalGeometry/positive_variety.jl	84.21%	3 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #4447      +/-   ##
==========================================
- Coverage   84.55%   84.43%   -0.13%     
==========================================
  Files         672      675       +3     
  Lines       88880    90060    +1180     
==========================================
+ Hits        75152    76039     +887     
- Misses      13728    14021     +293     

Files with missing lines	Coverage Δ
src/TropicalGeometry/positive_variety.jl	84.21% <84.21%> (ø)

... and 141 files with indirect coverage changes

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[YueRen]

@MateTelek Would you be able to review this pull request for me?

[lgoettgens]

It is a draft for now, as it is contingent on #4061 being merged. (I don't think it makes a lot of sense if tropical_variety returns a list of TropicalVariety while positive_tropical_variety returns a single TropicalVariety)

Is this comment obsolete or how did you proceed? I am asking since #4061 is not merged.

[fingolfin]

ping @YueRen

[YueRen]

I've reminded @MateTelek about the review, as the code builds on his work. Once he gives a green light, it should be good to go.

[MateTelek]

I apologize for the late reply. I like the code very much, I have only one question/comment that I couldn't make sense of. According to the documentation, the function positive_tropical_variety should work with both the trivial valuation and the t-adic valuation. When I run the code,

C = matrix(QQ,[[-3,1,-1,-2,2],[-1,1,-1,-1,1]])
K,t = rational_function_field(QQ,"t")
nu = tropical_semiring_map(K,t)
R,x = polynomial_ring(K,ncols(C))
Ilin = ideal(C*gens(R))  
TropL = positive_tropical_variety(Ilin,nu) 
n_maximal_polyhedra(TropL) == 4

C = matrix(QQ,[[-3,1,-1,-2,2],[-1,1,-1,-1,1]])
R,x = polynomial_ring(QQ,ncols(C))
nu = tropical_semiring_map(QQ)
I = ideal(C*gens(R))
TropPlusI = positive_tropical_variety(I,nu)
n_maximal_polyhedra(TropPlusI) == 5

once I get 4 maximal cones and other times 5, which is surprising since the tropicalization should not depend on the field extension. Is there an easy explanation for why this happens?

[YueRen]

@MateTelek: thanks for the review. The result is quite weird indeed, I will look into it!

[YueRen]

@MateTelek What you stumbled upon comes from the fact that a Bergman fan has multiple fan structures.

Let's forget about positivity and just look at the tropical linear space:

C = matrix(QQ,[[-3,1,-1,-2,2],[-1,1,-1,-1,1]])
K,t = rational_function_field(QQ,"t")
R,x = polynomial_ring(K,ncols(C))
I = ideal(C*gens(R))
S,x = polynomial_ring(QQ,ncols(C))
J = ideal(C*gens(S))

TropI = tropical_linear_space(I,tropical_semiring_map(K,t))
TropJ = tropical_linear_space(J,tropical_semiring_map(QQ))

If you look at their maximal polyhedra, you see that

1. Every maximal polyhedra in TropI has ray 1
2. Every maximal polyhedra in TropJ has ray 8
3. Ignoring the rays above, every other ray in TropI and TropJ connects to two others EXCEPT ray 5 in TropJ, which only connects to rays 3 and 4:

julia> maximal_polyhedra(IncidenceMatrix,TropI)
9×7 IncidenceMatrix
[1, 2, 4]
[1, 2, 5]
[1, 2, 6]
[1, 3, 4]
[1, 3, 5]
[1, 3, 6]
[1, 4, 7]
[1, 5, 7]
[1, 6, 7]

julia> maximal_polyhedra(IncidenceMatrix,TropJ)
10×8 IncidenceMatrix
[1, 2, 8]
[1, 3, 8]
[2, 4, 8]
[3, 5, 8]
[4, 5, 8]
[1, 6, 8]
[4, 6, 8]
[2, 7, 8]
[3, 7, 8]
[6, 7, 8]

A quick test shows that (modulo the lineality space), rays 3, 4, 5 in TropJ are colinear:

julia> rank(vcat(matrix(QQ,first(rays_modulo_lineality(TropJ)))[[3,4,5],:],ones_matrix(QQ,1,5)))
3

What you see here is a cone in the coarse structure TropI being subdivided into two cones in the fine structure TropJ. As to why TropI is coarse and TropJ is fine, that is most likely because TropI uses an optimized algorithm for linearly realizable matroids over QQ, while TropJ uses a more generic algorithm that works over any field.

Back to positive tropicalizations, you can see the exact same effect:

julia> TropPlusI = positive_tropical_variety(I,tropical_semiring_map(K,t))
Min tropical variety

julia> TropPlusJ = positive_tropical_variety(J,tropical_semiring_map(QQ))
Min tropical variety

julia> maximal_polyhedra(IncidenceMatrix,TropPlusI)
4×5 IncidenceMatrix
[1, 2, 3]
[1, 2, 4]
[1, 3, 5]
[1, 4, 5]

julia> maximal_polyhedra(IncidenceMatrix,TropPlusJ)
5×6 IncidenceMatrix
[1, 2, 3]
[1, 3, 4]
[2, 3, 5]
[3, 4, 6]
[3, 5, 6]

Notice how:

1. Every maximal polyhedra in TropPlusI has ray 1
2. Every maximal polyhedra in TropPlusJ has ray 3
3. Ignoring the rays above, TropPlusI is a circle with 4 edges (2-3-5-4-2) while TropPlusJ is a circle with 5 edges (1-2-5-6-4-1).

[YueRen]

I've emailed @MateTelek to see whether my explanation above makes sense. If he agrees, then this PR is ready to be merged.

[MateTelek]

@YueRen Thanks for the explanation. I agree with you, I missed that the Bergman fan can have different fan structures.