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