Embedded Picard-Vessiot extensions

TL;DR

The paper establishes the existence of Picard-Vessiot extensions within models of certain large, bounded fields with derivations, extending classical differential Galois theory to a broader model-theoretic context.

Contribution

It proves that Picard-Vessiot and strongly normal extensions can be embedded in models of a theory of large fields with derivation, under almost quantifier elimination.

Findings

Existence of Picard-Vessiot extensions in models of T_D.

Embedding of strongly normal extensions in these models.

Extension of differential Galois theory to model-theoretic frameworks.

Abstract

We prove that if T is a theory of large, bounded, fields of characteristic zero, with almost quantifier elimination, and T_D is the model companion of T + "D is a derivation", then for any model U of T_D, and differential subfield K of U whose field of constants is a model of T, and linear differential equation DY = AY over K, there is a Picard-Vessiot extension L of K for the equation which is embedded in U over K Likewise for logarithmic differential equations over K on connected algebraic groups over the constants of K and the corresponding strongly normal extensions of K.