On the computation of the differential Galois group
- Über die Berechnung der Differential-Galoisgruppe
Rettstadt, Daniel; Hartmann, Julia (Thesis advisor)
Aachen : Publikationsserver der RWTH Aachen University (2014)
Dissertation / PhD Thesis
Aachen, Techn. Hochsch., Diss., 2014
Let F be a field with a derivation d on F. The kernel of d is the field of constants C, which we require to be algebraically closed. A linear differential operator over F is a polynomial in the non-commutative ring F[d]. Every differential operator can be written as a product of irreducible differential operators. Such factorizations are unique up a certain equivalence relation, which is know as similarity. Furthermore every other factorization is obtained by a process we call transposition. To every differential operator L we can assign a differential equation L(y)=0. For a given differential operator L with coefficients in F, there exist differential field extensions of F containing n linearily independent solutions of L(y)=0, which has no new constants. A field that is in some sense minimal among these extensions will be called a Picard-Vessiot extension. For such a Picard-Vessiot extension K, we define the differential Galois group as the group of F-automorphisms of K that commute with the derivation. The differential Galois group has a representation on the C-vector space, which is spanned by the solutions of L(y)=0 in K. As an auxiliary results we present an algorithm that determines whether a transposition of two irreducible differential operators exists and computes it, if possible. Due to this algorithm we can determine all factorizations of linear differential operators, which have no pair of similar, irreducible factors that transpose. The main result is an algorithm that reduces the computation of the representation of the differential Galois group to the problem of computing all right hand factors of a certain, computable differential operator.