Effective descent for differential operators

الأصل · EN

A theorem of N. Katz Ka p.45, states that an irreducible differential operator L over a suitable differential field k, which has an isotypical decomposition over the algebraic closure of k, is a tensor product L=M⊗ₖ N of an absolutely irreducible operator M over k and an irreducible operator N over k having a finite differential Galois group. Using the existence of the tensor decomposition L=M⊗ N, an algorithm is given in C-W, which computes an absolutely irreducible factor F of L over a finite extension of k. Here, an algorithmic approach to finding M and N is given, based on the knowledge of F. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields k which are C₁-fields.