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Nilpotence of Orbits under Monodromy and the Length of Melnikov Functions

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

Abstract

Let $F\in\mathbb{C}[x,y]$ be a polynomial, $γ(z)\in π_1(F^{-1}(z))$ a non-trivial cycle in a generic fiber of $F$ and let $ω$ be a polynomial $1$-form, thus defining a polynomial deformation $dF+εω=0$ of the integrable foliation given by $F$. We study different invariants: the orbit depth $k$, the nilpotence class}$n$, the derivative length}$d$ associated with the couple $(F,γ)$. These invariants bound the length $\ell$ of the first nonzero Melnikov function of the deformation $dF+εω$ along $γ$. We study in detail a simple example of a polynomial $F$ given as product of four lines. We show how these invariants vary depending on the relative position of the four lines and relate it also to the length of the corresponding Godbillon-Vey sequence. We formulate a conjecture motivated by the study of this example.

Nilpotence of Orbits under Monodromy and the Length of Melnikov Functions

Abstract

Let be a polynomial, a non-trivial cycle in a generic fiber of and let be a polynomial -form, thus defining a polynomial deformation of the integrable foliation given by . We study different invariants: the orbit depth , the nilpotence class}, the derivative length} associated with the couple . These invariants bound the length of the first nonzero Melnikov function of the deformation along . We study in detail a simple example of a polynomial given as product of four lines. We show how these invariants vary depending on the relative position of the four lines and relate it also to the length of the corresponding Godbillon-Vey sequence. We formulate a conjecture motivated by the study of this example.
Paper Structure (7 sections, 6 theorems, 30 equations, 1 figure)

This paper contains 7 sections, 6 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction, Main Results and Conjectures
  2. Nilpotence class and derivative length
  3. Germs of diffeomorphisms
  4. Proof of Theorem \ref{['Thm:1']}
  5. Types of integrability of the deformations
  6. Degree on $F$ equals 1
  7. Proof of Theorem \ref{['Thm:2']}

Key Result

Proposition 1.2

Let $F\in\Bbb{R}[x,y]$ be a polynomial, $\gamma\in\pi_1(F^{-1}(z))$ be a real cycle and $\omega$ a polynomial $1$-form as above, and let $\pi_1^{\mathcal{O}-ab}$ be the group defined in G. Let $n$ be the nilpotence class of $\pi_1^{\mathcal{O}-ab}$, $d$ the derivative length of $\pi_1^{\mathcal{O}-a

Figures (1)

  • Figure 1: The real loops $\gamma(z)$ and $\gamma_1(z)$ and the complex vanishing loops $\delta_i(z)$ as elements of $\pi_1\left(F^{-1}(z),p_0\right)$.

Theorems & Definitions (18)

  • Conjecture 1.1
  • Proposition 1.2
  • Remark 1.3
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Conjecture 1.9
  • Definition 2.1
  • proof : Proof of Propostion \ref{['prop:ineq']}
  • ...and 8 more