Parameterized D-torsors in differential Galois theory

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.