On the Number of Isolated Zeros of Pseudo-Abelian Integrals: Degeneracies of the Cuspidal Type

Abstract

We consider a multivalued function of the form $H\_\epsilon =P\_{\epsilon }^{\alpha \_0}{\prod }^k\_i=1P\_i^{\alpha \_i},P\_i\in \mathbb{R}[x,y],\alpha \_i\in {\mathbb{R}}^{\ast }\_+$, which is a Darboux first integral of polynomial one-form $\omega =M\_\epsilon \frac{dH\_\epsilon }{H\_\epsilon }=0,M\_\epsilon =P\_\epsilon {\prod }^k\_i=1P\_i$. We assume, for $\epsilon =0$, that the polycyle $\{H\_0=H=0\}$ has only cuspidal singularity which we assume at the origin and other singularities are saddles. We consider families of Darboux first integrals unfolding $H\_\epsilon$ (and its cuspidal point) and pseudo-Abelian integrals associated to these unfolding. Under some conditions we show the existence of uniform local bound for the number of zeros of these pseudo-Abelian integrals.