2022/03/01 by Andy R. Magid, Magid, Andy R.
Computer Science · Mathematics · #12H05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2203.00705
openalex publication_date 2022/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F be a differential field with field of constants C. We assume C to be algebraically closed and of characteristic 0. The complete Picard--Vessiot closure of F is a differential field extension of F with the same constants C as F, which has no Picard--Vessiot extensions, and is minimal over F with these properties. There is a correspondence between subfields of the complete Picard--Vessiot closure and subgroups of its differential automorphism group, which arises because the complete Picard--Vessiot closure comes from F via repeated Picard--Vessiot extensions. This correspondence also obtains for certain normal subfields of the complete Picard--Vessiot closure, fields which can be characterized independently of their embedding in the complete Picard--Vessiot closure.