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

Abstract: We consider a multivalued function of the form $H_{\varepsilon}=P_{\varepsilon}^{\alpha_0}\prod^{k}_{i=1}P_i^{\alpha_i}, P_i\in\mathbb{R}[x,y], \alpha_i\in\mathbb{R}^{\ast}_+$, which is a Darboux first integral of polynomial one-form $\omega=M_{\varepsilon}\frac{dH_{\varepsilon}}{H_{\varepsilon}}=0, M_{\varepsilon}=P_{\varepsilon}\prod^{k}_{i=1}P_i$. We assume, for $\varepsilon=0$, 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 $H_{\varepsilon}$ (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.