Galois Groups of Linear Difference-Differential Equations

TL;DR

This paper explores the relationship between the Galois group of a linear difference-differential system and the Galois groups of its specialized difference and differential equations, providing structural insights and a criterion for linear dependence.

Contribution

It establishes that the Galois group of the system contains algebraic subgroups related to the specialized equations and describes its structure as a product of these subgroups, extending Kolchin's criterion.

Findings

Most groups in the union of specialized Galois groups are subgroups of the system's Galois group.

The system's Galois group can be expressed as a product of subgroups from the specialized groups.

A new criterion for testing linear dependence in difference-differential rings is proposed.

Abstract

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.