Linear estimate for the number of zeros of Abelian integrals

Abstract

We prove a linear in $\mathrm{deg}\ddot{I}‰$ upper bound on the number of real zeros of the Abelian integral $I(t)={\int }_{\hat{I}´(t)}\ddot{I}‰$, where $\hat{I}´(t)\subset {\mathbb{R}}^2$ is the real oval $x^{2}y(1-x-y)=t$ and $\ddot{I}‰$ is a one-form with polynomial coefficients.