Differential Galois cohomology and parameterized Picard-Vessiot extensions

Abstract

Assuming that the differential field $(K,\delta)$ is differentially large, in the sense of Le\'on S\'anchez and Tressl, and "bounded" as a field, we prove that for any linear differential algebraic group $G$ over $K$, the differential Galois (or constrained) cohomology set $H^1_\delta(K,G)$ is finite. This applies, among other things, to closed ordered differential fields $K$, in the sense of Singer, and to closed $p$-adic differential fields in the sense of Tressl. As an application, we prove a general existence result for parameterized Picard-Vessiot extensions within certain families of fields; if $(K,\delta_x,\delta_t)$ is a field with two commuting derivations, and $\delta_x Z = AZ$ is a parameterized linear differential equation over $K$, and $(K^{\delta_x},\delta_t)$ is "differentially large" and $K^{\delta_x}$ is bounded, and $(K^{\delta_x}, \delta_t)$ is existentially closed in $(K,\delta_t)$, then there is a PPV extension $(L,\delta_x,\delta_t)$ of $K$ for the equation such that $(K^{\delta_x},\delta_t)$ is existentially closed in $(L,\delta_t)$. For instance, it follows that if the $\delta_x$-constants of a formally real differential field $(K,\delta_x,\delta_t)$ is a closed ordered $\delta_t$-field, then for any homogeneous linear $\delta_x$-equation over $K$ there exists a PPV extension that is formally real. Similar observations apply to $p$-adic fields.