# Structure Operations on Differential Rings ## Category and Parent Differential Rings form the Magma category `RngDiff`. The notional power structures exist as parents of differential rings. ### `Category(R): RngDiff -> RngDiff` ### `Type(R): RngDiff -> RngDiff` The category, or type, of the differential ring $R$. ### `Parent(R): RngDiff -> PowStr` The power structure of the differential ring $R$. ## Related Structures The underlying ring and constant ring from which the differential ring was created can each be retrieved as described below. There is also the concept of a base ring. If one has created a differential extension $M/F$ in Magma, then $F$ is the *base ring* of $M$. ### `UnderlyingRing(R): RngDiff -> Rng` The underlying ring of the differential ring $R$. The type of the underlying ring indicates what ring $R$ inherits from. ### `UnderlyingField(R): RngDiff -> Rng` The underlying ring of the differential ring $R$, provided it is a field. ### `BaseRing(R): RngDiff -> Rng` The base ring of the differential ring $R$. ### `BaseField(R): RngDiff -> Rng` The base ring of the differential ring $R$, provided it is a field. ### `ConstantRing(R): RngDiff -> Rng` The constant ring of the differential ring $R$. The derivation of $R$ acts trivially on the constant ring. It is therefore contained in the differential ring of constants of $R$. ### `ConstantField(R): RngDiff -> Rng` The constant ring of the differential ring $R$, provided it is a field. ### `ExactConstantField(F): RngDiff -> RngDiff, Map` The exact constant field of $F$, i.e. the algebraic closure in $F$ of the constant field of $F$, together with the inclusion map to $F$. The field $F$ must be a function field. The differential field $F$ must have been created with respect to a differential. If the derivation of $F$ has been constructed with respect to a differential, then the exact constant field coincides with the differential field of constants of $F$. ### `Generators(R): RngDiff -> SeqEnum` The list of generators of the differential ring $R$. If there is no list assigned to $R$, one is constructed by default from the underlying ring of $R$. ### `Example: Diff Ring Related Structures (ex-64f77a)` First we construct the differential field $F={\mathbb{Q}}(z)$ with derivation ${d}/ {d}z$ and show what some of the related structures are. Then we construct the field extension $M={\mathbb{Q}}(z,\alpha)$, where $\alpha$ is a root of the polynomial $X^2-2$. We do this with the usual `ext< >` constructor. For $M$ we again derive some related structures. ```magma > F := RationalDifferentialField(Rationals()); > ConstantRing(F); Rational Field > UnderlyingRing(F); Algebraic function field defined over Rational Field by $.2 - 4711 > > _ := PolynomialRing(F); > M := ext< F | X^2-2 >; > BaseRing(M); Differential Ring of Algebraic function field defined over Rational Field by $.2 - 4711 with derivation given by (1) d(z) > BaseRing(M) eq F; true > ConstantRing(M); Rational Field > E := ExactConstantField(M); > E; Number Field with defining polynomial $.1^2 - 2 over the Rational Field > Generators(M); [ alpha ] ``` ### `Example: Diff Laur Ser Related Structures (ex-326abb)` Related structures also exist for differential Laurent series rings. ```magma > S:=DifferentialLaurentSeriesRing(Rationals()); > UnderlyingRing(S); Laurent series field in t over Rational Field > ConstantRing(S); Rational Field > Generators(S); [ t ] ``` ## Derivation and Differential The derivation of a differential ring and its differential, whenever applicable, can be retrieved as indicated below. ### `Derivation(R): RngDiff -> Map` The derivation of the differential ring $R$. ### `Differential(F): RngDiff -> DiffFunElt` The differential belonging to the derivation of the differential field $F$. The field $F$ must have been constructed in such a way that its derivation is defined by a differential. ### `Example: Diff Ring Derivation Differential (ex-627c5f)` ```magma > F := RationalDifferentialField(Rationals()); > Derivation(F); Mapping from: RngDiff: F to RngDiff: F given by a rule [no inverse] > Differential(F); (1) d(z) ``` ## Numerical Invariants ### `Ngens(R): RngDiff -> RngIntElt` The number of indeterminates associated with the differential ring $R$. ## Predicates and Booleans ### `R eq F: RngDiff, RngDiff -> BoolElt` Returns `true` if and only if the differential rings $R$ and $F$ are the same. ### `IsIdentical(R, F): RngDiff, RngDiff -> BoolElt` Returns `true` if and only if the differential rings $R$ and $F$ are identical. ### `IsDomain(R): RngDiff -> BoolElt` Returns `true` if and only if the differential ring $R$ is a domain. ### `IsField(R): RngDiff -> BoolElt` Returns `true` if and only if the differential ring $R$ is field. ### `IsDifferentialField(R): Rng -> BoolElt` Returns `true` if and only if the ring $R$ is a differential field. ### `IsAlgebraicDifferentialField(R): Rng -> BoolElt` Returns `true` if and only if the field structure of the differential ring $R$ is an algebraic function field. ### `IsDifferentialSeriesRing(R): Rng -> BoolElt` Returns `true` if and only if the underlying ring of the differential ring $R$ is a series ring. ### `IsDifferentialLaurentSeriesRing(R): Rng -> BoolElt` Returns `true` if and only if the underlying ring of the differential ring $R$ is a Laurent series ring and $R$ has been created with a known constant ring. ### `Example: Diff Rings Booleans (ex-913ce7)` This example shows some booleans for various differential rings. ```magma > F:=RationalDifferentialField(Rationals()); > S:=DifferentialLaurentSeriesRing(Rationals()); > IsAlgebraicDifferentialField(F); true > IsDifferentialSeriesRing(F); false > IsAlgebraicDifferentialField(S); false > IsDifferentialSeriesRing(S); true > IsDifferentialLaurentSeriesRing(S); true ``` ### `HasProjectiveDerivation(F): RngDiff -> BoolElt` Returns `true` if and only if $F$ is a differential ring with derivation weakly of the form $(F.1)\cdot d/d(F.1)$. ### `HasZeroDerivation(F): RngDiff -> BoolElt` Returns `true` if and only if the algebraic differential field or differential series ring $F$ has zero derivation. When $F$ is a series ring we relax being zero to being weakly zero. ### `Example: Diff Rings Booleans Derivation (ex-a3014d)` ```magma > F:=RationalDifferentialField(Rationals()); > S:=DifferentialLaurentSeriesRing(Rationals()); > HasProjectiveDerivation(F); false > HasProjectiveDerivation(ChangeDerivation(F,z)); true > HasZeroDerivation(F); false > HasProjectiveDerivation(S); true > HasProjectiveDerivation(ChangeDerivation(S,S!3)); false > HasZeroDerivation(S); false ``` ## Precision ### `RelativePrecision(F): RngDiff -> RngElt` Returns the relative precision of the underlying series ring of F. ### `RelativePrecisionOfDerivation(F): RngDiff -> RngElt` Given a differential Laurent series ring $F$, returns the relative precision of the ring derivative of $F.1$. ### `Example: Diff Rings Relative Precision (ex-4674cf)` This example illustrate the relative precision of differential rings. ```magma > S:=DifferentialLaurentSeriesRing(Rationals()); > Derivative(t); t > IsDifferentialLaurentSeriesRing(S); true > RelativePrecision(S); 20 > RelativePrecision(UnderlyingRing(S)); 20; > V:=DifferentialLaurentSeriesRing(Rationals():Precision:=30); > RelativePrecision(V); 30 > RelativePrecision(V) eq RelativePrecision(UnderlyingRing(V)); true ``` ### `Example Differential Ring Relative Precision Derivation (ex-617005)` ```magma > S := DifferentialLaurentSeriesRing(Rationals()); > RelativePrecisionOfDerivation(S); Infinity > V := ChangeDerivation(S,t+O(t^6)); > Derivation(V)(w); w^2 + O(w^7) > RelativePrecisionOfDerivation(V); 5 ``` ### `ChangePrecision(F, p): RngDiff, RngElt -> RngDiff, Map` Returns the differential series ring isomorphic to $F$ with relative precision $p$. The map returned is the induced map of $F$ to the new field. ### `Example: Diff Rings Change Precision (ex-d18d60)` ```magma > S:=DifferentialLaurentSeriesRing(Rationals()); > RelativePrecision(S); 20 > V,mp := ChangePrecision(S,10); > Type(V); RngDiff > IsDifferentialLaurentSeriesRing(V); true > RelativePrecision(V); 10 > RelativePrecision(1/(w-1)) eq 10; true > mp(t) eq w; true > w@@mp eq t; true > derivt := Derivation(S)(t); > derivt; t > derivw := Derivation(V)(w); > derivw; w > mp(derivt) eq Derivation(V)(w); true ```