Noetherianity and length of Melnikov functions
Pavao Mardesic, Dmitry Novikov, Laura Ortiz-Bobadilla, Jessie Pontigo-Herrera
TL;DR
The paper addresses the behavior of Melnikov functions for polynomial perturbations of foliations in ℂ^2 given by dH+εη=0, focusing on the Poincaré return map and the iterated-integral structure of Melnikov functions M_j^γ. It introduces a universal Noetherianity phenomenon: for any k there exists a universal index n_{H,γ}(k) such that vanishing of the first n_{H,γ}(k) Melnikov functions forces a uniform bound on the orbit-lengths of all Melnikov functions, independent of η, with the minimal such index denoted ν_{H,γ}(k). The main technique combines a structure theorem expressing Melnikov functions on an orbit in terms of a finite set of basic Melnikov functions and their derivatives with a differential-algebraic Noetherianity argument (Ritt-Raudenbush) applied to a universal holonomy algebra; a universal holonomy 𝒫 and differential-operator formalism yield a lower-triangular Toeplitz matrix representation of perturbations. The paper also develops a universal Lie-algebraic perspective for the principal diagonal, linking Melnikov data to a universal homomorphism gr π_1 → 𝔞 ⊂ 𝔄_1, and provides several explicit examples that compute n_{H,γ}(k) (with values like k, k+1, k+2 in different configurations) and discuss sharpness and potential conjectures. Overall, the work advances a Noetherian, universal framework for the length and structure of Melnikov functions, with implications for center-focus problems and Hilbert-type questions in perturbations of Hamiltonian foliations.
Abstract
We study foliations in $\mathbb{C}^2$ given by polynomial deformations of the form $dH+εη=0$, with $γ(t)\subset H^{-1}(t)$ a family of cycles. The \emph{Poincaré first return map} is of the form $P(t)=t+\sum_j ε^j M_j^γ(t).$ The functions $M_j^γ$ are called \emph{Melnikov functions} and are given by \emph{iterated integrals of orbit length} at most $j$. We show that, for each $k\in\mathbb{N}$, there exists a \emph{universal Noetherianity index} $n_{\scriptscriptstyle H,γ}(k)$, independent of the deformation $η$, such that, if $M_j^γ\equiv0$, for $j=1,\ldots,n_{ H,γ}(k)$, then $M_j^γ$ is of orbit length $j-k$, for any Melnikov function $M_j^γ$. We call the smallest index with this property just the \emph{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 $n_{H,γ}(k)$ in various nontrivial examples.