Degree localization of a rational function field

Degree localization

Given $k(x)$ a (univariate) rational function field, there are two rings of interest, both of which are Euclidean:

\[k[x]\]
k_\infty(x) = {a/b | a, b \in k[x] \;\;\mbox{where}\;\; \deg(a) \leq \deg(b)}

The second of these rings is the localization of $k[1/x]$ at $(1/x)$ inside the rational function field $k(x)$, i.e. the localization of the function field at the point at infinity, i.e. the valuation ring for valuation $-$degree$(x)$.

We refer to this ring as the degree localization of the rational function field $k(x)$.

Construction of the degree localization

The degree localization of a rational function field $k(x)$ can be constructed using a localization constructor, passing in the degree function as argument.

localization — Method

localization(K::RationalFunctionField{T}, ::typeof(degree)) where T <: FieldElement

Return the localization of $k[1/x]$ at $(1/x)$ inside the rational function field $k(x)$, i.e. the localization of the function field at the point at infinity, i.e. the valuation ring for valuation $-$degree$(x)$. This is the ring $k_\infty(x) = \{ f/g | \deg(f) \leq \deg(g)\}$.

Example +


julia> K, x = RationalFunctionField(FlintQQ, "x");
julia> R = localization(K, degree)Degree localization of Rational function field over QQ

Elements of the degree localization

Elements of the degree localization are created using the parent object $R$ representing the degree localization

Example +


julia> K, x = RationalFunctionField(FlintQQ, "x");
julia> R = localization(K, degree)Degree localization of Rational function field over QQ
julia> a = R()0
julia> b = R(1)1
julia> c = R((x + 1)//x)(x + 1)//x

Note that the degree of the denominator of the function field element passed to the constructor must be at least that of the numerator or an exception is raised.

Element functionality

degree — Method

 degree(a::KInftyElem)

Return the degree of the given element, i.e. degree(numerator) - degree(denominator).

valuation — Method

valuation(a::KInftyElem)

Return the degree valuation of the given element, i.e. -degree(a).

One can test whether a given element of a rational function field is in the degree localization.

in — Method

in(a::Generic.RationalFunctionFieldElem{T}, R::KInftyRing{T}) where T <: FieldElement

Return true if the given element of the rational function field is an element of k_\infty(x), i.e. if degree(numerator) <= degree(denominator).

All basic arithmetic operations are provided for elements of the degree localization.

As the degree localization is a Euclidean ring, all standard Euclidean functions, including div, divrem, mod, gcd, gcdx, are provided.