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.

engine-proto/Engine.Algebraic.jl

@@ -29,6 +29,18 @@ end
 dimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} =
   length(gens(base_ring(I))) - codimension(I, maxdepth)
 
+m2_ordering(R::MPolyRing) = Dict(
+  :lex => :Lex,
+  :deglex => :GLex,
+  :degrevlex => :GRevLex
+)[ordering(R)]
+
+string_m2(ring::MPolyRing) =
+  "QQ[$(join(symbols(ring), ", ")), MonomialOrder => $(m2_ordering(ring))]"
+
+string_m2(f::MPolyRingElem) =
+  replace(string(f), "//" => "/")
+
 # --- primitve elements ---
 
 abstract type Element{T} end
@@ -120,11 +132,11 @@ equation(rel::AlignsWithBy) = mprod(rel.elements[1].vec, rel.elements[2].vec) -
 # --- constructions ---
 
 mutable struct Construction{T}
-  points::Set{Point{T}}
+  points::Vector{Point{T}}
-  spheres::Set{Sphere{T}}
+  spheres::Vector{Sphere{T}}
-  relations::Set{Relation{T}}
+  relations::Vector{Relation{T}}
   
-  function Construction{T}(; elements = Set{Element{T}}(), relations = Set{Relation{T}}()) where T
+  function Construction{T}(; elements = Vector{Element{T}}(), relations = Vector{Relation{T}}()) where T
     allelements = union(elements, (rel.elements for rel in relations)...)
     new{T}(
       filter(elt -> isa(elt, Point), allelements),
@@ -149,7 +161,10 @@ function Base.push!(ctx::Construction{T}, rel::Relation{T}) where T
   end
 end
 
-function realize(ctx::Construction{T}) where T
+# output options:
+#  nothing - find a Gröbner basis
+#  :m2     - write a system of polynomials to a Macaulay2 file
+function realize(ctx::Construction{T}; output = nothing) where T
   # collect coordinate names
   coordnamelist = Symbol[]
   eltenum = enumerate(Iterators.flatten((ctx.spheres, ctx.points)))
@@ -197,8 +212,16 @@ function realize(ctx::Construction{T}) where T
     push!(eqns, sum(elt.vec[2] for elt in Iterators.flatten((ctx.points, ctx.spheres))) - n_elts)
   end
   
-  ## [test] (Generic.Ideal(coordring, eqns), eqns)
+  if output == :m2
-  (nothing, eqns)
+    file = open("macaulay2/construction.m2", "w")
+    write(file, string(
+      "coordring = $(string_m2(coordring))\n",
+      "eqns = {\n  $(join(string_m2.(eqns), ",\n  "))\n}"
+    ))
+    close(file)
+  else
+    return (Generic.Ideal(coordring, eqns), eqns)
+  end
 end
 
 end

engine-proto/Engine.Numerical.jl

@@ -1,5 +1,6 @@
 module Numerical
 
+using Random: default_rng
 using LinearAlgebra
 using AbstractAlgebra
 using HomotopyContinuation:
@@ -28,16 +29,16 @@ end
 
 # --- sampling ---
 
-function real_samples(F::AbstractSystem, dim)
+function real_samples(F::AbstractSystem, dim; rng = default_rng())
   # choose a random real hyperplane of codimension `dim` by intersecting
   # hyperplanes whose normal vectors are uniformly distributed over the unit
   # sphere
   # [to do] guard against the unlikely event that one of the normals is zero
   normals = transpose(hcat(
-    (normalize(randn(nvariables(F))) for _ in 1:dim)...
+    (normalize(randn(rng, nvariables(F))) for _ in 1:dim)...
   ))
   cut = LinearSubspace(normals, fill(0., dim))
-  filter(isreal, results(witness_set(F, cut)))
+  filter(isreal, results(witness_set(F, cut, seed = 0x1974abba)))
 end
 
 AbstractAlgebra.evaluate(pt::Point, vals::Vector{<:RingElement}) =

engine-proto/Engine.jl

@@ -23,106 +23,55 @@ using GLMakie
 
 CoeffType = Rational{Int64}
 
-##a = Engine.Point{CoeffType}()
+spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
-##s = Engine.Sphere{CoeffType}()
+tangencies = [
-##a_on_s = Engine.LiesOn{CoeffType}(a, s)
+  Engine.AlignsWithBy{CoeffType}(
-##ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
+    spheres[n],
-##ideal_a_s = Engine.realize(ctx)
+    spheres[mod1(n+1, length(spheres))],
-##println("A point on a sphere: $(Engine.dimension(ideal_a_s)) degrees of freedom")
+    CoeffType(1)
-
+  )
-##b = Engine.Point{CoeffType}()
+  for n in 1:3
-##b_on_s = Engine.LiesOn{CoeffType}(b, s)
+]
-##Engine.push!(ctx, b)
+ctx_tan_sph = Engine.Construction{CoeffType}(elements = spheres, relations = tangencies)
-##Engine.push!(ctx, s)
-##Engine.push!(ctx, b_on_s)
-##ideal_ab_s, eqns_ab_s = Engine.realize(ctx)
-##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 = [
-##  Engine.AlignsWithBy{CoeffType}(
-##    spheres[n],
-##    spheres[mod1(n+1, length(spheres))],
-##    CoeffType(-1//1)
-##  )
-##  for n in 1:3
-##]
-##tangencies = [
-  ##Engine.LiesOn{CoeffType}(points[1], spheres[2]),
-  ##Engine.LiesOn{CoeffType}(points[1], spheres[3]),
-  ##Engine.LiesOn{CoeffType}(points[2], spheres[3]),
-  ##Engine.LiesOn{CoeffType}(points[2], spheres[1]),
-  ##Engine.LiesOn{CoeffType}(points[3], spheres[1]),
-  ##Engine.LiesOn{CoeffType}(points[3], spheres[2])
-##]
-##ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
 ##ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
+Engine.realize(ctx_tan_sph, output = :m2)
 ##freedom = Engine.dimension(ideal_tan_sph)
 ##println("Three mutually tangent spheres: $freedom degrees of freedom")
 
-points = [Engine.Point{CoeffType}() for _ in 1:3]
-spheres = [Engine.Sphere{CoeffType}() for _ in 1:2]
-ctx_joined = Engine.Construction{CoeffType}(
-  elements = Set([points; spheres]),
-  relations= Set([
-    Engine.LiesOn{CoeffType}(pt, sph)
-    for pt in points for sph in spheres
-  ])
-)
-ideal_joined, eqns_joined = Engine.realize(ctx_joined)
-freedom = Engine.dimension(ideal_joined)
-println("$(length(points)) points on $(length(spheres)) spheres: $freedom degrees of freedom")
-
 # --- test rational cut ---
 
-coordring = base_ring(ideal_joined)
+##coordring = base_ring(ideal_tan_sph)
-vbls = Variable.(symbols(coordring))
+##vbls = Variable.(symbols(coordring))
 
 # test a random witness set
-system = CompiledSystem(System(eqns_joined, variables = vbls))
+##system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
-norm2 = vec -> real(dot(conj.(vec), vec))
+##norm2 = vec -> real(dot(conj.(vec), vec))
-Random.seed!(6071)
+##rng = MersenneTwister(6071)
-n_planes = 3
+##n_planes = 6
-samples = []
+##samples = []
-for _ in 1:n_planes
+##for _ in 1:n_planes
-  real_solns = solution.(Engine.Numerical.real_samples(system, freedom))
+##  real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng))
-  for soln in real_solns
+##  for soln in real_solns
-    if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
+##    if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
-      push!(samples, soln)
+##      push!(samples, soln)
-    end
+##    end
-  end
+##  end
-end
+##end
-println("$(length(samples)) sample solutions:")
+##println("Found $(length(samples)) sample solutions")
-for soln in samples
-  ## display([vbls round.(soln, digits = 6)]) ## [verbose]
-  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
 
 # show a sample solution
-function show_solution(ctx, vals)
+##function show_solution(ctx, vals)
-  # evaluate elements
+##  # evaluate elements
-  real_vals = real.(vals)
+##  real_vals = real.(vals)
-  disp_points = [Engine.Numerical.evaluate(pt, real_vals) for pt in ctx.points]
+##  disp_points = [Engine.Numerical.evaluate(pt, real_vals) for pt in ctx.points]
-  disp_spheres = [Engine.Numerical.evaluate(sph, real_vals) for sph in ctx.spheres]
+##  disp_spheres = [Engine.Numerical.evaluate(sph, real_vals) for sph in ctx.spheres]
-  
+##  
-  # create scene
+##  # create scene
-  scene = Scene()
+##  scene = Scene()
-  cam3d!(scene)
+##  cam3d!(scene)
-  scatter!(scene, disp_points, color = :green)
+##  scatter!(scene, disp_points, color = :green)
-  for sph in disp_spheres
+##  for sph in disp_spheres
-    mesh!(scene, sph, color = :gray)
+##    mesh!(scene, sph, color = :gray)
-  end
+##  end
-  scene
+##  scene
-end
+##end

engine-proto/macaulay2/engine.m2

@@ -0,0 +1,3 @@
+needsPackage "TriangularSets"
+
+mprod = (v, w) -> (v#0*w#1 + w#0*v#1) / 2 - v#2*w#2 - v#3*w#3 - v#4*w#4