Pseudo-abelian integrals: Unfolding generic exponential case

We consider functions of the form , with Pi, R, and , which are (generalized Darboux) first integrals of the polynomial system MdlogH0=0. We assume that H0 defines a family of real cycles in a region bounded by a polycycle. To each polynomial form η one can associate the pseudo-abelian integrals I(h) of M−1η along γ(h), which is the first order term of the displacement function of the orbits of MdH0+δη=0. We consider Darboux first integrals unfolding H0 (and its saddle-nodes) and pseudo-abelian integrals associated to these unfoldings. Under genericity assumptions we show the existence of a uniform local bound for the number of zeros of these pseudo-abelian integrals. The result is a part of a program to extend Varchenko–Khovanskiiʹs theorem from abelian integrals to pseudo-abelian integrals and prove the existence of a bound for the number of their zeros in function of the degree of the polynomial system only.