Noetherianity and length of Melnikov functions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917165727219712 |
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| author | Mardesic, Pavao Novikov, Dmitry Ortiz-Bobadilla, Laura Pontigo-Herrera, Jessie |
| author_facet | Mardesic, Pavao Novikov, Dmitry Ortiz-Bobadilla, Laura Pontigo-Herrera, Jessie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20045 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Noetherianity and length of Melnikov functions Mardesic, Pavao Novikov, Dmitry Ortiz-Bobadilla, Laura Pontigo-Herrera, Jessie Classical Analysis and ODEs 34C07, 34C05, 34C08, 14D05, 16P40 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. |
| title | Noetherianity and length of Melnikov functions |
| topic | Classical Analysis and ODEs 34C07, 34C05, 34C08, 14D05, 16P40 |
| url | https://arxiv.org/abs/2512.20045 |