2017/04/26 by Thierry Combot, Combot, Thierry
Mathematics · #37J30 #37J35 #37J40 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Meromorphic and Entire Functions #math.DS #msc:37J30 #msc:37J35 #msc:37J40
paper · pdf · doi:10.48550/arxiv.1704.08279
33 pages
arxiv created 2017/04/26 · openalex publication_date 2017/04/26 · arxiv updated 2017/04/28 · openalex created_date 2017/05/12 · openalex updated_date 2026/07/28
Let us consider a vector field X meromorphic on a neighbourhood of an algebraic curve Γ⊂ ℙn such that Γ is a particular solution of X. The vector field X is (l,n-l) integrable if it there exists Y1,…,Yl-1,X vector fields commuting pairwise, and F1,…,Fn-l common first integrals. The Ayoul-Zung Theorem gives necessary conditions in terms of Galois groups for meromorphic integrability of X in a neighbourhood of Γ. Conversely, if these conditions are satisfied, we prove that if the first normal variational equation NVE1 has a virtually diagonal monodromy group Mon(NVE1) with non resonance and Diophantine properties, X is meromorphically integrable on a finite covering of a neighbourhood of Γ. We then prove the same relaxing the non resonance condition but adding an additional Galoisian condition, which in fine is implied by the previous non resonance hypothesis. Using the same strategy, we then prove a linearisability result near 0 for a time dependant vector field X with X(0)=0 ∀ t.