Start interface to Macaulay2
I did this to try out Macaulay2's "triangularize" function, but that turns out to use Maple for rings with more than three variables.
This commit is contained in:
Aaron Fenyes
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3170a933e4
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74529048de
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@ -29,6 +29,18 @@ 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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m2_ordering(R::MPolyRing) = Dict(
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:lex => :Lex,
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:deglex => :GLex,
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:degrevlex => :GRevLex
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)[ordering(R)]
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string_m2(ring::MPolyRing) =
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"QQ[$(join(symbols(ring), ", ")), MonomialOrder => $(m2_ordering(ring))]"
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string_m2(f::MPolyRingElem) =
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replace(string(f), "//" => "/")
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# --- primitve elements ---
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abstract type Element{T} end
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@ -149,7 +161,10 @@ function Base.push!(ctx::Construction{T}, rel::Relation{T}) where T
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end
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end
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function realize(ctx::Construction{T}) where T
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# output options:
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# nothing - find a Gröbner basis
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# :m2 - write a system of polynomials to a Macaulay2 file
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function realize(ctx::Construction{T}; output = nothing) where T
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# collect coordinate names
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coordnamelist = Symbol[]
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eltenum = enumerate(Iterators.flatten((ctx.spheres, ctx.points)))
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@ -197,7 +212,16 @@ function realize(ctx::Construction{T}) where T
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push!(eqns, sum(elt.vec[2] for elt in Iterators.flatten((ctx.points, ctx.spheres))) - n_elts)
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end
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(Generic.Ideal(coordring, eqns), eqns)
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if output == :m2
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file = open("macaulay2/construction.m2", "w")
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write(file, string(
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"coordring = $(string_m2(coordring))\n",
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"eqns = {\n $(join(string_m2.(eqns), ",\n "))\n}"
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))
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close(file)
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else
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return (Generic.Ideal(coordring, eqns), eqns)
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end
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end
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end
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@ -33,44 +33,45 @@ tangencies = [
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for n in 1:3
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]
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ctx_tan_sph = Engine.Construction{CoeffType}(elements = spheres, relations = tangencies)
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ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
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freedom = Engine.dimension(ideal_tan_sph)
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println("Three mutually tangent spheres: $freedom degrees of freedom")
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##ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
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Engine.realize(ctx_tan_sph, output = :m2)
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##freedom = Engine.dimension(ideal_tan_sph)
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##println("Three mutually tangent spheres: $freedom degrees of freedom")
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# --- test rational cut ---
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coordring = base_ring(ideal_tan_sph)
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vbls = Variable.(symbols(coordring))
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##coordring = base_ring(ideal_tan_sph)
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##vbls = Variable.(symbols(coordring))
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# test a random witness set
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system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
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norm2 = vec -> real(dot(conj.(vec), vec))
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rng = MersenneTwister(6071)
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n_planes = 6
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samples = []
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for _ in 1:n_planes
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real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng))
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for soln in real_solns
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if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
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push!(samples, soln)
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end
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end
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end
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println("Found $(length(samples)) sample solutions")
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##system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
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##norm2 = vec -> real(dot(conj.(vec), vec))
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##rng = MersenneTwister(6071)
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##n_planes = 6
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##samples = []
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##for _ in 1:n_planes
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## real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng))
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## for soln in real_solns
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## if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
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## push!(samples, soln)
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## end
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## end
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##end
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##println("Found $(length(samples)) sample solutions")
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# show a sample solution
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function show_solution(ctx, vals)
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# evaluate elements
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real_vals = real.(vals)
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disp_points = [Engine.Numerical.evaluate(pt, real_vals) for pt in ctx.points]
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disp_spheres = [Engine.Numerical.evaluate(sph, real_vals) for sph in ctx.spheres]
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# create scene
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scene = Scene()
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cam3d!(scene)
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scatter!(scene, disp_points, color = :green)
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for sph in disp_spheres
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mesh!(scene, sph, color = :gray)
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end
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scene
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end
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##function show_solution(ctx, vals)
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## # evaluate elements
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## real_vals = real.(vals)
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## disp_points = [Engine.Numerical.evaluate(pt, real_vals) for pt in ctx.points]
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## disp_spheres = [Engine.Numerical.evaluate(sph, real_vals) for sph in ctx.spheres]
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##
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## # create scene
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## scene = Scene()
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## cam3d!(scene)
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## scatter!(scene, disp_points, color = :green)
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## for sph in disp_spheres
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## mesh!(scene, sph, color = :gray)
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## end
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## scene
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##end
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@ -0,0 +1,3 @@
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needsPackage "TriangularSets"
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mprod = (v, w) -> (v#0*w#1 + w#0*v#1) / 2 - v#2*w#2 - v#3*w#3 - v#4*w#4
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