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

Abstract

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.