Skip to content

TropicalGeometry: stable_intersection, added perturbation check #5272

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Draft
wants to merge 1 commit into
base: master
Choose a base branch
from
Draft

TropicalGeometry: stable_intersection, added perturbation check #5272

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
28 changes: 20 additions & 8 deletions src/TropicalGeometry/intersection.jl
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -17,14 +17,21 @@ function stable_intersection(TropV1::TropicalVarietySupertype{minOrMax,true}, Tr
perturbation = rand(Int,ambient_dim(TropV1))
end

# computing the expected dimension of the stable intersection
n = ambient_dim(TropV1)
expectedDimension = n - codim(TropV1) - codim(TropV2)

Sigma12 = Polyhedron{QQFieldElem}[]
mults12 = ZZRingElem[]

# todo: check that perturbation is actually generic
for (sigma1,m1) in maximal_polyhedra_and_multiplicities(TropV1)
for (sigma2,m2) in maximal_polyhedra_and_multiplicities(TropV2)
if dim(sigma1+sigma2)==n && intersect_after_perturbation(sigma1,sigma2,perturbation)
intersectAfterPerturbation, intersectionDimension = intersect_after_perturbation_with_dimension(sigma1,sigma2,perturbation)
if intersectAfterPerturbation && intersectionDimension>expectedDimension
# non-generic intersection, rerun everything with new random perturbation
println("non-generic intersection, rerunning with new perturbation")
return stable_intersection(TropV1, TropV2, rand(Int,ambient_dim(TropV1)))
end
if dim(sigma1+sigma2)==n && intersectAfterPerturbation
sigma12 = intersect(sigma1, sigma2)
i = findfirst(isequal(sigma12), Sigma12)
if isnothing(i)
Expand All @@ -47,10 +54,10 @@ function stable_intersection(TropV1::TropicalVarietySupertype{minOrMax,true}, Tr
return tropical_variety(Sigma12,mults12,convention(TropV1))
end

function intersect_after_perturbation(sigma1::Polyhedron{QQFieldElem}, sigma2::Polyhedron{QQFieldElem}, perturbation::Vector{Int})
function intersect_after_perturbation_with_dimension(sigma1::Polyhedron{QQFieldElem}, sigma2::Polyhedron{QQFieldElem}, perturbation::Vector{Int})

if dim(intersect(sigma1,sigma2))<0
return false
return false, -1
end

# construct sigma1Prime = sigma1 x RR in RR^(n+1)
Expand All @@ -66,14 +73,19 @@ function intersect_after_perturbation(sigma1::Polyhedron{QQFieldElem}, sigma2::P
N = affine_equation_matrix(affine_hull(sigma2))
N = hcat(N, zero_matrix(QQ, nrows(N), 1))
v = zero_matrix(QQ, 1, ncols(N))
v[1,end] = 1 # v = last unit vector
v[1,end] = 1 # v = last unit vector as a row vector
N = vcat(N, v)
sigma2Prime = polyhedron((M[:,2:end],QQFieldElem[-M[:,1]...]), (N[:,2:end],QQFieldElem[-N[:,1]...])) + convex_hull(zero_matrix(QQ, 1, ncols(N) - 1), matrix(QQ, [vcat(perturbation, [1])]))

sigma12Prime = intersect(sigma1Prime, sigma2Prime)
# @req codim(sigma1Prime)+codim(sigma2Prime)==codim(sigma12Prime) "perturbation "*string(perturbation)*" not generic"

return dim(sigma12Prime)>0 && last(relative_interior_point(sigma12Prime))>0
intersectAfterPerturbation = (last(relative_interior_point(sigma12Prime))>0)
intersectionDimension = dim(sigma12Prime)-1

@assert dim(sigma12Prime)>=0
@assert (!intersectAfterPerturbation || dim(sigma12Prime)>0)

return intersectAfterPerturbation, intersectionDimension
end


Expand Down
Loading