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An algebraic function g in a differential field extension M/F
satisfies a linear differential equation L(y)=0 with coefficients
in F⊂M.
If the minimal polynomial of g over F is of degree n, then the
order of L is at most n.
Given the irreducible polynomial f(X)∈F[X], return the
monic differential operator over F of minimal degree
to which a formal root of f is a solution.
The field F must be a differential field.
The base ring of the created differential operator is F.
The algorithm used in this function is straightforward.
If g is a root of an irreducible polynomial f(X)∈F[X], where
F is a differential field, then f(g)=0 induces a unique derivation
on g.
The field M=F(g) is an algebraic
differential field extension of F containing all derivatives of g.
If n is the degree of the polynomial f, then M/F is a field
extension of degree n.
This implies that there must be at least one non--trivial linear relation
between g, δM(g), ..., δMn(g).
The linear relation between these elements
involving the lowest powers of δMi gives exactly the
desired monic differential operator after a suitable normalization.
> F<z> := RationalDifferentialField(Rationals());
> _<X> := PolynomialRing(F);
> f := X^3-z;
> L := DifferentialOperator(f);
> L;
$.1 + -1/3/z
> M<alpha> := ext<F|f>;
> R<D> := DifferentialOperatorRing(M);
> Apply(R!L,alpha);
0
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