Hilbert series integration (#2698)

src/Modules/hilbert.jl

@@ -21,7 +21,7 @@ function HSNum_module(SubM::SubquoModule{T}, HSRing::Ring, backend::Symbol=:Abbo
     return multi_hilbert_series(quo(P,ideal(P,[1]))[1]; parent=HSRing, backend=backend)[1][1]
   end
   GensLM = gens(LM);
-  L = [[] for _ in 1:r];  # L[k] will be list of monomial gens for k-th cooord
+  L = [[] for _ in 1:r];  # L[k] will be list of monomial gens for k-th coord
   # Nested loop below extracts the coordinate monomial ideals -- is there a better way?
   for g in GensLM
     SR = coordinates(ambient_representative(g)); # should have length = 1
@@ -37,19 +37,12 @@ function HSNum_module(SubM::SubquoModule{T}, HSRing::Ring, backend::Symbol=:Abbo
   @vprintln :hilbert 1 "HSNum_module: shifts are $(shifts)";
   shift_expv = [gen_repr(d)  for d in shifts];
   @vprintln :hilbert 1 "HSNum_module: shift_expv are $(shift_expv)";
-###  HSeriesRing = parent(HSeriesList[1]);
-###  @vprintln :hilbert 1 "HSNum_module: HSeriesRing = $(HSeriesRing)";
   t = gens(HSRing);
   ScaleFactor = [_power_product(t,e)  for e in shift_expv];
   result = sum([ScaleFactor[k]*HSeriesList[k]  for k in 1:r]);
   return result;
 end
 
-# No longer needed
-# function HSNum_module(F::FreeMod{T}; parent::Union{Nothing,Ring} = nothing)  where T <: MPolyRingElem
-#   # ASSUME F is graded free module
-#   return HSNum_module(sub(F,gens(F))[1]; parent=parent)
-# end
 
 @doc raw"""
     multi_hilbert_series(M::SubquoModule; parent::Union{Nothing,Ring} = nothing)
@@ -93,6 +86,7 @@ julia> den
 function multi_hilbert_series(SubM::SubquoModule{T}; parent::Union{Nothing, Ring} = nothing, backend::Symbol = :Abbott)  where T <: MPolyRingElem
   R = base_ring(SubM)
   @req coefficient_ring(R) isa AbstractAlgebra.Field "The coefficient ring must be a field"
+  @req is_positively_graded(R) "ring must be positively graded"
 
   # Wrap the case where G is abstractly isomorphic to ℤᵐ, but not realized as a 
   # free Abelian group. 
@@ -126,6 +120,7 @@ function multi_hilbert_series(SubM::SubquoModule{T}; parent::Union{Nothing, Ring
 end
 
 function multi_hilbert_series(F::FreeMod{T}; parent::Union{Nothing,Ring} = nothing, backend::Symbol = :Abbott)  where T <: MPolyRingElem
+  @req is_positively_graded(base_ring(F)) "ring must be positively graded"
   return multi_hilbert_series(sub(F,gens(F))[1]; parent=parent, backend=backend)
 end
 

src/Rings/hilbert.jl

@@ -927,6 +927,29 @@ function separate_simple_pps(gens::Vector{PP})
 end
 
 
+# !!!OBSOLESCENT!!!   2023-08-17 this fn will no be needed after Wolfram's PR is merged
+# Returns nothing; throws if ker(W) contains a non-zero vector >= 0
+#function _hilbert_series_check_weights(W::Vector{Vector{Int}})
+#  # assumes W is rectangular (and at least 1x1)
+#  # Transpose while converting:
+#  ncols = length(W);
+#  nrows = length(W[1]);
+#  A = zero_matrix(FlintZZ, nrows,ncols);
+#  for i in 1:nrows  for j in 1:ncols  A[i,j] = W[j][i];  end; end;
+#  b = zero_matrix(FlintZZ, nrows,1);
+#  try
+#    solve_non_negative(A, b); # any non-zero soln gives rise to infinitely many, which triggers an exception
+#  catch e
+#    if !(e isa ArgumentError && e.msg == "Polyhedron not bounded")
+#      rethrow(e)
+#    end
+#    # solve_non_negative must have thrown because there is a non-zero soln
+#    error("given weights permit infinite dimensional homogeneous spaces")
+#  end
+#  return nothing # otherwise it returns the result of solve_non_negative (Doh!!)
+#end
+
+
 # Check args: either throws or returns nothing.
 function HSNum_check_args(gens::Vector{PP}, W::Vector{Vector{Int}})
   if isempty(W)

src/Rings/mpoly-affine-algebras.jl

@@ -544,6 +544,7 @@ function multi_hilbert_series(
   R = base_ring(A)
   I = modulus(A)
   @req coefficient_ring(R) isa AbstractAlgebra.Field "The coefficient ring must be a field"
+  @req is_positively_graded(A) "the ring must be positively graded"
 
   # Wrap the case where G is abstractly isomorphic to ℤᵐ, but not realized as a 
   # free Abelian group. 
@@ -561,7 +562,7 @@ function multi_hilbert_series(
     map_into_S = hom(R, S, gens(S))
     J = map_into_S(I)
     AA, _ = quo(S, J)
-    (numer, denom), _ = multi_hilbert_series(AA; algorithm, parent)
+    (numer, denom), _ = multi_hilbert_series(AA; algorithm, backend, parent)
     return (numer, denom), (H, iso)
   end
 
@@ -735,6 +736,7 @@ G
 ```
 """
 function multi_hilbert_series_reduced(A::MPolyQuoRing; algorithm::Symbol=:BayerStillmanA)
+  @req is_positively_graded(A) "ring must be positively graded"
    (p, q), (H, iso) = multi_hilbert_series(A, algorithm=algorithm)
    f = p//evaluate(q)
    p = numerator(f)

test/Modules/hilbert.jl

@@ -1,14 +1,14 @@
 @testset "Hilbert series and free resolution" begin
   R, _ = polynomial_ring(QQ, ["x", "y", "z"]);
   Z = abelian_group(0);
-  Rg, (x, y, z) = grade(R, [3*Z[1],Z[1],-Z[1]]);
+  Rg, (x, y, z) = grade(R, [3*Z[1],Z[1],Z[1]]);
   F = graded_free_module(Rg, 1);
   A = Rg[x; y];
   B = Rg[x^2+y^6; y^7; z^4];
   M = SubquoModule(F, A, B);
   fr = free_resolution(M)
 
-  # There is no length(fr), so instead we do the following
+  # 2023-08-18 There is no length(fr), so instead we do the following
   indexes = range(fr.C);
   fr_len = first(indexes)
 

test/Rings/hilbert.jl

@@ -113,3 +113,11 @@ end
   @test numer1 == evaluate(numer2, T)
   @test numer3 == evaluate(numer2, first(gens(parent(numer3))))
 end
+
+@testset "Hilbert series part 3" begin
+  R, (x, y) = graded_polynomial_ring(QQ, ["x", "y"], [1, -1]);
+  I = ideal(R, [x]);
+  RmodI, _ = quo(R, I);
+  @test_throws ArgumentError  hilbert_series(RmodI);  # weights must be non-neg
+  @test_throws ArgumentError  multi_hilbert_series(RmodI); # possible infinite dimension
+end