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