Order on RationalFunctionField

[lenweis]

The field of rational functions does not come with an order in Oscar which doesn't let me construct a polyhedron for example.

julia> RR, u = RationalFunctionField(QQ, :u)
(Rational function field over QQ, u)

julia> a = [u, u^2, u^3]
3-element Vector{AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem}}:
 u
 u^2
 u^3

julia> P = polyhedron(RR, [a], [u])
ERROR: MethodError: no method matching isless(::AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem}, ::AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem})

Closest candidates are:
  isless(::Any, ::EmbeddedElem)
   @ Hecke ~/.julia/packages/Hecke/D2kay/src/NumField/Embedded.jl:142
  isless(::EmbeddedElem, ::Any)
   @ Hecke ~/.julia/packages/Hecke/D2kay/src/NumField/Embedded.jl:140
  isless(::Any, ::Missing)
   @ Base missing.jl:88
  ...

As the field comes with a natural order ($\frac{f}{g} > 0$, when $\frac{lc(f)}{lc(g)} > 0$), it would be nice to have to this.

[fieker]

On Tue, Dec 05, 2023 at 01:58:55AM -0800, Lena wrote:

The field of rational functions does not come with an order in Oscar which doesn't let me construct a polyhedron for example.

julia> RR, u = RationalFunctionField(QQ, :u)
(Rational function field over QQ, u)

julia> a = [u, u^2, u^3]
3-element Vector{AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem}}:
 u
 u^2
 u^3

julia> P = polyhedron(RR, [a], [u])
ERROR: MethodError: no method matching isless(::AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem}, ::AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem})

Closest candidates are:
  isless(::Any, ::EmbeddedElem)
   @ Hecke ~/.julia/packages/Hecke/D2kay/src/NumField/Embedded.jl:142
  isless(::EmbeddedElem, ::Any)
   @ Hecke ~/.julia/packages/Hecke/D2kay/src/NumField/Embedded.jl:140
  isless(::Any, ::Missing)
   @ Base missing.jl:88
  ...

As the field comes with a natural order ($\frac{f}{g} > 0$, when $\frac{lc(f)}{lc(g)} > 0$), it would be nice to have to this.

The field comes with many orders, e.g. any irreducible in K[t] defines an exponential valuation, then there is the degree function...

What properties of an order are required here? Maybe we can pass an order into the constructor? Or do s.th. like the Embedded stuff where a new object with an order is created...

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[fingolfin]

The proposed "natural" order seems extremely unnatural to me. E.g. it would order $2x$ above $x^5$. Not saying this rules it out as a order, just that I'd be very careful with the term "natural".

Also, even if defining such ordering, I don't think you'll be able to do much with the resulting polyhedron. For geometry, you need to eventually fix an embedding into the real numbers.

[lgoettgens]

@benlorenz proposes to put a wrapper with an ordering around fractional function fields, like was done with embedded number fields