Inverse Problems

Abstract

In this chapter we continue the investigation of chap. 10 concerning the differential galois theory for special classes of differential modules. Recall that K is a. Differential field such that its field of constants C={a∈K|a′=0} has characteristic 0, is algebraically closed and different from K. Furthermore, C is a full subcategory of the category Diff _K of all differential modules over K, which is closed under all operations of linear algebra, i.e., kernels, cokernels, direct sums, and tensor products. Then C is a neutral tannakian category and thus isomorphic to Repr _G for some affine group scheme G over C. The inverse problem of differential Galois theory for the category C asks for a description of the linear algebraic groups H that occur as a differential galois group of some object in C C. We note that H occurs as a differential galois group if and only if there exists a surjective morphism G→H of affine group schemes over C. The very few examples where an explicit description of G is known are treated in chap. 10. In the present chapter we investigate the, a priori, easier inverse problem for certain categories C. This is a reworked version of [230].