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 $\mathcal{F}$ denote the foliation with triple point (assume it to be at the origin), and let $\mathcal{F}_\lambda = \{M_\lambda {dH_\lambda \over H_\lambda} = 0\}$, $M_\lambda$ is a integrating factor, be the close foliation. The main problem is that $\mathcal{F}_\lambda$ can have a small nest of cycles which shrinks to the origin as $\lambda\to 0$. A particular case of this situation, namely $H_\lambda = (x -\lambda)^\epsilon (y - x)^{\epsilon_+} (y + x)^{\epsilon_-}\Delta$ with $\Delta$ non-vanishing at the origin (and generic in appropriate sense).