[FTheoryTools] Improve Monte-Carlo method

Author: HereAround

experimental/FTheoryTools/src/HypersurfaceModels/attributes.jl

@@ -287,7 +287,7 @@ end
 
 Return the singular loci of the Weierstrass model equivalent to the given hypersurface model,
 along with the order of vanishing of ``(f, g, \Delta)`` at each locus and the corresponding
-refined Tate fiber type. See [singular_loci(w::WeierstrassModel)](@ref) for more details.
+refined Tate fiber type. See [`singular_loci(w::WeierstrassModel)`](@ref) for more details.
 
 Raises an error if no such Weierstrass model is known.
 
@@ -297,6 +297,7 @@ model (see [BMT25](@cite BMT25) for background), in order to demonstrate this fu
 !!! warning
     The classification of singularities is performed using a Monte Carlo algorithm, involving randomized sampling.
     While reliable in practice, this probabilistic method may occasionally yield non-deterministic results.
+    The random seed can be set with the optional argument `rng`.
 
 # Examples
 ```jldoctest
@@ -324,12 +325,14 @@ Weierstrass model over a concrete base
 
 julia> set_weierstrass_model(h, w)
 
-julia> length(singular_loci(h))
+julia> using Random;
+
+julia> length(singular_loci(h; rng = Random.Xoshiro(1234)))
 2
 ```
 """
-@attr Vector{<:Tuple{<:MPolyIdeal{<:MPolyRingElem}, Tuple{Int64, Int64, Int64}, String}} function singular_loci(h::HypersurfaceModel)
+@attr Vector{<:Tuple{<:MPolyIdeal{<:MPolyRingElem}, Tuple{Int64, Int64, Int64}, String}} function singular_loci(h::HypersurfaceModel; rng::AbstractRNG = Random.default_rng())
   @req base_space(h) isa NormalToricVariety "Singular loci currently only supported for toric varieties as base space"
   @req has_attribute(h, :weierstrass_model) || has_attribute(h, :global_tate_model) "No corresponding Weierstrass model or global Tate model is known"
-  return has_attribute(h, :weierstrass_model) ? singular_loci(weierstrass_model(h)) : singular_loci(global_tate_model(h))
+  return has_attribute(h, :weierstrass_model) ? singular_loci(weierstrass_model(h); rng) : singular_loci(global_tate_model(h); rng)
 end

experimental/FTheoryTools/src/LiteratureModels/constructors.jl

@@ -81,7 +81,9 @@ Construction over concrete base may lead to singularity enhancement. Consider co
 
 Weierstrass model over a concrete base -- U(1) Weierstrass model based on arXiv paper 1208.2695 Eq. (B.19)
 
-julia> length(singular_loci(w))
+julia> using Random;
+
+julia> length(singular_loci(w; rng = Random.Xoshiro(1234)))
 1
 ```
 
@@ -102,7 +104,9 @@ Construction over concrete base may lead to singularity enhancement. Consider co
 
 Weierstrass model over a concrete base -- U(1) Weierstrass model based on arXiv paper 1208.2695 Eq. (B.19)
 
-julia> length(singular_loci(w))
+julia> using Random;
+
+julia> length(singular_loci(w; rng = Random.Xoshiro(1234)))
 1
 ```
 

experimental/FTheoryTools/src/TateModels/attributes.jl

@@ -320,11 +320,12 @@ end
 
 Return the singular loci of the Weierstrass model equivalent to the given Tate model,
 along with the order of vanishing of ``(f, g, \Delta)`` at each locus and the corresponding
-refined Tate fiber type. See [singular_loci(w::WeierstrassModel)](@ref) for more details.
+refined Tate fiber type. See [`singular_loci(w::WeierstrassModel)`](@ref) for more details.
 
 !!! warning
     The classification of singularities is performed using a Monte Carlo algorithm, involving randomized sampling.
     While reliable in practice, this probabilistic method may occasionally yield non-deterministic results.
+    The random seed can be set with the optional argument `rng`.
 
 Below, we demonstrate this functionality by computing the singular loci of a Type ``III`` Tate model
 [KMSS11](@cite). In this case, the Tate sections are factored as follows:
@@ -360,13 +361,19 @@ Assuming that the first row of the given grading is the grading under Kbar
 
 Global Tate model over a not fully specified base
 
+<<<<<<< HEAD
 julia> sort([k[2:3] for k in singular_loci(t)])
+=======
+julia> using Random;
+
+julia> sort([k[2:3] for k in singular_loci(t; rng = Random.Xoshiro(1234))])
+>>>>>>> 267fca2ae9 ([FTheoryTools] Improve Monte-Carlo method)
 2-element Vector{Tuple{Tuple{Int64, Int64, Int64}, String}}:
  ((0, 0, 1), "I_1")
  ((1, 2, 3), "III")
 ```
 """
-@attr Vector{<:Tuple{<:MPolyIdeal{<:MPolyRingElem}, Tuple{Int64, Int64, Int64}, String}} function singular_loci(t::GlobalTateModel)
+@attr Vector{<:Tuple{<:MPolyIdeal{<:MPolyRingElem}, Tuple{Int64, Int64, Int64}, String}} function singular_loci(t::GlobalTateModel; rng::AbstractRNG = Random.default_rng())
   @req (base_space(t) isa NormalToricVariety || base_space(t) isa FamilyOfSpaces) "Singular loci of global Tate model currently only supported for toric varieties and families of spaces as base space"
-  return singular_loci(weierstrass_model(t))
+  return singular_loci(weierstrass_model(t); rng)
 end

experimental/FTheoryTools/src/TateModels/methods.jl

@@ -6,7 +6,7 @@ Determine the fiber of a (singular) global Tate model over a particular base loc
 !!! warning
     This method may run for very long time and is currently not tested as part of the regular OSCAR CI due to its excessive run times.
 """
-function analyze_fibers(model::GlobalTateModel, centers::Vector{<:Vector{<:Integer}})
+function analyze_fibers(model::GlobalTateModel, centers::Vector{<:Vector{<:Integer}}; rng::AbstractRNG = Random.default_rng())
 
   # This method only works if the model is defined over a toric variety over toric scheme
   @req base_space(model) isa NormalToricVariety "Analysis of fibers currently only supported for toric variety as base space"
@@ -23,7 +23,7 @@ function analyze_fibers(model::GlobalTateModel, centers::Vector{<:Vector{<:Integ
   lin = ideal_of_linear_relations(tas);
 
   # Singular loci
-  sing_loc = singular_loci(model)
+  sing_loc = singular_loci(model; rng)
 
   # Pick out the singular loci that are more singular than an I_1
   # Then keep only the locus and not the extra info about it

experimental/FTheoryTools/src/WeierstrassModels/attributes.jl

@@ -178,18 +178,21 @@ Advanced technical details are available in [BMT25](@cite BMT25).
 
 !!! warning
     The classification of singularities is based on a Monte Carlo algorithm, which involves random sampling.
-    While extensively tested and highly reliable, the method’s probabilistic nature may lead to non-deterministic results in rare cases.
+    While reliable in practice, this probabilistic method may occasionally yield non-deterministic results.
+    The random seed can be set with the optional argument `rng`.
 
 # Examples
 ```jldoctest
 julia> w =  weierstrass_model_over_projective_space(3)
 Weierstrass model over a concrete base
 
-julia> length(singular_loci(w))
+julia> using Random;
+
+julia> length(singular_loci(w; rng = Random.Xoshiro(1234)))
 1
 ```
 """
-@attr Vector{<:Tuple{<:MPolyIdeal{<:MPolyRingElem}, Tuple{Int64, Int64, Int64}, String}} function singular_loci(w::WeierstrassModel)
+@attr Vector{<:Tuple{<:MPolyIdeal{<:MPolyRingElem}, Tuple{Int64, Int64, Int64}, String}} function singular_loci(w::WeierstrassModel; rng::AbstractRNG = Random.default_rng())
   @req (base_space(w) isa NormalToricVariety || base_space(w) isa FamilyOfSpaces) "Singular loci of Weierstrass model is currently only supported for toric varieties and families of spaces as base space"
   B = irrelevant_ideal(base_space(w))
   d_primes = factor(discriminant(w))
@@ -202,7 +205,7 @@ julia> length(singular_loci(w))
     g_order = valuation(g, p)
     ords = (f_order, g_order, d_order)
     I = ideal([p])
-    kodaira_types[i] = (I, ords, _kodaira_type(I, ords, w))
+    kodaira_types[i] = (I, ords, _kodaira_type(I, ords, w; rng))
   end
   sort!(kodaira_types, by = x -> (x[2][2], x[2][3]))
   return kodaira_types

experimental/FTheoryTools/src/auxiliary.jl

@@ -114,7 +114,7 @@ _count_factors(poly::QQMPolyRingElem) = mapreduce(p -> p[end], +, absolute_prima
 
 _string_from_factor_count(poly::QQMPolyRingElem, string_list::Vector{String}) = string_list[_count_factors(poly)]
 
-function _kodaira_type(id::MPolyIdeal{<:MPolyRingElem}, ords::Tuple{Int64, Int64, Int64}, w::WeierstrassModel; rand_seed::Union{Int64, Nothing} = nothing)
+function _kodaira_type(id::MPolyIdeal{<:MPolyRingElem}, ords::Tuple{Int64, Int64, Int64}, w::WeierstrassModel; rng::AbstractRNG = Random.default_rng())
   f_ord = ords[1]
   g_ord = ords[2]
   d_ord = ords[3]
@@ -198,12 +198,10 @@ function _kodaira_type(id::MPolyIdeal{<:MPolyRingElem}, ords::Tuple{Int64, Int64
     # five times and see if we get agreement among all of the results
     num_gens = ngens(parent(f))
     gauge2s, f2s, g2s, d2s = [], [], [], []
-    if rand_seed != nothing
-      Random.seed!(rand_seed)
-    end
+    rng = MersenneTwister()
     for _ in 1:5
-      coord_inds = randperm(num_gens)[1:end-2]
-      rand_ints = rand(-100:100, num_gens - 2)
+      coord_inds = randperm(rng, num_gens)[1:end-2]
+      rand_ints = rand(rng, -100:100, num_gens - 2)
 
       push!(gauge2s, evaluate(forget_decoration(gens(id)[1]), coord_inds, rand_ints))
       push!(f2s, evaluate(forget_decoration(f), coord_inds, rand_ints))

experimental/FTheoryTools/test/hypersurface_models.jl

@@ -2,6 +2,9 @@
 # 1: Hypersurface models over concrete bases
 #############################################################
 
+using Random
+our_rng = Random.Xoshiro(1234)
+
 B3 = projective_space(NormalToricVariety, 3)
 ambient_space_of_fiber = weighted_projective_space(NormalToricVariety, [2,3,1])
 set_coordinate_names(ambient_space_of_fiber, ["x", "y", "z"])
@@ -25,7 +28,7 @@ end
   @test_throws ArgumentError weierstrass_model(h)
   @test_throws ArgumentError global_tate_model(h)
   @test_throws ArgumentError discriminant(h)
-  @test_throws ArgumentError singular_loci(h)
+  @test_throws ArgumentError singular_loci(h; rng = our_rng)
 end
 
 @testset "Saving and loading hypersurface models over concrete base space" begin

experimental/FTheoryTools/test/literature_models.jl

@@ -2,6 +2,9 @@
 # 1: Literature Tate model over concrete base
 #############################################################
 
+using Random
+our_rng = Random.Xoshiro(1234)
+
 B3 = projective_space(NormalToricVariety, 3)
 Kbar = anticanonical_divisor_class(B3)
 w = torusinvariant_prime_divisors(B3)[1]
@@ -16,7 +19,7 @@ D = classes_of_tunable_sections_in_basis_of_Kbar_and_defining_classes(t1)
   @test parent(tate_section_a6(t1)) == cox_ring(base_space(t1))
   @test parent(tate_polynomial(t1)) == cox_ring(ambient_space(t1))
   @test parent(discriminant(t1)) == cox_ring(base_space(t1))
-  @test length(singular_loci(t1)) == 2
+  @test length(singular_loci(t1; rng = our_rng)) == 2
   @test dim(base_space(t1)) == 3
   @test dim(ambient_space(t1)) == 5
   @test is_base_space_fully_specified(t1) == true
@@ -97,7 +100,7 @@ w1 = literature_model(arxiv_id = "1208.2695", equation = "B.19", base_space = B2
   @test parent(weierstrass_section_g(w1)) == cox_ring(base_space(w1))
   @test parent(weierstrass_polynomial(w1)) == cox_ring(ambient_space(w1))
   @test parent(discriminant(w1)) == cox_ring(base_space(w1))
-  @test length(singular_loci(w1)) == 1
+  @test length(singular_loci(w1; rng = our_rng)) == 1
   @test dim(base_space(w1)) == 2
   @test dim(ambient_space(w1)) == 4
   @test is_base_space_fully_specified(w1) == true
@@ -160,7 +163,7 @@ t3 = literature_model(arxiv_id = "1109.3454", equation = "3.1")
   @test parent(tate_section_a6(t3)) == cox_ring(base_space(t3))
   @test parent(tate_polynomial(t3)) == cox_ring(ambient_space(t3))
   @test parent(discriminant(t3)) == cox_ring(base_space(t3))
-  @test length(singular_loci(t3)) == 2
+  @test length(singular_loci(t3; rng = our_rng)) == 2
   @test dim(base_space(t3)) == 3
   @test dim(ambient_space(t3)) == 5
   @test is_base_space_fully_specified(t3) == false
@@ -261,7 +264,7 @@ w2 = literature_model(arxiv_id = "1208.2695", equation = "B.19", completeness_ch
   @test parent(weierstrass_section_g(w2)) == cox_ring(base_space(w2))
   @test parent(weierstrass_polynomial(w2)) == cox_ring(ambient_space(w2))
   @test parent(discriminant(w2)) == cox_ring(base_space(w2))
-  @test length(singular_loci(w2)) == 1
+  @test length(singular_loci(w2; rng = our_rng)) == 1
   @test dim(base_space(w2)) == 2
   @test dim(ambient_space(w2)) == 4
   @test is_base_space_fully_specified(w2) == false
@@ -326,7 +329,7 @@ b2 = anticanonical_divisor(B2)
 w5 = literature_model(arxiv_id = "1507.05954", equation = "A.1", completeness_check = false, base_space = B2, defining_classes = Dict("s8" => b2, "a1" => b, "a2" => b, "a3" => b))
 
 @testset "Test defining data for literature Weierstrass model over concrete base" begin
-  @test length(singular_loci(w4)) == 1
+  @test length(singular_loci(w4; rng = our_rng)) == 1
   @test dim(base_space(w4)) == 2
   @test dim(ambient_space(w4)) == 4
   @test is_base_space_fully_specified(w4) == true
@@ -344,7 +347,7 @@ b = torusinvariant_prime_divisors(B2)[1]
 w6 = literature_model(3, base_space = B2, defining_classes = Dict("b" => b), completeness_check = false)
 
 @testset "Test defining data for literature model defined by model index" begin
-  @test length(singular_loci(w6)) == 1
+  @test length(singular_loci(w6; rng = our_rng)) == 1
   @test dim(base_space(w6)) == 2
   @test dim(ambient_space(w6)) == 4
   @test is_base_space_fully_specified(w6) == true
@@ -783,20 +786,20 @@ foah16_B3_weier = literature_model(arxiv_id = "1408.4808", equation = "3.203", t
   @test parent(explicit_model_sections(foah14_B3_weier)["s7"]) == cox_ring(base_space(foah14_B3_weier))
   @test parent(explicit_model_sections(foah15_B3_weier)["s7"]) == cox_ring(base_space(foah15_B3_weier))
   @test parent(explicit_model_sections(foah16_B3_weier)["s7"]) == cox_ring(base_space(foah16_B3_weier))
-  @test [k[2:3] for k in singular_loci(foah1_B3_weier)] == [((0, 0, 1), "I_1")]
-  @test [k[2:3] for k in singular_loci(foah2_B3_weier)] == [((0, 0, 1), "I_1")]
-  @test [k[2:3] for k in singular_loci(foah3_B3_weier)] == [((0, 0, 1), "I_1")]
-  @test [k[2:3] for k in singular_loci(foah4_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2")]
-  @test [k[2:3] for k in singular_loci(foah5_B3_weier)] == [((0, 0, 1), "I_1")]
-  @test [k[2:3] for k in singular_loci(foah6_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2")]
-  @test [k[2:3] for k in singular_loci(foah7_B3_weier)] == [((0, 0, 1), "I_1")]
-  @test [k[2:3] for k in singular_loci(foah8_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2")]
-  @test [k[2:3] for k in singular_loci(foah9_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2")]
-  @test [k[2:3] for k in singular_loci(foah10_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 3), "Split I_3")]
-  @test [k[2:3] for k in singular_loci(foah11_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 3), "Split I_3")]
-  @test [k[2:3] for k in singular_loci(foah12_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2")]
-  @test [k[2:3] for k in singular_loci(foah13_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 4), "Split I_4")]
-  @test [k[2:3] for k in singular_loci(foah14_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 3), "Split I_3")]
-  @test [k[2:3] for k in singular_loci(foah15_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2")]
-  @test [k[2:3] for k in singular_loci(foah16_B3_weier)] == [((0, 0, 1), "I_1"), ((0, 0, 3), "Split I_3"), ((0, 0, 3), "Split I_3"), ((0, 0, 3), "Split I_3")]
+  @test [k[2:3] for k in singular_loci(foah1_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1")]
+  @test [k[2:3] for k in singular_loci(foah2_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1")]
+  @test [k[2:3] for k in singular_loci(foah3_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1")]
+  @test [k[2:3] for k in singular_loci(foah4_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2")]
+  @test [k[2:3] for k in singular_loci(foah5_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1")]
+  @test [k[2:3] for k in singular_loci(foah6_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2")]
+  @test [k[2:3] for k in singular_loci(foah7_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1")]
+  @test [k[2:3] for k in singular_loci(foah8_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2")]
+  @test [k[2:3] for k in singular_loci(foah9_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2")]
+  @test [k[2:3] for k in singular_loci(foah10_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 3), "Split I_3")]
+  @test [k[2:3] for k in singular_loci(foah11_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 3), "Split I_3")]
+  @test [k[2:3] for k in singular_loci(foah12_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2")]
+  @test [k[2:3] for k in singular_loci(foah13_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 4), "Split I_4")]
+  @test [k[2:3] for k in singular_loci(foah14_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 3), "Split I_3")]
+  @test [k[2:3] for k in singular_loci(foah15_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2"), ((0, 0, 2), "Non-split I_2")]
+  @test [k[2:3] for k in singular_loci(foah16_B3_weier; rng = our_rng)] == [((0, 0, 1), "I_1"), ((0, 0, 3), "Split I_3"), ((0, 0, 3), "Split I_3"), ((0, 0, 3), "Split I_3")]
 end