Real Liouvillian extensions of partial differential fields

The paper proves existence and uniqueness (up to K-differential isomorphism) of a real Picard–Vessiot extension for integrable linear partial differential systems over a real partial differential field with real closed constants (Theorem 2.1.3) by reducing to the ordinary case via Kolchin’s D-trick and then descending from a complexified Picard–Vessiot extension using complex conjugation . The candidate solution instead gives a direct descent over K(i) using Hilbert 90 for GL_r to produce a conjugation-fixed fundamental matrix and then shows the fixed field is the desired real Picard–Vessiot field, with uniqueness obtained by a conjugation-compatible isomorphism. Both arguments establish the same theorem; the paper’s proof sketch is shorter and references prior results, while the model’s proof fills in the conjugation descent step explicitly (which the paper only alludes to when asserting a fundamental matrix with entries in the fixed field exists). Minor notational slips and an under-justified descent step in the paper (existence of a fundamental matrix in the fixed field) are easily repaired by the model’s Hilbert 90 argument. Overall, both are correct, with different proof routes .