Galois Groups of Linear Difference-Differential Equations

We study the relation between the Galois group $G$ of a linear difference-differential system and two classes $\mathcal{C}_1$ and $\mathcal{C}_2$ 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 $\mathcal{C}_1\cup \mathcal{C}_2$ are algebraic subgroups of $G$, and there is a nonempty subset of $\mathcal{C}_1$ and a nonempty subset of $\mathcal{C}_2$ 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.