Add linear independence check to basis generation (#147)

src/DeformationBases/ArcDiagDeformBasis.jl

@@ -101,7 +101,11 @@ struct ArcDiagDeformBasis{C <: RingElem} <: DeformBasis{C}
         iter = Iterators.flatten(iters)
         if !no_normalize
             iter = unique(Iterators.filter(b -> !iszero(b), iter))
-            len = length(iter)
+            collected = Vector{DeformationMap{C}}(collect(iter))
+            _, rels = is_linearly_independent_with_relations(coefficient_ring(sp), collected)
+            inds = [findlast(!iszero, vec(rels[i, :]))::Int for i in 1:nrows(rels)]
+            deleteat!(collected, inds)
+            return new{C}(length(collected), collected, extra_data, normalize)
         end
         return new{C}(len, iter, extra_data, normalize)
     end

src/DeformationBases/PseudographDeformBasis.jl

@@ -62,7 +62,11 @@ struct PseudographDeformBasis{C <: RingElem} <: DeformBasis{C}
         iter = Iterators.flatten(iters)
         if !no_normalize
             iter = unique(Iterators.filter(b -> !iszero(b), iter))
-            len = length(iter)
+            collected = Vector{DeformationMap{C}}(collect(iter))
+            _, rels = is_linearly_independent_with_relations(coefficient_ring(sp), reverse(collected))
+            inds = [1 + ncols(rels) - (findfirst(!iszero, vec(rels[i, :]))::Int) for i in nrows(rels):-1:1]
+            deleteat!(collected, inds)
+            return new{C}(length(collected), collected, extra_data, normalize)
         end
         return new{C}(len, iter, extra_data, normalize)
     end

src/LinearIndependence.jl

@@ -0,0 +1,100 @@
+const _linear_independence_rref_cutoff = 0
+
+function column_rref!(mat::MatElem{T}) where {T <: FieldElem}
+    rk = rref!(AbstractAlgebra.Solve.lazy_transpose(mat))
+    return view(mat, :, 1:rk)
+end
+
+function column_rref!(mat::Nemo._MatTypes) # change to Nemo._FieldMatTypes
+    trmat = transpose!(mat)
+    rk = rref!(trmat)
+    return view(transpose!(trmat), :, 1:rk)
+end
+
+function is_linearly_independent(V::Vector{T}) where {T}
+    return is_linearly_independent_with_relations(V)[1]
+end
+
+function is_linearly_independent(F::Field, V::Vector{T}) where {T}
+    return is_linearly_independent_with_relations(F, V)[1]
+end
+
+function is_linearly_independent_with_relations(V::Vector{T}) where {T}
+    M = kernel(_linear_independence_coeff_matrix(V); side=:left)
+    return nrows(M) == 0, M
+end
+
+function is_linearly_independent_with_relations(F::Field, V::Vector{T}) where {T}
+    M = kernel(_linear_independence_coeff_matrix(F, V); side=:left)
+    return nrows(M) == 0, M
+end
+
+function _linear_independence_coeff_matrix(V::Vector{<:FieldElem})
+    @req length(V) > 0 "For empty vectors, the field needs to be specified"
+    return _linear_independence_coeff_matrix(parent(V[1]), V)
+end
+
+function _linear_independence_coeff_matrix(F::Field, V::Vector{<:FieldElem})
+    return matrix(F, length(V), 1, V)
+end
+
+function _linear_independence_coeff_matrix(F::Field, V::Vector{<:PolyRingElem})
+    @req isempty(V) || all(v -> parent(v) === parent(V[1]), V) "Incompatible polynomial rings"
+    return reduce(
+        hcat,
+        begin
+            mat = _linear_independence_coeff_matrix(F, [coeff(v, i) for v in V])
+            if ncols(mat) > _linear_independence_rref_cutoff * nrows(mat)
+                column_rref!(mat)
+            else
+                mat
+            end
+        end for i in 0:maximum(degree, V; init=-1);
+        init=zero_matrix(F, length(V), 0),
+    )
+end
+
+function _linear_independence_coeff_matrix(F::Field, V::Vector{<:MatElem})
+    n = length(V)
+    if n == 0
+        return zero_matrix(F, n, 0)
+    end
+    @req all(v -> parent(v) === parent(V[1]), V) "Incompatible matrix spaces"
+    return reduce(
+        hcat,
+        begin
+            mat = _linear_independence_coeff_matrix(F, [v[i] for v in V])
+            if ncols(mat) > _linear_independence_rref_cutoff * nrows(mat)
+                column_rref!(mat)
+            else
+                mat
+            end
+        end for i in eachindex(V[1]);
+        init=zero_matrix(F, n, 0),
+    )
+end
+
+function _linear_independence_coeff_matrix(F::Field, V::Vector{<:FreeAssAlgElem})
+    n = length(V)
+    if n == 0
+        return zero_matrix(F, n, 0)
+    end
+    R = parent(V[1])
+    C = base_ring(R)
+    @req all(v -> parent(v) === R, V) "Incompatible algebras"
+    coeff_maps = [Dict{Vector{Int}, elem_type(F)}(zip(exponent_words(v), coefficients(v))) for v in V]
+    support_words = reduce(union, keys.(coeff_maps))
+    return reduce(
+        hcat,
+        begin
+            mat = _linear_independence_coeff_matrix(F, [get(coeff_map, word, zero(C)) for coeff_map in coeff_maps])
+            if ncols(mat) > _linear_independence_rref_cutoff * nrows(mat)
+                column_rref!(mat)
+            else
+                mat
+            end
+        end for word in support_words;
+        init=zero_matrix(F, n, 0),
+    )
+end
+

src/PBWDeformations.jl

@@ -81,6 +81,8 @@ export general_linear_lie_algebra
 export inneighbor
 export inneighbors
 export is_crossing_free
+export is_linearly_independent
+export is_linearly_independent_with_relations
 export is_pbwdeformation
 export isomorphic_module_with_simple_structure
 export lookup_data
@@ -117,6 +119,7 @@ end
 
 include("OscarPatches.jl")
 
+include("LinearIndependence.jl")
 include("ModuleSimpleStructure.jl")
 
 include("DeformationBases/DeformBasis.jl")

test/DeformationBases-test.jl

@@ -45,15 +45,12 @@
                 @test all_pbwdeformations(sp, b) == collect(b)
 
                 b = ArcDiagDeformBasis{QQFieldElem}(sp, 0:3)
-                @test length(collect(b)) == 4
-                @test all_pbwdeformations(sp, b; special_return=SMat)[1] ==
-                      matrix(QQ, 4, 3, [1, 0, 0, 0, -3 // 2, 1 // 2, 0, 1, 0, 0, 0, 1])
+                @test length(collect(b)) == 3
+                @test all_pbwdeformations(sp, b; special_return=SMat)[1] == matrix(QQ, [1 0; 0 -3//2; 0 1])
                 ms = all_pbwdeformations(sp, b)
-                @test length(ms) == 3
+                @test length(ms) == 2
                 @test ms[1] == collect(b)[1]
                 @test 2 * ms[2] == -3 * collect(b)[2] + 2 * collect(b)[3]
-                @test 2 * ms[3] == 1 * collect(b)[2] + 2 * collect(b)[4]
-                @test iszero(ms[2] + ms[3]) # TODO: Check result for linear independence
             end
 
             @testset "SO_5, ⋀²V" begin
@@ -139,15 +136,12 @@
                 @test all_pbwdeformations(sp, b) == collect(b)
 
                 b = PseudographDeformBasis{QQFieldElem}(sp, 0:3)
-                @test length(collect(b)) == 4
-                @test all_pbwdeformations(sp, b; special_return=SMat)[1] ==
-                      matrix(QQ, 4, 3, [1, 0, 0, 0, -3 // 2, 1 // 2, 0, 1, 0, 0, 0, 1])
+                @test length(collect(b)) == 3
+                @test all_pbwdeformations(sp, b; special_return=SMat)[1] == matrix(QQ, [1 0; 0 -3//2; 0 1])
                 ms = all_pbwdeformations(sp, b)
-                @test length(ms) == 3
+                @test length(ms) == 2
                 @test ms[1] == collect(b)[1]
                 @test 2 * ms[2] == -3 * collect(b)[2] + 2 * collect(b)[3]
-                @test 2 * ms[3] == 1 * collect(b)[2] + 2 * collect(b)[4]
-                @test iszero(ms[2] + ms[3])
             end
 
             @testset "SO_5, ⋀²V" begin