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.
> 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.
> 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.
> 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.
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.
The identity element of the differential ring R.
Returns the separating element of the algebraic differential field F.
> 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