Differential Galois cohomology and parameterized Picard-Vessiot extensions

Abstract

Assuming that the differential field $(K,\delta )$ is differentially large, in the sense of León Sánchez 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 $H_{\delta }^1(K,G)$ 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.