2025/12/23 by Pavao Mardešić, Dmitry Novikov, Mardesic, Pavao +5
Mathematics · #14D05 #16P40 #34C05 #34C07 #34C08 #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Meromorphic and Entire Functions
paper · doi:10.48550/arxiv.2512.20045
openalex publication_date 2025/12/23 · openalex created_date 2025/12/25 · openalex updated_date 2026/07/28
We study foliations in ℂ2 given by polynomial deformations of the form dH+εη=0, with γ(t)⊂ H-1(t) a family of cycles. The Poincaré first return map is of the form P(t)=t+∑j εj Mjγ(t). The functions Mjγ are called Melnikov functions and are given by iterated integrals of orbit length at most j. We show that, for each k∈ℕ, there exists a universal Noetherianity index n\scriptscriptstyle H,γ(k), independent of the deformation η, such that, if Mjγ≡0, for j=1,…,n H,γ(k), then Mjγ is of orbit length j-k, for any Melnikov function Mjγ. We call the smallest index with this property just the Noetherianity index ν\scriptscriptstyle H,γ(k). In order to prove this theorem, we develop a structure theorem for Melnikov functions and use the Ritt-Raudenbush differential algebra theorem. We calculate the universal Noetherianity index nH,γ(k) in various nontrivial examples.