Darboux integrable system with a triple point and pseudo-abelian integrals

TL;DR

This paper studies the behavior of pseudo-abelian integrals near a Darboux integrable system with a triple point, proving that the number of zeros remains uniformly bounded under small perturbations.

Contribution

It establishes a uniform bound on the zeros of pseudo-abelian integrals for perturbations of Darboux integrable foliations with a triple point, addressing degeneracies of the third type.

Findings

Number of zeros of pseudo-abelian integrals is uniformly bounded.

Analyzes perturbations near a triple point in Darboux integrable systems.

Provides bounds for small nests of cycles shrinking to the origin.

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).