Differential Operators of Algebraic Functions
An algebraic function \(g\) in a differential field extension \(M/F\) satisfies a linear differential equation \(L(y)=0\) with coefficients in \(F\subset M\). If the minimal polynomial of \(g\) over \(F\) is of degree \(n\), then the order of \(L\) is at most \(n\).
- DifferentialOperator(f): RngUPolElt -> RngDiffOpElt
Given the irreducible polynomial \(f(X)\in F[X]\), 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 \(f(X)\in F[X]\), 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 \(g,\delta_M(g),\ldots,\delta_M^n(g)\). The linear relation between these elements involving the lowest powers of \(\delta_M^i\) gives exactly the desired monic differential operator after a suitable normalization.