Parameterized D-torsors in differential Galois theory

TL;DR

This paper explores the Galois theory of parameterized differential equations on D-torsors using model-theoretic methods, providing new cohomological criteria and extending classical results to a parameterized setting.

Contribution

It introduces a parameterized version of Kolchin's differential cohomology theorem and characterizes Galois extensions of generalized strongly normal extensions via cohomology.

Findings

Parameterized Galois extensions correspond to D-torsors.

A cohomological criterion determines when an extension is a Galois extension.

Extension of classical theorems to the parameterized differential Galois context.

Abstract

In the context of differential fields of characteristic zero with several commuting derivations, we discuss the notion of $\#$-differential equations on parameterized D-torsors and their associated Galois extensions. Using model-theoretic methods, we observe that any generalized strongly normal extension (in the sense of Pillay [14] and, more generally, Le\'on S\'anchez [9]) is the Galois extension of a parameterized D-torsor. Furthermore, we prove a parameterized version of a theorem of Kolchin on differential cohomology, itself of independent interest, and use it to provide a necessary and sufficient cohomological condition for when a generalized strongly normal extension is the Galois extension for a log-differential equation on its Galois group (as a parameterized D-group). We also present general model-theoretic versions of some of the main results.