  We consider an integrable polynomial system with generalized Darboux first
integral H_0. We assume that it defines a family of real cycles in a region
bounded by a polycycle.
  To any polynomial form \eta one can associate the pseudo-abelian integrals
I(h), which is the first order term of the displacement function of the system
perturbed by \eta.
  We consider Darboux first integrals unfolding H_0 (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 part of a program to extend Varchenko-Khovanskii's 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.
