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]. KeywordsGalois GroupFull SubcategoryDifferential ModuleLinear Algebraic GroupZariski TopologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.