denominator for polynomials

[SoongNoonien]

I was again playing around with the universal polynomial rings and wondered why there is no denominator function for them. But then I noticed that with plain AbstractAlgebra there is not even a denominator function for univariate or multivariate polynomials:

julia> using AbstractAlgebra

julia> pkgversion(AbstractAlgebra)
v"0.41.9"

julia> R,(x,y)=QQ[:x,:y]
(Multivariate polynomial ring in 2 variables over rationals, AbstractAlgebra.Generic.MPoly{Rational{BigInt}}[x, y])

julia> denominator(1//2*x+1//3*x^2)
ERROR: MethodError: no method matching denominator(::AbstractAlgebra.Generic.MPoly{Rational{BigInt}})

Closest candidates are:
  denominator(::AbstractAlgebra.Generic.RationalFunctionFieldElem, ::Bool)
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/EYoda/src/generic/RationalFunctionField.jl:90
  denominator(::AbstractAlgebra.Generic.RationalFunctionFieldElem)
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/EYoda/src/generic/RationalFunctionField.jl:90
  denominator(::AbstractAlgebra.Generic.TotFrac, ::Bool)
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/EYoda/src/generic/TotalFraction.jl:73
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[3]:1

julia> R,x=QQ[:x]
(Univariate polynomial ring in x over rationals, x)

julia> denominator(1//2*x+1//3*x^2)
ERROR: MethodError: no method matching denominator(::AbstractAlgebra.Generic.Poly{Rational{BigInt}})

Closest candidates are:
  denominator(::AbstractAlgebra.Generic.RationalFunctionFieldElem, ::Bool)
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/EYoda/src/generic/RationalFunctionField.jl:90
  denominator(::AbstractAlgebra.Generic.RationalFunctionFieldElem)
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/EYoda/src/generic/RationalFunctionField.jl:90
  denominator(::AbstractAlgebra.Generic.TotFrac, ::Bool)
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/EYoda/src/generic/TotalFraction.jl:73
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[6]:1

julia> R=universal_polynomial_ring(QQ)
Universal Polynomial Ring over Rationals

julia> x=gen(R,:x)
x

julia> denominator(1//2*x+1//3*x^2)
ERROR: MethodError: no method matching denominator(::AbstractAlgebra.Generic.UnivPoly{Rational{BigInt}, AbstractAlgebra.Generic.MPoly{Rational{BigInt}}})

Closest candidates are:
  denominator(::AbstractAlgebra.Generic.RationalFunctionFieldElem, ::Bool)
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/EYoda/src/generic/RationalFunctionField.jl:90
  denominator(::AbstractAlgebra.Generic.RationalFunctionFieldElem)
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/EYoda/src/generic/RationalFunctionField.jl:90
  denominator(::AbstractAlgebra.Generic.TotFrac, ::Bool)
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/EYoda/src/generic/TotalFraction.jl:73
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[10]:1

julia> 

In all of these cases I would expect a value of 6.

[SoongNoonien]

I think 1//content(f) should be what I thought of. But content is not defined for universal polynomials.

[lgoettgens]

I think 1//content(f) should be what I thought of. But content is not defined for universal polynomials.

#1970 will add it.

[SoongNoonien]

Nice, thank you!

[fingolfin]

1//content(f) is not the same as denominator(f). Try e.g. 2//3*x.

[fingolfin]

Perhaps denominator(f) = numerator(content(f)) might work as a definition. But this notion really only makes sense for polynomials over the rationals. But what are "rationals"? In the context of AA this means something else than in the context of Nemo and Oscar. We could add such dedicated methods for PolyRingElem{<:Rational} inside AA but in practice we'd hopefully never use them.

I suggest that instead we (or rather, you, @SoongNoonien ;-) ) add a bunch of specialized methods for certain UniversalPolyRingElem specializations to Nemo, say in a new src/UnivPoly.jl there. It could contain e.g. a method

denominator(f::UniversalPolyRingElem{QQFieldElem}) = denominator(data(f))

[SoongNoonien]

1//content(f) is not the same as denominator(f). Try e.g. 2//3*x.

Yes, of course I missed that... My examples were a bit too simple.