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There are two ways to create a differential ring.
The first creation is a general creation of a differential ring,
for which the user specifies the ring and its derivation.
The second creates a
differential field which has the structure of a
rational function field of transcendence degree 1 over its base field.
Its derivation is specified by a differential.
Once a differential ring is created one can ask for its
ring or field of fractions.
Given a ring P and derivation f acting on P,
return the differential ring isomorphic to P,
with induced derivation f acting
on it, and ring of constants C.
The ring C should be a subring of P on which
f is zero.
Here we illustrate the creation and printing of a general differential ring.
> P := PolynomialRing(Rationals());
> f := map<P->P | a:->5*Derivative(a)>;
> R := DifferentialRing(P, f, Rationals());
> R;
Differential Ring of Univariate Polynomial Ring over
Rational Field with derivation given by Mapping
from: RngUPol: P to RngUPol: P given by a rule [no inverse]
The differential field in one variable over the constant field C.
If this field is called F, say, then
the derivation on F is given by
(d) / (1) (d) (F.1),
where F.1 is the variable of F, and (1) (d) (F.1) is its
differential in the differential space of F.
Any exact field with polynomial GCD is valid input for C.
Here we illustrate the creation and printing of the differential field
obtained from the command RationalDifferentialField.
> F<z> := RationalDifferentialField(Rationals());
> F;
Differential Ring of Algebraic function field defined over
Rational Field by $.2 - 4711 with
derivation given by (1) d(z)
The differential Laurent series ring (in one variable) over the constant
field C.
If this field is called F, say, then
the derivation on F is given by
F.1 .(d) / (d) (F.1),
where F.1 is the variable of F.
This example illustrates the creation and printing of the differential Laurent
series ring obtained from the command DifferentialLaurentSeriesRing.
> S<t> := DifferentialLaurentSeriesRing(Rationals());
> S;
Differential Ring of Laurent series field in t over Rational Field
with derivation given by Mapping from: Laurent series field in t over Rational
Field to Laurent series field in t over Rational Field given by a rule [no
inverse]
Returns the differential ring
R[r - 1: r ∈R (not a zero divisor)]
of fractions of the differential ring R, together with the inclusion map
from R to the newly created ring.
Returns the differential field of fractions of the differential ring R,
together with the inclusion map from R to the newly created field.
Given a differential ring R with n indeterminates and a sequence S
of n strings, assign the elements of S to the names of the variables
of R.
This procedure only changes the names used in the printing of the elements
of R.
The easiest way to create an element in a given ring is to use the angle
bracket construction to attach names to the indeterminates of the ring.
Others are given below.
R . i : RngDiff, RngIntElt -> RngDiffElt
The i-th indeterminate of the differential ring R, where i is between
1 and the number of generators of R.
Coerce the element s in the differential ring R.
Elements that are coercible are elements that are
coercible in the underlying ring of the differential
ring R.
The zero element of the differential ring R.
Identity(R) : RngDiff -> RngDiffElt
The identity element of the differential ring R.
Returns the separating element of the algebraic differential field F.
We construct the differential field F=Q(z) with derivation
(d)/ (d)z and show some of the elements that can be created.
> F<z> := RationalDifferentialField(Rationals());
> F.1;
z
> two := F!2;
> two;
2
> Parent(two) eq F;
true
> Zero(F); One(F);
0
1
> Parent(Zero(F)) eq F and Parent(Identity(F)) eq F;
true
> elt := SeparatingElement(F);
> elt;
z
> ISA(Type(elt),RngDiffElt);
true
> Parent(elt) eq F;
true
> elt eq F!SeparatingElement(UnderlyingRing(F));
true
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