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Find witnesses on random rational hyperplanes 

Choose hyperplanes that go through the trivial solution.
This commit is contained in:
Aaron Fenyes 2024-02-09 23:44:10 -05:00
parent 95c0ff14b2
commit 34358a8728
1 changed files with 60 additions and 23 deletions
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83
engine-proto/Engine.jl
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@ -225,6 +225,8 @@ end
# ~~~ sandbox setup ~~~
using Random
using Distributions
using AbstractAlgebra
using HomotopyContinuation
@ -243,7 +245,8 @@ Engine.push!(ctx, b)
Engine.push!(ctx, s)
Engine.push!(ctx, b_on_s)
ideal_ab_s, eqns_ab_s = Engine.realize(ctx)
println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of freedom")
freedom = Engine.dimension(ideal_ab_s)
println("Two points on a sphere: ", freedom, " degrees of freedom")
##spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
##tangencies = [
@ -260,33 +263,67 @@ println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of f
# --- test rational cut ---
cut_coeffs = [
1 1 1 0 0 0 1 1 1 1 1;
2 1 1 0 0 0 1 2 1 1 1;
1 2 0 0 0 0 1 1 0 1 2
]
cut = [
sum(vcat([1, 1, 1] .* a.coords, [1, 1, 1, 1, 1] .* (s.coords - [0, 0, 0, 0, 1])))
sum(vcat([2, 1, 1] .* a.coords, [1, 2, 1, 1, 1] .* (s.coords - [0, 0, 0, 0, 1])))
sum(vcat([1, 2, 0] .* a.coords, [1, 1, 0, 1, 2] .* (s.coords - [0, 0, 0, 0, 1])))
sum(vcat(cf[1:3] .* a.coords, cf[4:6] .* b.coords, cf[7:end] .* (s.coords - [0, 0, 0, 0, 1])))
for cf in eachrow(cut_coeffs)
]
cut_ideal_ab_s = Generic.Ideal(base_ring(ideal_ab_s), [gens(ideal_ab_s); cut])
cut_dim = Engine.dimension(cut_ideal_ab_s)
println("Two points on a sphere, after cut: ", cut_dim, " degrees of freedom")
if cut_dim == 0
vbls = Variable.(symbols(base_ring(ideal_ab_s)))
cut_freedom = Engine.dimension(cut_ideal_ab_s)
println("Two points on a sphere, after cut: ", cut_freedom, " degrees of freedom")
if cut_freedom == 0
coordring = base_ring(ideal_ab_s)
vbls = Variable.(symbols(coordring))
cut_system = System([eqns_ab_s; cut], variables = vbls)
cut_result = HomotopyContinuation.solve(cut_system)
println("non-singular solutions:")
for soln in solutions(cut_result)
display(soln)
end
println("singular solutions:")
for sing in singular(cut_result)
display(sing.solution)
end
##cut_result = HomotopyContinuation.solve(cut_system)
##println("non-singular solutions:")
##for soln in solutions(cut_result)
## display(soln)
##end
##println("singular solutions:")
##for sing in singular(cut_result)
## display(sing.solution)
##end
# test corresponding witness set
cut_matrix = [1 1 1 1 0 1 1 0 1 1 0; 1 2 1 2 0 1 1 0 1 1 0; 1 1 0 1 0 1 2 0 2 0 0]
cut_subspace = LinearSubspace(cut_matrix, [1, 1, 2])
witness = witness_set(System(eqns_ab_s, variables = vbls), cut_subspace)
# test a random witness set
max_slope = 2
binom = Binomial(2max_slope, 1/2)
Random.seed!(6071)
samples = []
for _ in 1:3
cut_matrix = rand(binom, freedom, length(gens(coordring))) .- max_slope
##cut_matrix = [
## 1 1 1 1 0 1 1 0 1 1 0;
## 1 2 1 2 0 1 1 0 1 1 0;
## 1 1 0 1 0 1 2 0 2 0 0
##]
sph_z_ind = indexin([sph.coords[5] for sph in ctx.spheres], gens(coordring))
cut_offset = [sum(cf[sph_z_ind]) for cf in eachrow(cut_matrix)]
println("sphere z variables: ", vbls[sph_z_ind])
display(cut_matrix)
display(cut_offset)
cut_subspace = LinearSubspace(cut_matrix, cut_offset)
wtns = witness_set(System(eqns_ab_s, variables = vbls), cut_subspace)
append!(samples, solution.(filter(isreal, results(wtns))))
end
println("witness solutions:")
for wtns in solutions(witness)
display(wtns)
for soln in samples
display([vbls round.(soln, digits = 6)])
k_sq = abs2(soln[1])
if abs2(soln[end-2]) > 1e-12
if k_sq < 1e-12
println("center at infinity: z coordinates $(round(soln[end], digits = 6)) and $(round(soln[end-1], digits = 6))}")
else
sum_sq = soln[4]^2 + soln[7]^2 + soln[end-2]^2 / k_sq
println("center on z axis: r² = $(round(1/k_sq, digits = 6)), x² + y² + h² = $(round(sum_sq, digits = 6))")
end
else
sum_sq = sum(soln[[4, 7, 10]] .^ 2)
println("center at origin: r² = $(round(1/k_sq, digits = 6)); x² + y² + z² = $(round(sum_sq, digits = 6))")
end
end
end