The first differential ring and field extensions we consider are the ones induced by a differential operator. Given a differential operator L = anDn + an - 1Dn - 1 + ... + a1D + a0, annot=0 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.
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[Y1, Y2, ..., Yn], with indeterminates Y1, Y2, ..., Yn. If L is written as anDn + an - 1Dn - 1 + ... + a1D + a0∈F[D], then the derivation of P is induced by the differential operator L as follows: δP(Yi)=Yi + 1, for i<n and anδP(Yn)= - an - 1Yn - 1 - ... - a2 Y2 - a1Y1. With this construction Y1 mimics a solution of L(y)=0, and all the others are its derivatives.
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(Y1, Y2, ..., Yn), with n indeterminates Y1, Y2, ..., Yn is created. The action of the derivation of M on Y1, Y2, ..., Yn is as described in DifferentialRingExtension(L).
> 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
> 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
The differential field extension F(α) of the differential field F, where α is a root of the irreducible polynomial f over F. The angle bracket notation may be used to assign the root α to an identifier.
> 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; zThe differential of M is the differential (d)z of the differential space of F lifted to the space of differentials of M.
Returns the differential field F(E) as an extension of F, such that the derivation of E is f.E. The parent of f must be F.
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.
> 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
Creates a purely ramified field extension M of the differential field F with respect to the purely ramified polynomial f∈F[X]. By definition, such a polynomial f is of the form Xn - a.(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.
> 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;
> 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
> 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)