Implement syzygy_generators for lists of module elements (oscar-system#4204)

Co-authored-by: Tommy Hofmann <[email protected]>

Author: HechtiDerLachs

src/Modules/UngradedModules/FreeMod.jl

@@ -308,3 +308,67 @@ end
 
 rels(F::FreeMod) = elem_type(F)[]
 
+function syzygy_generators(
+    g::Vector{T};
+    parent::Union{<:FreeMod, Nothing}=nothing
+  ) where {T<:FreeModElem}
+  isempty(g) && return Vector{T}()
+  F = Oscar.parent(first(g))
+  @assert all(Oscar.parent(x) === F for x in g) "parent mismatch"
+  R = base_ring(F)
+  m = length(g)
+  G = (parent === nothing ?  FreeMod(R, m) : parent)::typeof(F)
+  @req ngens(G) == m "given parent does not have the correct number of generators"
+  phi = hom(G, F, g)
+  K, _ = kernel(phi)
+  return ambient_representatives_generators(K)
+end
+
+@doc raw"""
+    syzygy_generators(
+        a::Vector{T};
+        parent::Union{FreeMod{T}, Nothing} = nothing
+      ) where {T<:RingElem}
+
+Return generators for the syzygies on the polynomials given as elements of `a`.
+The optional keyword argument can be used to specify the parent of the output.
+
+# Examples
+```jldoctest
+julia> R, (x, y) = polynomial_ring(QQ, [:x, :y])
+(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
+
+julia> S = syzygy_generators([x^3+y+2,x*y^2-13*x^2,y-14])
+3-element Vector{FreeModElem{QQMPolyRingElem}}:
+ (-y + 14)*e[2] + (-13*x^2 + x*y^2)*e[3]
+ (-169*y + 2366)*e[1] + (-13*x*y + 182*x - 196*y + 2744)*e[2] + (13*x^2*y^2 - 2548*x^2 + 196*x*y^2 + 169*y + 338)*e[3]
+ (-13*x^2 + 196*x)*e[1] + (-x^3 - 16)*e[2] + (x^4*y + 14*x^4 + 13*x^2 + 16*x*y + 28*x)*e[3]
+```
+"""
+function syzygy_generators(
+    a::Vector{T};
+    parent::Union{FreeMod{T}, Nothing} = nothing
+  ) where {T<:RingElem}
+  isempty(a) && return Vector{FreeModElem{T}}()
+  R = Oscar.parent(first(a))
+  @assert all(Oscar.parent(x) === R for x in a) "parent mismatch"
+  F = FreeMod(R, 1)
+  return syzygy_generators(elem_type(F)[x*F[1] for x in a]; parent)
+end
+
+function syzygy_generators(
+    a::Vector{T};
+    parent::Union{FreeMod{T}, Nothing} = nothing
+  ) where {CT <: Union{<:FieldElem, ZZRingElem, QQPolyRingElem}, # Can be adjusted to whatever is digested by Singular
+           T<:MPolyRingElem{CT}}
+  isempty(a) && return Vector{FreeModElem{T}}()
+  R = Oscar.parent(first(a))
+  @assert all(Oscar.parent(x) === R for x in a) "parent mismatch"
+  I = ideal(R, a)
+  s = Singular.syz(singular_generators(I))
+  F = (parent === nothing ? FreeMod(R, length(a)) : parent)::FreeMod{T}
+  @req ngens(F) == length(a) "parent does not have the correct number of generators"
+  @assert rank(s) == length(a)
+  return elem_type(F)[F(s[i]) for i=1:Singular.ngens(s)]
+end
+

src/Modules/UngradedModules/SubquoModule.jl

@@ -754,7 +754,11 @@ function set_default_ordering!(M::SubquoModule, ord::ModuleOrdering)
   set_default_ordering!(M.sum, ord)
 end
 
-function standard_basis(M::SubquoModule; ordering::ModuleOrdering = default_ordering(M))
+function standard_basis(M::SubquoModule; ordering::Union{ModuleOrdering, Nothing} = nothing)
+  error("standard basis computation is not supported for modules over rings of type $(typeof(base_ring(M)))")
+end
+
+function standard_basis(M::SubquoModule{<:MPolyRingElem{T}}; ordering::ModuleOrdering = default_ordering(M)) where {T<:Union{<:FieldElem, ZZRingElem}}
   @req is_exact_type(elem_type(base_ring(M))) "This functionality is only supported over exact fields."
   if !haskey(M.groebner_basis, ordering)
     if isdefined(M, :quo)

src/Rings/groebner.jl

@@ -506,30 +506,7 @@ function groebner_basis_with_transformation_matrix(I::MPolyIdeal; ordering::Mono
 end
 
 # syzygies #######################################################
-@doc raw"""
-    syzygy_generators(G::Vector{<:MPolyRingElem})
-
-Return generators for the syzygies on the polynomials given as elements of `G`.
-
-# Examples
-```jldoctest
-julia> R, (x, y) = polynomial_ring(QQ, [:x, :y])
-(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
-
-julia> S = syzygy_generators([x^3+y+2,x*y^2-13*x^2,y-14])
-3-element Vector{FreeModElem{QQMPolyRingElem}}:
- (-y + 14)*e[2] + (-13*x^2 + x*y^2)*e[3]
- (-169*y + 2366)*e[1] + (-13*x*y + 182*x - 196*y + 2744)*e[2] + (13*x^2*y^2 - 2548*x^2 + 196*x*y^2 + 169*y + 338)*e[3]
- (-13*x^2 + 196*x)*e[1] + (-x^3 - 16)*e[2] + (x^4*y + 14*x^4 + 13*x^2 + 16*x*y + 28*x)*e[3]
-```
-"""
-function syzygy_generators(a::Vector{<:MPolyRingElem})
-  I = ideal(a)
-  s = Singular.syz(singular_generators(I))
-  F = free_module(parent(a[1]), length(a))
-  @assert rank(s) == length(a)
-  return [F(s[i]) for i=1:Singular.ngens(s)]
-end
+# See src/Modules/UngradedModules/FreeMod.jl for the implementation. 
 
 # leading ideal #######################################################
 @doc raw"""

test/Modules/UngradedModules.jl

@@ -1275,3 +1275,41 @@ end
   F = graded_free_module(Pn, 1)
   dualFIC = hom(FI.C, F)
 end
+
+@testset "issue 4203" begin
+  R, (x,y) = polynomial_ring(GF(2), ["x","y"]);
+  
+  g = [x+1, y+1]
+  s = syzygy_generators(g)
+  F = parent(first(s))
+  s2 = syzygy_generators(g; parent=F)
+  @test s == s2
+  s3 = syzygy_generators(g)
+  @test s2 != s3
+  
+  A, _ = quo(R, ideal(R, [x^2+1, y^2+1]));
+  
+  g = A.([x+1, y+1])
+  s = syzygy_generators(g)
+  F = parent(first(s))
+  s2 = syzygy_generators(g; parent=F)
+  @test s == s2
+  s3 = syzygy_generators(g)
+  @test s2 != s3
+  
+  F = free_module(A,2);
+  M,_ = sub(F, [F([A(x+1),A(y+1)])]);
+  FF = free_module(A, ngens(M)); 
+  phi = hom(FF, M, gens(M)); 
+  K, = kernel(phi); 
+  @test ambient_representatives_generators(K) == [(x*y + x + y + 1)*FF[1]]
+  @test_throws ErrorException standard_basis(K)
+  s1 = syzygy_generators(ambient_representatives_generators(M))
+  G = parent(first(s1))
+  s2 = syzygy_generators(ambient_representatives_generators(M); parent=G)
+  @test s1 == s2
+  s3 = syzygy_generators(ambient_representatives_generators(M))
+  @test s3 != s2
+  @test_throws ArgumentError syzygy_generators(ambient_representatives_generators(M); parent=F)
+end
+