Galois Groups of Linear Difference-Differential Equations

10/18/2022

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by   Ruyong Feng, et al.

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We study the relation between the Galois group G of a linear difference-differential system and two classes 𝒞_1 and 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 𝒞_1∪𝒞_2 are algebraic subgroups of G, and there is a nonempty subset of 𝒞_1 and a nonempty subset of 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.