Real Liouvillian Extensions of Partial Differential Fields

Abstract

In this paper, we consider real partial differential fields with a real closed field of constants. For an integrable differential system defined over such a field, we prove that there exists a real Picard-Vessiot extension. Moreover, two real Picard-Vessiot extensions for the same system with orderings restricting to the same ordering on the base field are isomorphic as differential extensions. We obtain a Galois fundamental theorem in this setting. We characterise real Liouvillian extensions of real partial differential fields by means of split solvable linear algebraic groups. We present some examples of real dynamical systems showing the value of developing Picard-Vessiot theory for real partial differential fields.