Extend AbstractAlgebra ideals to rational coefficients

The extension should also let us work over finite fields of prime order,
although we don't need to do that.

engine-proto/engine.jl

@@ -7,6 +7,26 @@ using LinearAlgebra
 using AbstractAlgebra
 using Groebner
 
+# --- commutative algebra ---
+
+# as of version 0.36.6, AbstractAlgebra only supports ideals in multivariate
+# polynomial rings when coefficients are integers. in `reduce_gens`, the
+# `lmnode` constructor requires < to be defined on the coefficients, and the
+# `reducer_size` heuristic requires `ndigits` to be defined on the coefficients.
+# this patch for `reducer_size` removes the `ndigits` dependency
+##function Generic.reducer_size(f::T) where {U <: MPolyRingElem{<:FieldElement}, V, N, T <: Generic.lmnode{U, V, N}}
+##   if f.size != 0.0
+##      return f.size
+##   end
+##   return 0.0 + sum(j^2 for j in 1:length(f.poly))
+##end
+
+# as of version 0.36.6, AbstractAlgebra only supports ideals in multivariate
+# polynomial rings when the coefficients are integers. we use Groebner to extend
+# support to rationals and to finite fields of prime order
+Generic.reduce_gens(I::Generic.Ideal{U}) where {T <: FieldElement, U <: MPolyRingElem{T}} =
+  Generic.Ideal{U}(base_ring(I), groebner(gens(I)))
+
 # --- primitve elements ---
 
 abstract type Element{T} end
@@ -23,8 +43,6 @@ mutable struct Point{T} <: Element{T}
   ) where T = new(coords, vec, nothing)
 end
 
-##coordnames(_::Point) = [:xₚ, :yₚ, :zₚ]
-
 function buildvec!(pt::Point)
   coordring = parent(pt.coords[1])
   pt.vec = [one(coordring), dot(pt.coords, pt.coords), pt.coords...]
@@ -43,8 +61,6 @@ mutable struct Sphere{T} <: Element{T}
   ) where T = new(coords, vec, rel)
 end
 
-##coordnames(_::Sphere) = [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ]
-
 function buildvec!(sph::Sphere)
   coordring = parent(sph.coords[1])
   sph.vec = sph.coords
@@ -130,10 +146,6 @@ function realize(ctx::Construction{T}) where T
     end
   end
   
-  display(collect(elemenum))
-  display(coordnamelist)
-  println()
-  
   # construct coordinate ring
   coordring, coordqueue = polynomial_ring(parent_type(T)(), coordnamelist, ordering = :degrevlex)
   
@@ -150,16 +162,14 @@ function realize(ctx::Construction{T}) where T
   # construct coordinate vectors
   for (_, elem) in elemenum
     buildvec!(elem)
-    display(elem.coords)
-    display(elem.vec)
-    println()
   end
   
   # turn relations into equations
-  vcat(
+  eqns = vcat(
     equation.(ctx.relations),
     [elem.rel for elem in ctx.elements if !isnothing(elem.rel)]
   )
+  Generic.Ideal(coordring, eqns)
 end
 
 end
@@ -172,22 +182,23 @@ a = Engine.Point{CoeffType}()
 s = Engine.Sphere{CoeffType}()
 a_on_s = Engine.LiesOn{CoeffType}(a, s)
 ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
-eqns_a_s = Engine.realize(ctx)
+ideal_a_s = Engine.realize(ctx)
 
 b = Engine.Point{CoeffType}()
 b_on_s = Engine.LiesOn{CoeffType}(b, s)
 Engine.push!(ctx, b)
 Engine.push!(ctx, s)
 Engine.push!(ctx, b_on_s)
-eqns_ab_s = Engine.realize(ctx)
+ideal_ab_s = Engine.realize(ctx)
 
 spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
 tangencies = [
   Engine.AlignsWithBy{CoeffType}(
     spheres[n],
     spheres[mod1(n+1, length(spheres))],
-    -1//1
+    CoeffType(-1//1)
   )
   for n in 1:3
 ]
-ctx_chain = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
+ctx_chain = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
+ideal_chain = Engine.realize(ctx_chain)