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polynomialrings.jl's Issues

How to tell if polynomials are linearly independent in a quotient ring?

I have a vanishing ideal V and I'm trying to compute the dimension of degree d polynomials in the quotient ring ℚ[x,y] / V. For example, consider the circle:

julia> @ring! ℚ[x,y]
@ring(ℚ[x,y])

julia> V = 1-x^2-y^2
-x^2 + -y^2 + 1//1

julia> @test rem(y^2, V) == y^2 && div(y^2, V) == 0
Test Passed

julia> @test rem(x*y, V) == x*y && div(x*y, V) == 0
Test Passed

julia> @test div(x^2, V) ≠ 0 # lies in the span of the other two
Test Passed

That is, we know x^2 = 1-y^2 hence [1,x^2,y^2] are linearly dependent.

Is there a systematic way of doing this?

I see theres a QuotientRing type but no documentation...

rem(::Integer, x) errors

I would expect integers to behave like 0-degree polynomials but:

julia> @ring! ℚ[x,y]
@ring(ℚ[x,y])

julia> rem(1, x)
ERROR: MethodError: no method matching rem(::Int64, ::Generator{@variable(x), @ring(ℚ[x,y])})

Closest candidates are:
  rem(::Any, ::Any, ::RoundingMode{:ToZero})
   @ Base div.jl:97
  rem(::Any, ::Any, ::RoundingMode{:Down})
   @ Base div.jl:98
  rem(::Any, ::Any, ::RoundingMode{:Up})
   @ Base div.jl:99
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[14]:1

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Linear algebra doesn't work

I think we just need to overload iterate and adjoint to fix this:

julia> dot([x,y], [-y,x])
ERROR: MethodError: no method matching iterate(::@ring(ℚ[x,y]))

Closest candidates are:
  iterate(::Union{LinRange, StepRangeLen})
   @ Base range.jl:880
  iterate(::Union{LinRange, StepRangeLen}, ::Integer)
   @ Base range.jl:880
  iterate(::T) where T<:Union{Base.KeySet{<:Any, <:Dict}, Base.ValueIterator{<:Dict}}
   @ Base dict.jl:698
  ...

Stacktrace:
 [1] dot(x::@ring(ℚ[x,y]), y::@ring(ℚ[x,y]))
   @ LinearAlgebra ~/Projects/julia-1.9/usr/share/julia/stdlib/v1.9/LinearAlgebra/src/generic.jl:848
 [2] dot(x::Vector{@ring(ℚ[x,y])}, y::Vector{@ring(ℚ[x,y])})
   @ LinearAlgebra ~/Projects/julia-1.9/usr/share/julia/stdlib/v1.9/LinearAlgebra/src/generic.jl:886
 [3] top-level scope
   @ REPL[24]:1

julia> [x,y]'* [-y,x]
ERROR: MethodError: no method matching adjoint(::@ring(ℚ[x,y]))

Closest candidates are:
  adjoint(::Union{QR, LinearAlgebra.QRCompactWY, QRPivoted})
   @ LinearAlgebra ~/Projects/julia-1.9/usr/share/julia/stdlib/v1.9/LinearAlgebra/src/qr.jl:517
  adjoint(::Union{Cholesky, CholeskyPivoted})
   @ LinearAlgebra ~/Projects/julia-1.9/usr/share/julia/stdlib/v1.9/LinearAlgebra/src/cholesky.jl:556
  adjoint(::LQ)
   @ LinearAlgebra ~/Projects/julia-1.9/usr/share/julia/stdlib/v1.9/LinearAlgebra/src/lq.jl:138
  ...

Stacktrace:
  [1] getindex
    @ ~/Projects/julia-1.9/usr/share/julia/stdlib/v1.9/LinearAlgebra/src/adjtrans.jl:302 [inlined]
  [2] iterate
    @ ./abstractarray.jl:1220 [inlined]
  [3] iterate
    @ ./abstractarray.jl:1218 [inlined]
  [4] _zip_iterate_some
    @ ./iterators.jl:424 [inlined]
  [5] _zip_iterate_all
    @ ./iterators.jl:416 [inlined]
  [6] iterate
    @ ./iterators.jl:406 [inlined]
  [7] _foldl_impl
    @ ./reduce.jl:56 [inlined]
  [8] foldl_impl(op::Base.MappingRF{LinearAlgebra.var"#13#14", Base.BottomRF{typeof(Base.add_sum)}}, nt::Base._InitialValue, itr::Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}})
    @ Base ./reduce.jl:48
  [9] mapfoldl_impl(f::typeof(identity), op::typeof(Base.add_sum), nt::Base._InitialValue, itr::Base.Generator{Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}}, LinearAlgebra.var"#13#14"})
    @ Base ./reduce.jl:44
 [10] mapfoldl(f::Function, op::Function, itr::Base.Generator{Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}}, LinearAlgebra.var"#13#14"}; init::Base._InitialValue)
    @ Base ./reduce.jl:170
 [11] mapfoldl(f::Function, op::Function, itr::Base.Generator{Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}}, LinearAlgebra.var"#13#14"})
    @ Base ./reduce.jl:170
 [12] mapreduce(f::Function, op::Function, itr::Base.Generator{Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}}, LinearAlgebra.var"#13#14"}; kw::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ Base ./reduce.jl:302
 [13] mapreduce(f::Function, op::Function, itr::Base.Generator{Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}}, LinearAlgebra.var"#13#14"})
    @ Base ./reduce.jl:302
 [14] sum(f::Function, a::Base.Generator{Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}}, LinearAlgebra.var"#13#14"}; kw::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ Base ./reduce.jl:530
 [15] sum(f::Function, a::Base.Generator{Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}}, LinearAlgebra.var"#13#14"})
    @ Base ./reduce.jl:530
 [16] sum(a::Base.Generator{Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}}, LinearAlgebra.var"#13#14"}; kw::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ Base ./reduce.jl:559
 [17] sum(a::Base.Generator{Base.Iterators.Zip{Tuple{Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, Vector{@ring(ℚ[x,y])}}}, LinearAlgebra.var"#13#14"})
    @ Base ./reduce.jl:559
 [18] _dot_nonrecursive(u::Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, v::Vector{@ring(ℚ[x,y])})
    @ LinearAlgebra ~/Projects/julia-1.9/usr/share/julia/stdlib/v1.9/LinearAlgebra/src/adjtrans.jl:428
 [19] *(u::Adjoint{Union{}, Vector{@ring(ℚ[x,y])}}, v::Vector{@ring(ℚ[x,y])})
    @ LinearAlgebra ~/Projects/julia-1.9/usr/share/julia/stdlib/v1.9/LinearAlgebra/src/adjtrans.jl:435
 [20] top-level scope
    @ REPL[25]:1

Why does QuotientRing use a dictionary of IDs?

The design of QuotientRing is very non-standard. At the moment it stores the ring in the type information as an ID to a dictionary, a pattern I've never seen in a Julia package before:

PolynomialRings.jl/src/CommutativeAlgebras/QuotientRings.jl

Line 28 in 7c86134

struct QuotientRing{P<:Polynomial, ID}

A more standard design would have just had a field pointing to the relevant ring.

Can you explain the motivation? Note the fact that the ID is in the type does not appear to be used anywhere.

`rem` errors over the reals 

julia> @ring! ℝ[x,y]
@ring(ℝ[x,y])

julia> I = x*(1-x^2-y^2)
-x^3 + -x*y^2 + x

julia> rem(x*(1-x^2-y^2), I)
ERROR: MethodError: no method matching //(::BigFloat, ::BigFloat)

Closest candidates are:
  //(::AbstractArray, ::Number)
   @ Base rational.jl:82
  //(::PolynomialRings.QuotientRings.QuotientRing, ::Number)
   @ PolynomialRings ~/.julia/packages/PolynomialRings/JNZGk/src/CommutativeAlgebras/QuotientRings.jl:152
  //(::NumberField, ::Number)
   @ PolynomialRings ~/.julia/packages/PolynomialRings/JNZGk/src/CommutativeAlgebras/NumberFields.jl:343
  ...

Stacktrace:
  [1] maybe_div(a::(Term over BigFloat in @degrevlex(x > y)), b::(Term over BigFloat in @degrevlex(x > y)))
    @ PolynomialRings.Terms ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Terms.jl:108
  [2] #one_step_div!#4
    @ ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Operators.jl:96 [inlined]
  [3] one_step_div!
    @ ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Operators.jl:85 [inlined]
  [4] rem!(f::@ring(ℝ[x,y]), G::Vector{@ring(ℝ[x,y])}; order::typeof(@degrevlex(x > y)), redtype::PolynomialRings.Operators.Lead)
    @ PolynomialRings.Reductions ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Reductions.jl:102
  [5] rem!
    @ ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Reductions.jl:87 [inlined]
  [6] rem!(f::@ring(ℝ[x,y]), G::Vector{@ring(ℝ[x,y])}; order::typeof(@degrevlex(x > y)), redtype::PolynomialRings.Operators.Full)
    @ PolynomialRings.Reductions ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Reductions.jl:89
  [7] rem!
    @ ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Reductions.jl:87 [inlined]
  [8] #rem#6
    @ ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Reductions.jl:351 [inlined]
  [9] rem
    @ ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Reductions.jl:348 [inlined]
 [10] #rem#37
    @ ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Reductions.jl:435 [inlined]
 [11] rem(f::@ring(ℝ[x,y]), g::@ring(ℝ[x,y]))
    @ PolynomialRings.Reductions ~/.julia/packages/PolynomialRings/JNZGk/src/PolynomialRings/Reductions.jl:434
 [12] top-level scope
    @ REPL[10]:1

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