We study foliations in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {C}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> given by polynomial deformations of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$dH+\epsilon \eta =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mi>H</mi> <mo>+</mo> <mi>ϵ</mi> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma (t)\subset H^{-1}(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> <msup> <mi>H</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> a family of cycles. The Poincaré first return map is of the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$P(t)=t+\sum _j \epsilon ^j M_j^\gamma (t).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>t</mi> <mo>+</mo> <msub> <mo>∑</mo> <mi>j</mi> </msub> <msup> <mi>ϵ</mi> <mi>j</mi> </msup> <msubsup> <mi>M</mi> <mi>j</mi> <mi>γ</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The functions <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M_j^\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>M</mi> <mi>j</mi> <mi>γ</mi> </msubsup> </math></EquationSource> </InlineEquation> are called Melnikov functions and are given by iterated integrals of orbit length at most j. We show that, for each <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists a universal Noetherianity index <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n_{\scriptscriptstyle H,\gamma }(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mstyle displaystyle="false" scriptlevel="2"> <mrow> <mi>H</mi> <mo>,</mo> <mi>γ</mi> </mrow> </mstyle> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, independent of the deformation <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>, such that, if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$M_j^\gamma \equiv 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>M</mi> <mi>j</mi> <mi>γ</mi> </msubsup> <mo>≡</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$j=1,\ldots ,n_{ H,\gamma }(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow> <mi>H</mi> <mo>,</mo> <mi>γ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$M_j^\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>M</mi> <mi>j</mi> <mi>γ</mi> </msubsup> </math></EquationSource> </InlineEquation> is of orbit length <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$j-k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>-</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>, for any Melnikov function <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$M_j^\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>M</mi> <mi>j</mi> <mi>γ</mi> </msubsup> </math></EquationSource> </InlineEquation>. We call the smallest index with this property just the Noetherianity index <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\nu _{\scriptscriptstyle H,\gamma }(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ν</mi> <mstyle displaystyle="false" scriptlevel="2"> <mrow> <mi>H</mi> <mo>,</mo> <mi>γ</mi> </mrow> </mstyle> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. To prove this theorem, we develop a structure theorem for Melnikov functions and use the Ritt–Raudenbush differential algebra theorem. We calculate the universal Noetherianity index <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$n_{H,\gamma }(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mrow> <mi>H</mi> <mo>,</mo> <mi>γ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in various nontrivial examples.

Noetherianity and Length of Melnikov Functions

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2026年7月20日

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Pavao Mardešić,
Dmitry Novikov,
Laura Ortiz-Bobadilla,
Jessie Pontigo-Herrera

摘要

We study foliations in $\mathbb {C}^2$ $C^{2}$ given by polynomial deformations of the form $dH+\epsilon \eta =0$ $d H + ϵ η = 0$ , with $\gamma (t)\subset H^{-1}(t)$ $γ (t) \subset H^{- 1} (t)$ a family of cycles. The Poincaré first return map is of the form $P(t)=t+\sum _j \epsilon ^j M_j^\gamma (t).$ $P (t) = t + \sum_{j} ϵ^{j} M_{j}^{γ} (t) .$ The functions $M_j^\gamma $ $M_{j}^{γ}$ are called Melnikov functions and are given by iterated integrals of orbit length at most j. We show that, for each $k\in \mathbb {N}$ $k \in N$ , there exists a universal Noetherianity index $n_{\scriptscriptstyle H,\gamma }(k)$ $n_{H, γ} (k)$ , independent of the deformation $\eta $ $η$ , such that, if $M_j^\gamma \equiv 0$ $M_{j}^{γ} \equiv 0$ , for $j=1,\ldots ,n_{ H,\gamma }(k)$ $j = 1, \dots, n_{H, γ} (k)$ , then $M_j^\gamma $ $M_{j}^{γ}$ is of orbit length $j-k$ $j - k$ , for any Melnikov function $M_j^\gamma $ $M_{j}^{γ}$ . We call the smallest index with this property just the Noetherianity index $\nu _{\scriptscriptstyle H,\gamma }(k)$ $ν_{H, γ} (k)$ . 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 $n_{H,\gamma }(k)$ $n_{H, γ} (k)$ in various nontrivial examples.