Linear estimate for the number of zeros of Abelian integrals

Abstract

We prove a linear in $\degω$ upper bound on the number of real zeros of the Abelian integral $I(t)=\int_{δ(t)}ω$, where $δ(t)\subset\R^2$ is the real oval $x^2y(1-x-y)=t$ and $ω$ is a one-form with polynomial coefficients.