标题： 阿贝尔积分对于在$\mathbb{C}^2$上具有平凡全局单值性的多项式 显示英文标题

标题： Abelian integrals for polynomials with trivial global monodromy on $\mathbb{C}^2$

摘要： We consider infinitesimal perturbations of Hamiltonian differential equations $dH + \varepsilon \omega =0$ on the complex plane $\mathbb{C}^2$, where $H$ is a polynomial of degree $m+1$ and $\omega$ is a non-exact polynomial 1-form of degree $n$. In order to study these perturbed differential equations, the associated Abelian integrals $I(c)=\int_{\gamma(c)} \omega$ are valuable tools. 我们假设多项式$H$是本原的且具有平凡的全局单值性。 对于这些多项式，W. D. Neumann 和 P. Norbury 提供了在代数等价下的三大类分类。 对这些族的了解使我们能够证明第一个主要结果，即相应的阿贝尔积分$I(c)$是变量$c$的多项式函数，并找到它们零点数量的精确显式上界。 这些上界取决于$m$、$n$以及泛纤维的基本群的生成元数量$H$。 这些上界适用于多个新的哈密顿微分方程的无穷小扰动族。 在平凡的全局单值性下，对于$H$的所有一般纤维的基本群存在规范的全局生成元$BC(H)= \{ \gamma_{\tt i}(c)\}$，它们是$dH=0$的复周期。 作为第二个主要结果；我们计算了从$BC(H)$中的复周期产生的$dH+ \varepsilon\omega=0$的复极限周期的数量。 提供了几个准确的例子。 显示英文摘要

摘要： We consider infinitesimal perturbations of Hamiltonian differential equations $dH + \varepsilon \omega =0$ on the complex plane $\mathbb{C}^2$, where $H$ is a polynomial of degree $m+1$ and $\omega$ is a non-exact polynomial 1-form of degree $n$. In order to study these perturbed differential equations, the associated Abelian integrals $I(c)=\int_{\gamma(c)} \omega$ 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 $BC(H)= \{ \gamma_{\tt i}(c)\}$ 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 $dH+ \varepsilon\omega=0$ which originate from complex cycles in $BC(H)$. Several accurate examples are provided.