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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2508.15925 |
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| _version_ | 1866914000548134912 |
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| author | Muciño-Raymundo, Jesús Rebollo-Perdomo, Salomón |
| author_facet | Muciño-Raymundo, Jesús Rebollo-Perdomo, Salomón |
| contents | We consider infinitesimal perturbations of Hamiltonian differential equations $dH + \varepsilon ω=0$ on the complex plane $\mathbb{C}^2$, where $H$ is a polynomial of degree $m+1$ and $ω$ is a non-exact polynomial 1-form of degree $n$. In order to study these perturbed differential equations, the associated Abelian integrals $I(c)=\int_{γ(c)} ω$ are valuable tools. We assume that the polynomials $H$ are primitive with trivial global monodromy. For these polynomials, W. D. Neumann and P. Norbury provided a classification in three large families, up to algebraic equivalence. The knowledge of these families allows us to prove as first main result, that the respective Abelian integrals $I(c)$ are polynomial functions of the variable $c$, and to find sharp explicit upper bounds for the number of their zeros. The bounds depend on $m$, $n$ and the number of the generators of the fundamental group of the generic fibers of $H$. These upper bounds works for several new families of infinitesimal perturbations of Hamiltonian differential equations. Under trivial global monodromy, there exist canonical global generators $BC(H)= \{ γ_{\tt i}(c)\}$ of the fundamental groups for all the generic fibers of $H$, which are complex cycles of $dH=0$. As second main result; we compute the number of complex limit cycles of $dH+ \varepsilonω=0$ which originate from complex cycles in $BC(H)$. Several accurate examples are provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15925 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abelian integrals for polynomials with trivial global monodromy on $\mathbb{C}^2$ Muciño-Raymundo, Jesús Rebollo-Perdomo, Salomón Dynamical Systems 34M45 We consider infinitesimal perturbations of Hamiltonian differential equations $dH + \varepsilon ω=0$ on the complex plane $\mathbb{C}^2$, where $H$ is a polynomial of degree $m+1$ and $ω$ is a non-exact polynomial 1-form of degree $n$. In order to study these perturbed differential equations, the associated Abelian integrals $I(c)=\int_{γ(c)} ω$ are valuable tools. We assume that the polynomials $H$ are primitive with trivial global monodromy. For these polynomials, W. D. Neumann and P. Norbury provided a classification in three large families, up to algebraic equivalence. The knowledge of these families allows us to prove as first main result, that the respective Abelian integrals $I(c)$ are polynomial functions of the variable $c$, and to find sharp explicit upper bounds for the number of their zeros. The bounds depend on $m$, $n$ and the number of the generators of the fundamental group of the generic fibers of $H$. These upper bounds works for several new families of infinitesimal perturbations of Hamiltonian differential equations. Under trivial global monodromy, there exist canonical global generators $BC(H)= \{ γ_{\tt i}(c)\}$ of the fundamental groups for all the generic fibers of $H$, which are complex cycles of $dH=0$. As second main result; we compute the number of complex limit cycles of $dH+ \varepsilonω=0$ which originate from complex cycles in $BC(H)$. Several accurate examples are provided. |
| title | Abelian integrals for polynomials with trivial global monodromy on $\mathbb{C}^2$ |
| topic | Dynamical Systems 34M45 |
| url | https://arxiv.org/abs/2508.15925 |