DIFFERENTIAL GALOIS COHOMOLOGY AND PARAMETERIZED PICARD-VESSIOT EXTENSIONS

Assuming that the differential field (K, δ) is differentially large, in the sense of Le´on S´anchez and Tressl [18], and “bounded” as a field, we prove that for any linear differential algebraic group G over K, the differential Galois (or constrained) cohomology set H1 δ (K, G) is finite. This applies, among other things, to closed ordered differential fields in the sense of Singer [27], and to closed p-adic differential fields in the sense of Tressl [28]. As an application, we prove a general existence result for parameterized Picard-Vessiot extensions within certain families of fields; if (K, δx, δt) is a field with two commuting derivations, and δxZ = AZ is a parameterized linear differential equation over K, and (Kδx , δt) is “differentially large” and Kδx is bounded, and (Kδx , δt) is existentially closed in (K, δt), then there is a PPV extension (L, δx, δt) of K for the equation such that (Kδx , δt) is existentially closed in (L, δt). For instance, it follows that if the δx-constants of a formally real differential field (K, δx, δt) is a closed ordered δt-field, then for any homogeneous linear δx-equation over K there exists a PPV extension that is formally real. Similar observations apply to p-adic fields.