Galois groups of linear difference-differential equations

Abstract

We study the relation between the Galois group G of a linear difference-differential system and two classes C1 and C2 of groups that are the Galois groups of the specializations of the linear difference equation and the linear differential equation in this system respectively. We show that almost all groups in C1∪C2 are algebraic subgroups of G, and there is a nonempty subset of C1 and a nonempty subset of C2 such that G is the product of any pair of groups from these two subsets. These results have potential application to the computation of the Galois group of a linear difference-differential system. We also give a criterion for testing linear dependence of elements in a simple difference-differential ring, which generalizes Kolchin's criterion for partial differential fields.