Pseudo-Abelian integrals: unfolding generic exponential case

TL;DR

This paper establishes a uniform bound on the number of zeros of pseudo-abelian integrals arising from unfolded integrable polynomial systems with Darboux first integrals, extending classical results to a more general setting.

Contribution

It introduces a bound for the zeros of pseudo-abelian integrals in unfolded systems, generalizing Varchenko-Khovanskii's theorem from abelian to pseudo-abelian integrals.

Findings

Existence of a uniform local bound for zeros of pseudo-abelian integrals.

Extension of classical theorems to more general integrable systems.

Application to systems with generalized Darboux first integrals.

Abstract

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.