PolyhedralGeometry: more operators on fans and complexes

Author: YueRen

src/PolyhedralGeometry/PolyhedralComplex/standard_constructions.jl

@@ -1,7 +1,7 @@
 @doc raw"""
     common_refinement(PC1::PolyhedralComplex{T},PC2::PolyhedralComplex{T}) where T<:scalar_types
 
-Return the common refinement of two polyhedral complexes. 
+Return the common refinement of two polyhedral complexes.
 
 # Examples
 ```jldoctest
@@ -75,3 +75,68 @@ function k_skeleton(PC::PolyhedralComplex{T}, k::Int) where {T<:scalar_types}
   )
   return PolyhedralComplex{T}(ksk, coefficient_field(PC))
 end
+
+
+###############################################################################
+## Scalar product
+###############################################################################
+
+function *(c::QQFieldElem, Sigma::PolyhedralComplex)
+    # if scalar is zero, return polyhedral complex consisting only of the origin
+    if iszero(c)
+        return polyhedral_complex(convex_hull(zero_matrix(QQ,1,ambient_dim(Sigma))))
+    end
+
+    # if scalar is non-zero, multiple all vertices and rays by said scalar
+    SigmaVertsAndRays = vertices_and_rays(Sigma)
+    SigmaRayIndices = findall(vr -> vr isa RayVector, SigmaVertsAndRays)
+    SigmaLineality = lineality_space(Sigma)
+    SigmaIncidence = maximal_polyhedra(IncidenceMatrix,Sigma)
+    return polyhedral_complex(SigmaIncidence, multiply_by_nonzero_scalar.(SigmaVertsAndRays,c), SigmaRayIndices, SigmaLineality)
+end
+*(c::Union{ZZRingElem,Rational,Int}, Sigma::PolyhedralComplex) = QQ(c)*Sigma
+*(Sigma::PolyhedralComplex, c::Union{QQFieldElem,ZZRingElem,Rational,Int}) = c*Sigma
+
+
+###############################################################################
+## Negation
+###############################################################################
+
+function -(Sigma::PolyhedralComplex)
+    SigmaVertsAndRays = vertices_and_rays(Sigma)
+    SigmaRayIndices = findall(vr -> vr isa RayVector, SigmaVertsAndRays)
+    SigmaLineality = lineality_space(Sigma)
+    SigmaIncidence = maximal_polyhedra(IncidenceMatrix,Sigma)
+    return polyhedral_complex(SigmaIncidence, -SigmaVertsAndRays, SigmaRayIndices, SigmaLineality)
+end
+
+###############################################################################
+## Translation
+###############################################################################
+
+function translate_by_vector(u::PointVector{QQFieldElem}, v::Vector{QQFieldElem})
+    return u .+ v
+end
+function translate_by_vector(u::RayVector{QQFieldElem}, ::Vector{QQFieldElem})
+    return u
+end
+function +(v::Vector{QQFieldElem}, Sigma::PolyhedralComplex)
+    @req length(v)==ambient_dim(Sigma) "ambient dimension mismatch"
+    SigmaVertsAndRays = vertices_and_rays(Sigma)
+    SigmaRayIndices = findall(vr -> vr isa RayVector, SigmaVertsAndRays)
+    SigmaLineality = lineality_space(Sigma)
+    SigmaIncidence = maximal_polyhedra(IncidenceMatrix,Sigma)
+    return polyhedral_complex(SigmaIncidence, translate_by_vector.(SigmaVertsAndRays,Ref(v)), SigmaRayIndices, SigmaLineality)
+end
++(v::Vector{ZZRingElem}, Sigma::PolyhedralComplex) = QQ.(v)+Sigma
++(v::Vector{Rational}, Sigma::PolyhedralComplex) = QQ.(v)+Sigma
++(v::Vector{Int}, Sigma::PolyhedralComplex) = QQ.(v)+Sigma
+
++(Sigma::PolyhedralComplex, v::Vector{QQFieldElem}) = v+Sigma
++(Sigma::PolyhedralComplex, v::Vector{ZZRingElem}) = QQ.(v)+Sigma
++(Sigma::PolyhedralComplex, v::Vector{Rational}) = QQ.(v)+Sigma
++(Sigma::PolyhedralComplex, v::Vector{Int}) = QQ.(v)+Sigma
+
+# Vector addition for polyhedral fans
++(Sigma::PolyhedralFan, v::Vector) = polyhedral_complex(Sigma)+v
++(v::Vector, Sigma::PolyhedralFan) = v+polyhedral_complex(Sigma)

src/PolyhedralGeometry/PolyhedralFan/standard_constructions.jl

@@ -406,3 +406,40 @@ function arrangement_polynomial(
   F = parent(first(A))
   return arrangement_polynomial(ring, matrix(F, A); hyperplanes)
 end
+
+###############################################################################
+## Scalar multiplication
+###############################################################################
+
+function multiply_by_nonzero_scalar(u::PointVector{QQFieldElem}, c::QQFieldElem)
+  return u .* c
+end
+function multiply_by_nonzero_scalar(u::RayVector{QQFieldElem}, c::QQFieldElem)
+  return (c<0 ? -u : u)
+end
+
+function *(c::QQFieldElem, Sigma::PolyhedralFan)
+  # if scalar is zero, return polyhedral complex consisting only of the origin
+  if iszero(c)
+    return polyhedral_complex(convex_hull(zero_matrix(QQ,1,ambient_dim(Sigma))))
+  end
+
+  # if scalar is non-zero, multiple all vertices and rays by said scalar
+  SigmaRays = first(rays_modulo_lineality(Sigma))
+  SigmaLineality = lineality_space(Sigma)
+  SigmaIncidence = maximal_cones(IncidenceMatrix,Sigma)
+  return polyhedral_fan(SigmaIncidence, multiply_by_nonzero_scalar.(SigmaRays,c), SigmaLineality)
+end
+*(c::Union{ZZRingElem,Rational,Int}, Sigma::PolyhedralFan) = QQ(c)*Sigma
+*(Sigma::PolyhedralFan,c::Union{QQFieldElem,ZZRingElem,Rational,Int}) = c*Sigma
+
+###############################################################################
+## Negation
+###############################################################################
+
+function -(Sigma::PolyhedralFan)
+  SigmaRays = first(rays_modulo_lineality(Sigma))
+  SigmaLineality = lineality_space(Sigma)
+  SigmaIncidence = maximal_cones(IncidenceMatrix,Sigma)
+  return polyhedral_fan(SigmaIncidence, -SigmaRays, SigmaLineality)
+end

test/PolyhedralGeometry/polyhedral_complex.jl

@@ -142,4 +142,28 @@
       @test n_maximal_polyhedra(vrep) == n_maximal_polyhedra(hrep)
     end
   end
+
+  @testset "Operators" begin
+    PC = normal_fan(cube(2))
+    PCshifted = PC + [1, 1]
+    @test dim(PCshifted) == dim(PC)
+    @test ambient_dim(PCshifted) == ambient_dim(PC)
+    @test lineality_dim(PCshifted) == lineality_dim(PC)
+    @test issetequal(rays(PCshifted), rays(PC))
+    @test n_maximal_polyhedra(PCshifted) == n_maximal_cones(PC)
+
+    PCscaled = PCshifted*2
+    @test dim(PCscaled) == dim(PCshifted)
+    @test ambient_dim(PCscaled) == ambient_dim(PCshifted)
+    @test lineality_dim(PCscaled) == lineality_dim(PCshifted)
+    @test issetequal(rays(PCscaled), rays(PCshifted))
+    @test n_maximal_polyhedra(PCscaled) == n_maximal_polyhedra(PCshifted)
+
+    PCnegated = -PCshifted
+    @test dim(PCnegated) == dim(PCshifted)
+    @test ambient_dim(PCnegated) == ambient_dim(PCshifted)
+    @test lineality_dim(PCnegated) == lineality_dim(PCshifted)
+    @test issetequal(rays(PCnegated), rays(PCshifted))
+    @test n_maximal_polyhedra(PCnegated) == n_maximal_polyhedra(PCshifted)
+  end
 end

test/PolyhedralGeometry/polyhedral_fan.jl

@@ -218,6 +218,15 @@ end
   sff1 = star_subdivision(ff, w1)
   @test number_of_maximal_cones(sff0) == 2
   @test number_of_maximal_cones(sff1) == 3
+
+  f = normal_fan(cube(2))
+  fMinus = -f
+  @test n_rays(fMinus) == n_rays(f)
+  @test n_maximal_cones(fMinus) == n_maximal_cones(f)
+
+  fMinus = -1*f
+  @test n_rays(fMinus) == n_rays(f)
+  @test n_maximal_cones(fMinus) == n_maximal_cones(f)
 end
 
 @testset "Defining polynomial of hyperplane arrangement from matrix" begin