Restricted version of the infinitesimal Hilbert 16th problem

Abstract

The paper deals with the {\it infinitesimal Hilbert 16th problem}: to find an upper estimate of the number of zeros of an Abelian integral regarded as a function of a parameter. In more details, consider a real polynomial $ H$ of degree $ n+1 $ in the plane, and a continuous family of ovals $γ_t$ (compact components of level curves $ H = t$) of this polynomial. Consider a polynomial 1-form $ω$ with coefficients of degree at most $n.$ Let I(t) = \int_{γ_t} ω. \label{I} The problem is to give an upper estimate of the number of zeros of this integral. We solve a {\it restricted version} of this problem. Namely, the form $ ω$ is {\it arbitrary,}, and the polynomial $ H$, though having an arbitrary degree, is not too close to the hypersurface of degenerate (non ultra-Morse) polynomials. We hope that the solution of the restricted version of the problem is a step to the solution of the complete (nonrestricted) version.