Type(s) : RngDiffElt -> RngDiffElt
The category, or type, of the differential ring element s.
The parent of the differential ring element s.
All the usual arithmetic operations are possible for differential ring
elements.
The sum of the two differential ring elements s and t.
The negation of the differential ring element s.
The difference between the differential ring elements s and t.
The product of the differential ring elements s and t.
Given a differential ring element s and an integer n,
return the n-th power of s.
If s is invertible, n may be negative.
Given the differential ring elements s and t,
return the exact division of s by t, if s
is divisible by t.
Given the differential field elements s and t,
return s divided by t.
Return true iff the differential ring elements s and t are
exactly the same.
Return true iff the differential ring element s is the zero
element of its parent.
Return true iff the differential ring element s is the unity
element of its parent.
Return true if and only if the differential ring element s is
weakly equal to the differential ring element t.
Return true if and only if the differential ring element s is weakly
equal to the zero element of its parent.
IsOrderTerm(s) : RngSerElt -> BoolElt
Return true if and only if the differential ring element s is purely an order
term of a differential series ring.
This examples shows the booleans for various differential rings.
> F<z> := RationalDifferentialField(Rationals());
> S<t> := DifferentialLaurentSeriesRing(Rationals());
> IsOne(F!1);
true
> t eq t+O(t^2);
false
> IsWeaklyEqual(t, t+O(t^2));
true
> IsWeaklyZero(t^(-1));
false
> IsWeaklyZero(O(t));
true
> IsOrderTerm(t+O(t^2));
false
> IsOrderTerm(O(t));
true
Creates the order term of the differential series s.
The known part of the differential series s.
Returns the coefficients of the differential ring element s.
Exponents(s) : RngSerElt -> SeqEnum
Returns the interval from the valuation of s to (including) the degree of s.
> F<z> := RationalDifferentialField(Rationals());
> _<X> := PolynomialRing(F);
> K<x>, mp := ext<F|X^2+X+1>;
> seq := Eltseq(x^2);
> seq;
[ -1, -1 ]
> Universe(seq) eq F;
true
> S<t> := DifferentialLaurentSeriesRing(Rationals());
> O(t+t^2);
O(t)
> Parent(O(t)) eq S;
true
> trunc := Truncate(t^(-1)+5*t^2 +O(t^4));
> trunc;
t^-1 + 5*t^2
> Parent(trunc) eq S;
true
> seq := Eltseq(trunc);
> seq;
[ 1, 0, 0, 5 ]
> Universe(seq) eq Rationals();
true
> Exponents(trunc);
[ -1 .. 2 ]
The minimal polynomial of the differential field element s
over the base field.
> F<z> := RationalDifferentialField(Rationals());
> P<X> := PolynomialRing(F);
> K<x>, mp := ext<F|X^2+X+1>;
> f := MinimalPolynomial(x^2);
> f;
X^2 + X + 1
> Parent(f) eq P;
true
> g := MinimalPolynomial(x+3/2);
> g;
X^2 + -2*X + 7/4
The image of s under the derivation of the parent of s.
Notice that it can be different to the "usual" derivative, as it
relies on the defined derivation.
Returns the differential of s in the algebraic differential
field F, as a differential in the differential space of
the underlying ring of F.
> F<z> := RationalDifferentialField(Rationals());
> Derivative(z^2 + 7/z);
(2*z^3 - 7)/z^2
> Differential(z);
(1) d(z)
> Differential(1/z+6+5*z);
((5*z^2 - 1)/z^2) d(z)
> S<t> := DifferentialLaurentSeriesRing(Rationals());
> Derivative(5 + 2*t + 3*t^2);
2*t + 6*t^2
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|