Noetherianity and length of Melnikov functions

Abstract

We study foliations in ${\mathbb{C}}^2$ given by polynomial deformations of the form $dH+\hat{I}\mu \hat{I}\cdot =0$, with ${\hat{I}}^3(t)\subset H^{-1}(t)$ a family of cycles. The \emph{Poincaré first return map} is of the form $P(t)=t+{\sum }_j\hat{I}{\mu }^j{M}_j^3(t).$ The functions $M_j^3$ 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} $n_{H,{\hat{I}}^3}(k)$, independent of the deformation $\hat{I}\cdot$, such that, if $M_j^3\equiv 0$, for $j=1,\dots ,n_{H,{\hat{I}}^3}(k)$, then $M_j^3$ is of orbit length $j-k$, for any Melnikov function $M_j^3$. We call the smallest index with this property just the \emph{Noetherianity index} $\hat{I}{½}_{H,{\hat{I}}^3}(k)$. 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 $n_{H,{\hat{I}}^3}(k)$ in various nontrivial examples.

22 pages and 4 figures