Noetherianity and length of Melnikov functions

Abstract

We study foliations in $\mathbb{C}^2$ given by polynomial deformations of the form $dH+\epsilon \eta=0$, with $\gamma(t)\subset H^{-1}(t)$ a family of cycles. The \emph{Poincar\'e first return map} is of the form $P(t)=t+\sum_j \epsilon^j M_j^\gamma(t).$ The functions $M_j^\gamma$ 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} $n_{\scriptscriptstyle H,\gamma}(k)$, independent of the deformation $\eta$, such that, if $M_j^\gamma\equiv0$, for $j=1,\ldots,n_{ H,\gamma}(k)$, then $M_j^\gamma$ is of orbit length $j-k$, for any Melnikov function $M_j^\gamma$. We call the smallest index with this property just the \emph{Noetherianity index} $\nu_{\scriptscriptstyle H,\gamma}(k)$. 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 $n_{H,\gamma}(k)$ in various nontrivial examples.