Seek sample solutions by cutting with a hyperplane
The example hyperplane yields a single solution, with multiplicity six. You can find it analytically by hand, and homotopy continuation finds it numerically.
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Aaron Fenyes
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@ -2,12 +2,13 @@ include("HittingSet.jl")
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module Engine
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export Construction, mprod
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export Construction, mprod, codimension, dimension
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import Subscripts
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using LinearAlgebra
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using AbstractAlgebra
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using Groebner
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using HomotopyContinuation: Variable, Expression, System
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using ..HittingSet
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# --- commutative algebra ---
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@ -27,6 +28,34 @@ end
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dimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} =
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length(gens(base_ring(I))) - codimension(I, maxdepth)
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# hat tip Sascha Timme
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# https://github.com/JuliaHomotopyContinuation/HomotopyContinuation.jl/issues/520#issuecomment-1317681521
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function Base.convert(::Type{Expression}, f::MPolyRingElem)
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variables = Variable.(symbols(parent(f)))
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f_data = zip(coefficients(f), exponent_vectors(f))
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sum(cf * prod(variables .^ exp_vec) for (cf, exp_vec) in f_data)
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end
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# create a ModelKit.System from an ideal in a multivariate polynomial ring. the
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# variable ordering is taken from the polynomial ring
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function System(I::Generic.Ideal)
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eqns = Expression.(gens(I))
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variables = Variable.(symbols(base_ring(I)))
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System(eqns, variables = variables)
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end
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## [to do] not needed right now
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# create a ModelKit.System from a list of elements of a multivariate polynomial
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# ring. the variable ordering is taken from the polynomial ring
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##function System(eqns::AbstractVector{MPolyRingElem})
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## if isempty(eqns)
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## return System([])
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## else
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## variables = Variable.(symbols(parent(f)))
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## return System(Expression.(eqns), variables = variables)
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## end
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##end
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# --- primitve elements ---
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abstract type Element{T} end
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@ -189,39 +218,75 @@ function realize(ctx::Construction{T}) where T
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append!(eqns, [sum(sph.coords[k] for sph in ctx.spheres) for k in 3:4])
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end
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Generic.Ideal(coordring, eqns)
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(Generic.Ideal(coordring, eqns), eqns)
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end
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end
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# ~~~ sandbox setup ~~~
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using AbstractAlgebra
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using HomotopyContinuation
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CoeffType = Rational{Int64}
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a = Engine.Point{CoeffType}()
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s = Engine.Sphere{CoeffType}()
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a_on_s = Engine.LiesOn{CoeffType}(a, s)
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ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
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ideal_a_s = Engine.realize(ctx)
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println("A point on a sphere: ", Engine.dimension(ideal_a_s), " degrees of freeom")
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##ideal_a_s = Engine.realize(ctx)
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##println("A point on a sphere: ", Engine.dimension(ideal_a_s), " degrees of freedom")
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b = Engine.Point{CoeffType}()
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b_on_s = Engine.LiesOn{CoeffType}(b, s)
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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 = Engine.realize(ctx)
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println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of freeom")
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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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spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
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tangencies = [
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Engine.AlignsWithBy{CoeffType}(
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spheres[n],
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spheres[mod1(n+1, length(spheres))],
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CoeffType(-1//1)
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)
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for n in 1:3
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##spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
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##tangencies = [
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## Engine.AlignsWithBy{CoeffType}(
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## spheres[n],
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## spheres[mod1(n+1, length(spheres))],
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## CoeffType(-1//1)
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## )
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## for n in 1:3
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##]
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##ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
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##ideal_tan_sph = Engine.realize(ctx_tan_sph)
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##println("Three mutually tangent spheres: ", Engine.dimension(ideal_tan_sph), " degrees of freedom")
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# --- test rational cut ---
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cut = [
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sum(vcat(a.coords, (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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]
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ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
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ideal_tan_sph = Engine.realize(ctx_tan_sph)
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println("Three mutually tangent spheres: ", Engine.dimension(ideal_tan_sph), " degrees of freeom")
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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_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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# 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, 1])
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witness = witness_set(System(eqns_ab_s, variables = vbls), cut_subspace)
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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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end
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end
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