Minimal associated primes

src/Rings/mpoly-ideals.jl

@@ -1068,6 +1068,12 @@ julia> L = minimal_primes(I)
 function minimal_primes(I::MPolyIdeal; algorithm::Symbol = :GTZ, cache::Bool=true)
   has_attribute(I, :minimal_primes) && return get_attribute(I, :minimal_primes)::Vector{typeof(I)}
   R = base_ring(I)
+  if coefficient_ring(R) isa QQField && is_zero(dim(I))
+    L = Singular.LibAssprimeszerodim.assPrimes(singular_generators(I))
+    result = typeof(I)[ideal(R, q) for q in L]
+    cache && set_attribute!(I, :minimal_primes=>result)
+    return result
+  end
   if isa(base_ring(R), NumField) && !isa(base_ring(R), AbsSimpleNumField)
     A, mA = absolute_simple_field(base_ring(R))
     mp = minimal_primes(map_coefficients(pseudo_inv(mA), I); algorithm = algorithm)
@@ -1133,18 +1139,34 @@ function minimal_primes(
 
   # This will in many cases lead to an easy simplification of the problem
   if factor_generators
-    J = typeof(I)[ideal(R, elem_type(R)[])]
+    J = [ideal(R, gens(I))] # A copy of I as initialization
     for g in gens(I)
       K = typeof(I)[]
       is_zero(g) && continue
       for (b, k) in factor(g)
+        # Split the already collected components with b
         for j in J
           push!(K, j + ideal(R, b))
         end
       end
       J = K
     end
-    result = unique!(filter!(!is_one, vcat([minimal_primes(j; algorithm, factor_generators=false) for j in J]...)))
+
+    unique_comp = typeof(I)[]
+    for q in J
+      is_one(q) && continue
+      q in unique_comp && continue
+      push!(unique_comp, q)
+    end
+    J = unique_comp
+    result = typeof(I)[]
+    # `unique!` does not work for lists of ideals. I don't know why, but for the moment we need the 
+    # following workaround.
+    for p in filter!(!is_one, vcat([minimal_primes(j; algorithm, factor_generators=false) for j in J]...))
+      p in result && continue
+      push!(result, p)
+    end
+
     # The list might not consist of minimal primes only. We have to discard the embedded ones
     final_list = typeof(I)[]
     for p in result
@@ -2032,7 +2054,8 @@ function small_generating_set(
   computed_gb = IdealGens(ring, sing_gb, true)
   if !haskey(I.gb,computed_gb.ord)
   # if not yet present, store gb for later use
-    I.gb[computed_gb.ord] = computed_gb
+    I.gb[computed_gb.ord]      = computed_gb
+    I.gb[computed_gb.ord].isGB = true
   end
 
   # we do not have a notion of minimal generating set in this context!
@@ -2241,3 +2264,4 @@ function flag_pluecker_ideal(ring::MPolyRing{<: FieldElem}, dimensions::Vector{I
                    isReduced=true,
                    isGB=true))
 end
+

test/Rings/mpoly.jl

@@ -220,6 +220,19 @@ end
   R, (x, y) = polynomial_ring(QQ, ["x", "y"])
   I = ideal(R, [one(R)])
   @test is_prime(I) == false
+
+  J = ideal(R, [x*(x-1), y*(y-1), x*y])
+  l = minimal_primes(J)
+  @test length(l) == 3
+
+  QQt, t = QQ[:t]
+  kk, a = extension_field(t^2 + 1)
+
+  R, (x, y) = kk[:x, :y]
+  J = ideal(R, [x^2 + 1, y^2 + 1, (x - a)*(y - a)])
+  l = minimal_primes(J)
+  @test length(l) == 3
+
 end
 
 @testset "Groebner" begin
@@ -574,3 +587,5 @@ end
     @test default_ordering(T) == old_default
   end
 end
+
+