Type: Article

Abelian integrals for polynomials with trivial global monodromy on C2

Muciño-Raymundo, Jesús; Rebollo-Perdomo, Salomón;

Abstract:

We consider infinitesimal perturbations of Hamiltonian differential equations on the complex plane , where H is a polynomial of degree and ? is a non-exact polynomial 1-form of degree n. In order to study these perturbed differential equations, the associated Abelian integrals 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 are polynomial functions of the variable c, and to find sharp explicit upper bounds for the number of their zeros. These upper bounds depend on m, n and the number of the generators of the fundamental group of the generic fibers of H, and they work for several new families of infinitesimal perturbations of Hamiltonian differential equations. Under trivial global monodromy, there exist canonical global generators of the fundamental groups for all the generic fibers of H, which are complex cycles of . As second main result; we compute the number of complex limit cycles of which originate from complex cycles in . Several accurate examples are provided.