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
@@ -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)))
@@ -197,7 +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
   
-  (Generic.Ideal(coordring, eqns), eqns)
+  if output == :m2
+    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.jl

@@ -33,44 +33,45 @@ tangencies = [
   for n in 1:3
 ]
 ctx_tan_sph = Engine.Construction{CoeffType}(elements = spheres, relations = tangencies)
-ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
-freedom = Engine.dimension(ideal_tan_sph)
-println("Three mutually tangent spheres: $freedom degrees of freedom")
+##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")
 
 # --- test rational cut ---
 
-coordring = base_ring(ideal_tan_sph)
-vbls = Variable.(symbols(coordring))
+##coordring = base_ring(ideal_tan_sph)
+##vbls = Variable.(symbols(coordring))
 
 # test a random witness set
-system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
-norm2 = vec -> real(dot(conj.(vec), vec))
-rng = MersenneTwister(6071)
-n_planes = 6
-samples = []
-for _ in 1:n_planes
-  real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng))
-  for soln in real_solns
-    if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
-      push!(samples, soln)
-    end
-  end
-end
-println("Found $(length(samples)) sample solutions")
+##system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
+##norm2 = vec -> real(dot(conj.(vec), vec))
+##rng = MersenneTwister(6071)
+##n_planes = 6
+##samples = []
+##for _ in 1:n_planes
+##  real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng))
+##  for soln in real_solns
+##    if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
+##      push!(samples, soln)
+##    end
+##  end
+##end
+##println("Found $(length(samples)) sample solutions")
 
 # show a sample solution
-function show_solution(ctx, vals)
-  # evaluate elements
-  real_vals = real.(vals)
-  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]
-  
-  # create scene
-  scene = Scene()
-  cam3d!(scene)
-  scatter!(scene, disp_points, color = :green)
-  for sph in disp_spheres
-    mesh!(scene, sph, color = :gray)
-  end
-  scene
-end
+##function show_solution(ctx, vals)
+##  # evaluate elements
+##  real_vals = real.(vals)
+##  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]
+##  
+##  # create scene
+##  scene = Scene()
+##  cam3d!(scene)
+##  scatter!(scene, disp_points, color = :green)
+##  for sph in disp_spheres
+##    mesh!(scene, sph, color = :gray)
+##  end
+##  scene
+##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