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Sometimes whilst working with a differential
ring R, one might wish to consider the same ring, but with a different
derivation or with a larger constant ring.
It is a consequence of the creation of a differential ring,
that its constant ring may actually be smaller than its differential ring
of constants.
To alter the settings defined by the creation of a differential
ring or field the following functions are available.
Returns a differential ring isomorphic to R, but whose derivation is
the map f.Derivation(R) induced by the isomorphism.
The ring element f must be non--zero.
The isomorphism of R to the new differential ring
is also returned.
The new differential ring has the same underlying ring as R.
> F<z> := RationalDifferentialField(Rationals());
> Derivative(z^2);
2*z
> K, toK := ChangeDerivation(F, z);
> K;
Differential Ring of Algebraic function field defined over Rational Field by
$.2 - 4711
with derivation given by (1/z) d(z)
> toK;
Mapping from: RngDiff: F to RngDiff: K given by a rule
> Derivative(toK(z^2));
2*z^2
> UnderlyingRing(F) eq UnderlyingRing(K);
true
Notice that the differential of K is (1/z) (d) (z),
so that the derivation of K is z.(d)/(d)z,
as requested.
Returns the algebraic differential field, whose underlying ring is the
one of F, but with derivation with respect to the differential df.
The map returned is the bijective map from F into the
new algebraic differential field.
> F<z> := RationalDifferentialField(Rationals());
> df := Differential(1/z);
> df in DifferentialSpace(UnderlyingRing(F));
true
> M<u>, mp := ChangeDifferential(F,df);
> IsAlgebraicDifferentialField(M);
true
> Domain(mp) eq F and Codomain(mp) eq M;
true
> Differential(M);
(-1/u^2) d(u)
> mp(z);
u
> Derivation(M)(u);
u^2
> Derivation(F)(z);
1
> dg := Differential(z^3+5);
> N<v>, mp := ChangeDifferential(F,dg);
> Differential(M);
(3*v^2) d(v)
> mp(z);
v
> Derivation(N)(mp(z));
1/3/v^2
Returns the differential field isomorphic
to the differential field F, but whose constant field is the extension C,
and the isomorphism from F to
the new field.
The differential field F must be an algebraic function field.
> F<z> := RationalDifferentialField(Rationals());
> _<X> := PolynomialRing(F);
> M := ext< F | X^2-2 >;
> ConstantField(M);
Rational Field
> _<x>:=PolynomialRing(Rationals());
> C := NumberField(x^2-2);
> Mext, toMext := ConstantFieldExtension(M, C);
> ConstantField(Mext);
Number Field with defining polynomial x^2 - 2 over the Rational Field
> toMext;
Mapping from: RngDiff: M to RngDiff: Mext given by a rule
> S<t>:=DifferentialLaurentSeriesRing(Rationals());
> P<T> := PolynomialRing(Rationals());
> Cext := ext<Rationals()|T^2+1>;
> Sext<text>, mp := ConstantFieldExtension(S,Cext);
> IsDifferentialLaurentSeriesRing(Sext);
true
> ConstantRing(Sext) eq Cext;
true
> Derivative(text^(-2)+7+2*text^3+O(text^6));
-2*text^-2 + 6*text^3 + O(text^6);
> mp;
Mapping from: RngDiff: S to RngDiff: Sext given by a rule
> mp(t);
text
Precision: RngIntElt Default: ∞
The completion of the differential field F with respect to the place p.
The place p should be an element of the set of places of F.
The derivation of the completion is the one naturally induced by
the derivation of F.
The map returned is the embedding of F into the completion.
Upon creation one can set the precision by using Precision.
If no precision is given, then a default value is taken.
This example illustrates the creation of the differential Laurent
series ring by using the command Completion.
> F<z> := RationalDifferentialField(Rationals());
> pl := Zeros(z)[1];
> S<t>, mp := Completion(F,pl: Precision := 5);
> IsDifferentialLaurentSeriesRing(S);
true
> mp;
Mapping from: RngDiff: F to RngDiff: S given by a rule
> Domain(mp) eq F, Codomain(mp) eq S;
true true
> Derivation(S)(t);
1
> 1/(1-t);
1 + t + t^2 + t^3 + t^4 + O(t^5)
This example shows that one does not have to restrict to differential fields
of genus 0 to use Completion.
> F<z> := RationalDifferentialField(Rationals());
> P<Y> := PolynomialRing(F);
> K<y> := ext<F|Y^2-z^3+z+1>;
> Genus(UnderlyingRing(K));
1
> pl:=Zeros(K!z)[1];
> Degree(pl);
2
> S<t>, mp := Completion(K,pl);
> IsDifferentialLaurentSeriesRing(S);
true
> C<c> := ConstantRing(S);
> C;
Number Field with defining polynomial $.1^2 + 1 over the Rational Field
> mp(y) + O(t^4);
c - t - 4*t^3 + O(t^4)
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