Ring and Field Extensions#
The first differential ring and field extensions we consider are the ones induced by a differential operator. Given a differential operator
in a differential operator ring \(F[D]\) with coefficients in a differential field \(F\), we construct a ring or field extension of degree \(n\) over \(F\), whose indeterminates play the role of a formal solution of \(L(y)=0\) and its derivatives.
Given a differential field \(F\), it is also possible to construct differential extensions of the form \(F[X]/f(X)\), where \(f(X)\) is an irreducible polynomial over \(F\).
- DifferentialRingExtension(L): RngDiffOpElt -> RngDiff#
Constructs a differential ring extension of the base ring of the differential operator \(L\), by adding a formal solution of \(L\) and its formal derivatives as indeterminates.
Let \(P\) denote the new differential ring, and \(F\) the coefficient ring of \(L\). The ring \(F\) is a differential field. If \(n\) is the degree of \(L\), the underlying ring of \(P\) is a multivariate polynomial ring of degree \(n\) over \(F\). We thus have \(P=F[Y_1,Y_2,\ldots,Y_n]\), with indeterminates \(Y_1,Y_2,\ldots,Y_n\). If \(L\) is written as \(a_nD^n+a_{n-1}D^{n-1}+\cdots+a_1D+a_0\in F[D]\), then the derivation of \(P\) is induced by the differential operator \(L\) as follows: \(\delta_P(Y_i)=Y_{i+1}\), for \(i<n\) and \(a_n\delta_P(Y_{n})=-a_{n-1}Y_{n-1}-\cdots- a_2 Y_2-a_1Y_1.\) With this construction \(Y_1\) mimics a solution of \(L(y)=0\), and all the others are its derivatives.
- DifferentialFieldExtension(L): RngDiffOpElt -> RngDiff#
Constructs a differential field extension of the base ring of the differential operator \(L\), by adding a formal solution of \(L\) and its formal derivatives as indeterminates.
The construction of the new differential field is completely analogous to the differential ring created by
DifferentialRingExtension(L). The only difference is that now a differential field \(M=F(Y_1,Y_2,\ldots,Y_n)\), with \(n\) indeterminates \(Y_1,Y_2,\ldots,Y_n\) is created. The action of the derivation of \(M\) on \(Y_1,Y_2,\ldots,Y_n\) is as described inDifferentialRingExtension(L).
- Example: Diff Ring Ext Operator (ex-434a22)#
> F<z> := RationalDifferentialField(Rationals()); > R<D> := DifferentialOperatorRing(F); > L := z^2*D^2-z*D+1; > P<Y1,Y2> := DifferentialRingExtension(L); > P; Differential Ring Extension over F with derivation given by Mapping from: Polynomial ring of rank 2 over F to Polynomial ring of rank 2 over F given by a rule [no inverse] > Derivative(Y1); Y2 > Derivative(Y2); -1/z^2*Y1 + 1/z*Y2
- Example: Diff Field Ext Operator (ex-5ef2f3)#
> F<z> := RationalDifferentialField(Rationals()); > R<D> := DifferentialOperatorRing(F); > L := z^2*D^2-1; > M<Y,DY> := DifferentialFieldExtension(L); > IsDifferentialField(M); true > Derivative(Y); DY > Derivative(DY); -1/z^2*Y
- ext< F | f >: RngDiff, RngUPolElt -> RngDiff#
The differential field extension \(F(\alpha)\) of the differential field \(F\), where \(\alpha\) is a root of the irreducible polynomial \(f\) over \(F\). The angle bracket notation may be used to assign the root \(\alpha\) to an identifier.
- Example: Diff Field Ext Ext (ex-8a3a0e)#
> F<z> := RationalDifferentialField(Rationals()); > _<X> := PolynomialRing(F); > M<alpha> := ext< F | X^2-z >; > M; Differential Ring Extension over F by $.1^2 - z with derivation given by (1) d(z) > alpha^2; z
The differential of \(M\) is the differential \({d}z\) of the differential space of \(F\) lifted to the space of differentials of \(M\).
- ExponentialFieldExtension(F, f): RngDiff, RngDiffElt -> RngDiff#
Returns the differential field \(F(E)\) as an extension of \(F\), such that the derivation of \(E\) is \(f\cdot E\). The parent of \(f\) must be \(F\).
- LogarithmicFieldExtension(F, f): RngDiff, RngDiffElt -> RngDiff#
Returns the differential field \(F(L)\) as an extension of \(F\), such that the derivation of \(L\) is \(F(L)!f\). The parent of \(f\) must be \(F\).
- Example: Diff Field Ext Exp Log (ex-0b50af)#
> F<z> := RationalDifferentialField(Rationals()); > K<E> := ExponentialFieldExtension(F, z); > K; Differential Ring Extension over F with derivation given by Mapping from: Multivariate Rational function field of rank 1 over F to Multivariate Rational function field of rank 1 over F given by a rule [no inverse] > Derivative(E); z*E > _<L> := LogarithmicFieldExtension(F, 1/z); > Derivative(L); 1/z > Parent($1) eq Parent(L); true
- PurelyRamifiedExtension(f): RngUPolElt[RngDiff] -> RngDiff, Map#
Creates a purely ramified field extension \(M\) of the differential field \(F\) with respect to the purely ramified polynomial \(f\in F[X]\). By definition, such a polynomial \(f\) is of the form \(X^n-a\cdot(F.1)\) for some constant element \(a\) in \(F\) and positive integer \(n\). The returned extension field \(M\) is of the same type as \(F\). The allowed differential fields are algebraic differential fields and differential Laurent series rings. When \(F\) is a differential Laurent series ring, its derivation is required to be weakly of the form \(c*(F.1)*d/d(F.1)\) for some constant \(c\). The relative precision of \(M\) is then \(n\) times the relative precision of \(F\). The second argument returned is the embedding map of \(F\) into \(M\). The inverse map acts on elements for which it is defined. Otherwise it returns \(0\).
- Example: Diff Field Purely Ramified Ext ADF (ex-838a6e)#
A purely ramified extension of an algebraic differential field is constructed in this example.
> F<z> := RationalDifferentialField(Rationals()); > _<X> := PolynomialRing(F); > Fext<v>, mp := PurelyRamifiedExtension(X^2-5*z); > IsAlgebraicDifferentialField(Fext); true > mp(z) eq 1/5*v^2; true > Parent(mp(z)) eq Fext; true > Derivation(Fext)(mp(z)); 1 > Derivation(Fext)(v); 1/2/z*v > Derivation(Fext)(v^2) eq Fext!5; true > Inverse(mp)(v^2); 5*z;
- Example: Diff Field Purely Ramified Ext DLSR 1 (ex-efad09)#
A differential Laurent series ring with a derivation without an order term is considered in this example.
> S<t>:=DifferentialLaurentSeriesRing(Rationals()); > _<T>:=PolynomialRing(S); > pol := T^4-5*t; > Sext<r>,mp := PurelyRamifiedExtension(pol); > IsDifferentialLaurentSeriesRing(Sext); true > BaseRing(Sext) eq S and ConstantField(Sext) eq ConstantField(S); true > RelativePrecision(Sext); 80 > RelativePrecisionOfDerivation(Sext); Infinity > Derivation(S)(t); t > mp(t); 1/5*r^4 > Derivation(Sext)(mp(t)); 1/5*r^4 > mp(Derivation(S)(t)); 1/5*r^4 > x := 4+6*t+O(t^6); > mp(x); 4 + 6/5*r^4 + O(r^24) > Derivation(Sext)(mp(x)); 6/5*r^4 + O(r^24) > mp(Derivation(S)(x)); 6/5*r^4 + O(r^24) > Inverse(mp)(r^4-r^8); 5*t - 25*t^2 > Inverse(mp)(r^4+O(r^5)); 5*t + O(t^2) > Derivation(Sext)(r); 1/4*r
- Example: Diff Field Purely Ramified Ext DLSR 2 (ex-da702b)#
The ring in this example has an order term in its derivation. Therefore, taking a derivative of an element \(x\) is of influence on the relative precision of the image of \(x\).
> F<z> := RationalDifferentialField(Rationals()); > FF<z>:=ChangeDerivation(RationalDifferentialField(Rationals()),z); > RR<DD>:=DifferentialOperatorRing(FF); > RS<DS>, mpRRtoRS :=Completion(RR,Zeros(z)[1]); > S<t>:=BaseRing(RS); > IsDifferentialLaurentSeriesRing(S); true > _<T> := PolynomialRing(S); > E<r>, mp := PurelyRamifiedExtension(T^3-5*t); > IsDifferentialLaurentSeriesRing(E); true > RelativePrecision(E); 60 > RelativePrecisionOfDerivation(E); 60 > Derivation(E)(r); 1/3*r + O(r^61); > mp(t); 1/5*r^3 > Derivation(S)(t); t + O(t^21) > Derivation(E)(mp(t)); 1/5*r^3 + O(r^63) > mp(Derivation(S)(t)); 1/5*r^3 + O(r^63) > x:=t^(-2) +7+t^3 +O(t^15); > Derivation(S)(x); -2*t^-2 + 3*t^3 + O(t^15) > Derivation(E)(mp(x)); -50*r^-6 + 3/125*r^9 + O(r^45) > mp(Derivation(S)(x)); -50*r^-6 + 3/125*r^9 + O(r^45) > y := 2*t+O(t^25); > Derivation(S)(y); 2*t + O(t^21) > Derivation(E)(mp(y)) eq mp(Derivation(S)(y)); true > Derivation(E)(mp(y)); 2/5*r^3 + O(r^63)