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

TL;DR

This paper proves that Abelian integrals associated with certain polynomials are polynomial functions and provides explicit bounds on their zeros, aiding the study of perturbed Hamiltonian systems with trivial global monodromy.

Contribution

It establishes that Abelian integrals for polynomials with trivial global monodromy are polynomial functions and derives explicit bounds on their zeros, extending understanding of Hamiltonian perturbations.

Findings

Abelian integrals are polynomial functions of the parameter c.

Explicit upper bounds for the number of zeros of Abelian integrals are provided.

The results apply to several new families of Hamiltonian perturbations.

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.