Darboux Integrable System with a Triple Point and Pseudo-Abelian Integrals

Abstract

In this paper we consider the degeneracies of the third type. More exact, the perturbations of the Darboux integrable foliation with a triple point, i.e. the case where three of the curves {P i = 0} meet at one point, are considered. Assuming that this is the only non-genericity, we prove that the number of zeros of the corresponding pseudo-abelian integrals is bounded uniformly for close Darboux integrable foliations. Let F denote the foliation with triple point (assume it to be at the origin), and let F λ = {M λ dH λ H λ = 0}, M λ is a integrating factor, be the close foliation. The main problem is that F λ can have a small nest of cycles which shrinks to the origin as λ → 0. A particular case of this situation, namely H λ = (x − λ) ǫ (y − x) ǫ + (y + x) ǫ − ∆ with ∆ non-vanishing at the origin (and generic in appropriate sense). Mathematics subject classification: 34C07, 34C08