Add more monomial and module orderings (#723)

docs/src/CommutativeAlgebra/ideals.md

@@ -175,7 +175,62 @@ m_{i-1}\alpha\ =m_{i-1}\beta,\ m_i\alpha>m_i\beta$
 
 To create matrix orderings, OSCAR allows for matrices with integer coefficients as input matrices.
 
-### Normal Forms
+#### Functions for creating orderings
+
+When computing Gröbner bases an ordering must be supplied. Standard Singular
+orderings, including block orderings, weighted orderings and local orderings
+are available.
+
+The basic orderings are `:lex`, `:revlex`, `:deglex`, `:degrevlex`,
+`:neglex`, `:negrevlex`, `:negdeglex`, `:negdegrevlex`, `:wdeglex`,
+`:wdegrevlex`, `:negwdeglex` and `:negwdegrevlex`.
+
+The orderings starting with `w` are weighted orderings.
+
+The following functions exist for creating orderings:
+
+```@docs
+lex(::AbstractVector{<:MPolyElem})
+revlex(::AbstractVector{<:MPolyElem})
+deglex(::AbstractVector{<:MPolyElem})
+degrevlex(::AbstractVector{<:MPolyElem})
+neglex(::AbstractVector{<:MPolyElem})
+negrevlex(::AbstractVector{<:MPolyElem})
+negdeglex(::AbstractVector{<:MPolyElem})
+negdegrevlex(::AbstractVector{<:MPolyElem})
+```
+
+```@docs
+wdeglex(::AbstractVector{<:MPolyElem}, ::Vector{Int})
+wdegrevlex(::AbstractVector{<:MPolyElem}, ::Vector{Int})
+negwdeglex(::AbstractVector{<:MPolyElem}, ::Vector{Int})
+negwdegrevlex(::AbstractVector{<:MPolyElem}, ::Vector{Int})
+```
+
+Block orderings can be obtained by concatening monomial orderings using the `*`
+operator.
+
+Term over position and position over term module orderings are also available.
+These are also specified byy concatenation using the `*` operator. One creates
+the requisite module ordering (`lex` or `revlex`) for the generators of the
+free module.
+
+Term over position is specified by appending the module ordering to the
+monomial ordering and position over term by prepending the module ordering.
+
+###### Examples
+
+```@repl oscar
+R, (x, y, s, t, u) = PolynomialRing(QQ, ["x", "y", "s", "t", "u"])
+O1 = degrevlex(gens(R))
+O2 = lex([x, y])*deglex([s, t, u])
+O3 = wdeglex(gens(R), [2, 3, 5, 7, 3])
+
+K = FreeModule(R, 3)
+O4 = revlex(gens(K))*degrevlex(gens(R))
+```
+
+## Normal Forms
 
 ```@docs
 normal_form(f::T, J::MPolyIdeal) where { T <: MPolyElem }

experimental/ModStd/ModStdQ.jl

@@ -112,11 +112,11 @@ function Oscar.groebner_assure(I::MPolyIdeal{fmpq_mpoly}, ord::Symbol = :degrevl
   end
 end
 
-function Oscar.groebner_basis_with_transform(I::MPolyIdeal{fmpq_mpoly}; ordering::Symbol = :degrevlex, complete_reduction::Bool = true, use_hilbert::Bool = false)
-
+function groebner_basis_with_transform_inner(I::MPolyIdeal{fmpq_mpoly}, ord::MonomialOrdering; complete_reduction::Bool = true, use_hilbert::Bool = false)
   if iszero(I)
     I.gb = BiPolyArray(base_ring(I), fmpq_mpoly[], isGB = true, keep_ordering = false)
-    singular_assure(I.gb)
+    I.gb.ord = ord.o
+    singular_assure(I.gb, ord)
     return fmpq_mpoly[], matrix(base_ring(I), ngens(I), 0, fmpq_mpoly[])
   end
 
@@ -147,7 +147,7 @@ function Oscar.groebner_basis_with_transform(I::MPolyIdeal{fmpq_mpoly}; ordering
     R = GF(p)
     Rt, t = PolynomialRing(R, [string(s) for s = symbols(Qt)], cached = false)
     @vtime :ModStdQ 3 Ip = Oscar.BiPolyArray([Rt(x) for x = gI], keep_ordering = false)
-    Gp, Tp = Oscar.groebner_basis_with_transform(Ip, ord = ordering, complete_reduction = complete_reduction)
+    Gp, Tp = Oscar.groebner_basis_with_transform(Ip, ord; complete_reduction = complete_reduction)
     length_gc = length(Gp)
     Jp = vcat(map(x->lift(Zt, x), Gp), map(x->lift(Zt, x), reshape(collect(Tp), :)))
 
@@ -193,10 +193,13 @@ function Oscar.groebner_basis_with_transform(I::MPolyIdeal{fmpq_mpoly}; ordering
           T = matrix(Qt, length_gc, length(gI), gd[length_gc+1:end])
           #at this point we SHOULD have T*gens(I) == G...
           if T*matrix(Qt, length(gI), 1, gI) == matrix(Qt, length_gc, 1, G)
-            if ordering == :degrevlex && !isdefined(I, :gb)
+            if !isdefined(I.gens, :ord)
+               I.gens.ord = ord.o
+            end
+            if ord.o == I.gens.ord && !isdefined(I, :gb)
               I.gb = BiPolyArray(gd[1:length_gc], keep_ordering = false)
+              I.gb.isGB = true
               singular_assure(I.gb)
-              I.gb.S.isGB = true
             end
             return G, T
           else
@@ -209,6 +212,14 @@ function Oscar.groebner_basis_with_transform(I::MPolyIdeal{fmpq_mpoly}; ordering
   end
 end
 
+function Oscar.groebner_basis_with_transform(I::MPolyIdeal{fmpq_mpoly}; ordering::Symbol = :degrevlex, complete_reduction::Bool = true, use_hilbert::Bool = false)
+   ord = Oscar.Orderings.MonomialOrdering(base_ring(I), Oscar.Orderings.ordering(gens(base_ring(I)), ordering))
+   return groebner_basis_with_transform_inner(I, ord; complete_reduction=complete_reduction, use_hilbert=use_hilbert)
+end
+
+function Oscar.groebner_basis_with_transform(I::MPolyIdeal{fmpq_mpoly}, ord::MonomialOrdering; complete_reduction::Bool = true, use_hilbert::Bool = false)
+   return groebner_basis_with_transform_inner(I, ord; complete_reduction=complete_reduction, use_hilbert=use_hilbert)
+end
 
 function Oscar.lift(R::Nemo.Ring, f::Union{gfp_mpoly, nmod_mpoly})
   g = MPolyBuildCtx(R)

src/Modules/FreeModules-graded.jl

@@ -1,5 +1,5 @@
 #TODO make d and S a function optionally - to support HUGE degree
-export presentation, grading_group
+export presentation, grading_group, ModuleOrdering
 
 abstract type ModuleFP_dec{T} end
 
@@ -117,7 +117,7 @@ end
 dim(F::FreeModule_dec)  = length(F.d)
 ngens(F::FreeModule_dec) = dim(F)
 
-struct FreeModuleElem_dec{T}
+struct FreeModuleElem_dec{T} <: AbstractAlgebra.ModuleElem{T}
   r::SRow{T}
   parent::FreeModule_dec{T}
 end

src/Oscar.jl

@@ -228,6 +228,7 @@ include("Groups/GrpAb.jl")
 
 include("Rings/integer.jl")
 include("Rings/rational.jl")
+include("Rings/orderings.jl")
 include("Rings/mpoly.jl")
 include("Rings/mpoly-graded.jl")
 include("Rings/mpoly-ideals.jl")

src/Rings/MPolyQuo.jl

@@ -575,6 +575,7 @@ function simplify(f::MPolyQuoElem)
   g = f.f
   return R(I.gens.Ox(reduce(Sx(g), I.gb.S)))
 end
+
 function simplify!(f::MPolyQuoElem)
   R = parent(f)
   I = R.I

src/Rings/groebner.jl

@@ -2,7 +2,8 @@ export  groebner_basis, groebner_basis_with_transformation_matrix, leading_ideal
 
 # groebner stuff #######################################################
 @doc Markdown.doc"""
-    groebner_assure(I::MPolyIdeal)
+    groebner_assure(I::MPolyIdeal; complete_reduction::Bool = false)
+    groebner_assure(I::MPolyIdeal, ord::MonomialOrdering; complete_reduction::Bool = false)
 
 **Note**: Internal function, subject to change, do not use.
 
@@ -27,19 +28,15 @@ julia> Oscar.groebner_assure(I)
  x^3 - 9//2*x
 
 julia> I.gb
-Oscar.BiPolyArray{fmpq_mpoly}(fmpq_mpoly[x*y - 3*x, y^3 - 6*x^2, x^3 - 9//2*x], Singular ideal over Singular Polynomial Ring (QQ),(x,y),(dp(2),C) with generators (x*y - 3*x, y^3 - 6*x^2, x^3 - 9//2*x), Multivariate Polynomial Ring in x, y over Rational Field, Singular Polynomial Ring (QQ),(x,y),(dp(2),C), false, #undef, false)
+Oscar.BiPolyArray{fmpq_mpoly}(fmpq_mpoly[x*y - 3*x, y^3 - 6*x^2, x^3 - 9//2*x], Singular ideal over Singular Polynomial Ring (QQ),(x,y),(dp(2),C) with generators (x*y - 3*x, y^3 - 6*x^2, x^3 - 9//2*x), Multivariate Polynomial Ring in x, y over Rational Field, Singular Polynomial Ring (QQ),(x,y),(dp(2),C), true, #undef, false)
 ```
 """
 function groebner_assure(I::MPolyIdeal; complete_reduction::Bool = false)
   if !isdefined(I, :gb)
     singular_assure(I)
-#    @show "std on", I.gens.S
-    if complete_reduction
-      i = Singular.std(I.gens.S, complete_reduction = complete_reduction)
-      I.gb = BiPolyArray(base_ring(I), i)
-    else
-      I.gb = BiPolyArray(base_ring(I), Singular.std(I.gens.S))
-    end
+    i = Singular.std(I.gens.S; complete_reduction = complete_reduction)
+    I.gb = BiPolyArray(base_ring(I), i)
+    I.gb.isGB = true
     I.gb.O = [I.gb.Ox(x) for x = gens(I.gb.S)]
   end
 end
@@ -82,7 +79,7 @@ function groebner_basis(B::BiPolyArray; ordering::Symbol = :degrevlex, complete_
 end
 
 #= @doc Markdown.doc"""
- =     groebner_basis(I::MPolyIdeal)
+ =     groebner_basis(I::MPolyIdeal; ordering::Symbol = :degrevlex, complete_reduction::Bool = false)
  =
  = Compute a Groebner basis w.r.t. the given monomial ordering of the polynomial ring.
  =
@@ -108,7 +105,8 @@ end
 
 @doc Markdown.doc"""
     groebner_basis(I::MPolyIdeal; ordering::Symbol = :degrevlex, complete_reduction::Bool = false)
-
+    groebner_basis(I::MPolyIdeal, ord::MonomialOrdering; complete_reduction::Bool=false)
+ 
 Given an ideal `I` and optional parameters monomial ordering `ordering` and `complete_reduction`,
 compute a Groebner basis (if `complete_reduction = true` the reduced Groebner basis) of `I`
     w.r.t. the given monomial ordering `ordering` (as default `:degrevlex`).
@@ -121,7 +119,7 @@ julia> R, (x, y) = PolynomialRing(QQ, ["x", "y"], ordering=:degrevlex)
 julia> I = ideal([x*y-3*x,y^3-2*x^2*y])
 ideal(x*y - 3*x, -2*x^2*y + y^3)
 
-julia> H = groebner_basis(I, ordering=:lex)
+julia> H = groebner_basis(I; ordering=:lex)
 3-element Vector{fmpq_mpoly}:
  y^4 - 3*y^3
  x*y - 3*x
@@ -136,7 +134,8 @@ function groebner_basis(I::MPolyIdeal; ordering::Symbol = :degrevlex, complete_r
 end
 
 @doc Markdown.doc"""
-    groebner_basis_with_transform(B::BiPolyArray; ord::Symbol = :degrevlex, complete_reduction::Bool = false)
+    groebner_basis_with_transform(B::BiPolyArray; ordering::Symbol = :degrevlex, complete_reduction::Bool = false)
+    groebner_basis_with_transform(B::BiPolyArray, ord::MonomialOrdering; complete_reduction::Bool = false)
 
 **Note**: Internal function, subject to change, do not use.
 
@@ -173,9 +172,9 @@ function groebner_basis_with_transform(B::BiPolyArray; ord::Symbol = :degrevlex,
   return BiPolyArray(B.Ox, i), map_entries(x->B.Ox(x), m)
 end
 
-
 @doc Markdown.doc"""
     groebner_basis_with_transformation_matrix(I::MPolyIdeal; ordering::Symbol = :degrevlex, complete_reduction::Bool=false)
+    groebner_basis_with_transformation_matrix(I::MPolyIdeal, ord::MonomialOrdering; complete_reduction::Bool=false)
 
 Return a pair `G, m` where `G` is a Groebner basis of the ideal `I` with respect to the
 monomial ordering `ordering`, and `m` is a transformation matrix from `gens(I)` to `G`. If
@@ -197,7 +196,7 @@ true
 ```
 """
 function groebner_basis_with_transformation_matrix(I::MPolyIdeal; ordering::Symbol = :degrevlex, complete_reduction::Bool=false)
-  G, m = Oscar.groebner_basis_with_transform(I, ordering=ordering, complete_reduction=complete_reduction)
+  G, m = Oscar.groebner_basis_with_transform(I; ordering=ordering, complete_reduction=complete_reduction)
   return G, Array(m)
 end
 
@@ -277,6 +276,7 @@ end
 
 @doc Markdown.doc"""
     leading_ideal(I::MPolyIdeal)
+    leading_ideal(I::MPolyIdeal, ord::MonomialOrdering)
 
 Given a multivariate polynomial ideal `Ì` this function returns the
 leading ideal for `I`. This is done w.r.t. the given monomial ordering
@@ -300,7 +300,14 @@ function leading_ideal(I::MPolyIdeal)
   singular_assure(I.gb)
   return MPolyIdeal(base_ring(I), Singular.Ideal(I.gb.Sx, [Singular.leading_monomial(g) for g in gens(I.gb.S)]))
 end
-
+
+function leading_ideal(I::MPolyIdeal, ord::MonomialOrdering)
+  singular_assure(I, ord)
+  groebner_assure(I, ord)
+  singular_assure(I.gb, ord)
+  return MPolyIdeal(base_ring(I), Singular.Ideal(I.gb.Sx, [Singular.leading_monomial(g) for g in gens(I.gb.S)]))
+end
+
 @doc Markdown.doc"""
     leading_ideal(I::MPolyIdeal, ordering::Symbol)
 
@@ -320,7 +327,7 @@ ideal(y^7, x*y^2, x^3)
 ```
 """
 function leading_ideal(I::MPolyIdeal, ordering::Symbol)
-  return leading_ideal(groebner_basis(I, ordering=ordering), ordering)
+  return leading_ideal(groebner_basis(I; ordering=ordering), ordering)
 end
 
 @doc Markdown.doc"""
@@ -454,17 +461,37 @@ function groebner_assure(I::MPolyIdeal, ord::MonomialOrdering; complete_reductio
 end
 
 function groebner_basis(B::BiPolyArray, ord::MonomialOrdering; complete_reduction::Bool = false)
-    R = singular_ring(B.Ox, ord.o)
-    i = Singular.Ideal(R, [R(x) for x = B])
-    i = Singular.std(i, complete_reduction = complete_reduction)
-    return BiPolyArray(B.Ox, i)
+   singular_assure(B, ord)
+   R = B.Sx
+   !Oscar.Singular.has_global_ordering(R) && error("The ordering has to be a global ordering.")
+   I = Singular.Ideal(R, gens(B.S)...)
+   i = Singular.std(I, complete_reduction = complete_reduction)
+   return BiPolyArray(B.Ox, i)
 end
 
 function groebner_basis(I::MPolyIdeal, ord::MonomialOrdering; complete_reduction::Bool=false)
-  R = singular_ring(base_ring(I), ord.o)
-  !Oscar.Singular.has_global_ordering(R) && error("The ordering has to be a global ordering.")
-  i = Singular.std(Singular.Ideal(R, [R(x) for x = gens(I)]), complete_reduction = complete_reduction)
-  return collect(BiPolyArray(base_ring(I), i))
+  return collect(groebner_basis(I.gens, ord))
 end
 
-
+function groebner_basis_with_transform(B::BiPolyArray, ord::MonomialOrdering; complete_reduction::Bool = false)
+   if !isdefined(B, :ord)
+      singular_assure(B, ord)
+   elseif ord != B.ord
+     R = singular_ring(B.Ox, ord)
+     i = Singular.Ideal(R, [R(x) for x = B])
+     i, m = Singular.lift_std(i, complete_reduction = complete_reduction)
+     return BiPolyArray(B.Ox, i), map_entries(x->B.Ox(x), m)
+   end
+
+   if !isdefined(B, :S)
+     B.S = Singular.Ideal(B.Sx, [B.Sx(x) for x = B.O])
+   end
+
+   i, m = Singular.lift_std(B.S, complete_reduction = complete_reduction)
+   return BiPolyArray(B.Ox, i), map_entries(x->B.Ox(x), m)
+ end
+
+ function groebner_basis_with_transformation_matrix(I::MPolyIdeal, ord::MonomialOrdering; complete_reduction::Bool=false)
+   G, m = Oscar.groebner_basis_with_transform(I, ord; complete_reduction=complete_reduction)
+   return G, Array(m)
+ end