Pseudo-Abelian integrals: unfolding generic exponential case

Details Pseudo-Abelian integrals: unfolding generic exponential case Pseudo-Abelian integrals: unfolding generic exponential case

2009-03-15

arxiv.org/abs/0903.2673v1

Mathematics

Dynamical Systems

Scientific paper

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.

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