Linear estimate for the number of zeros of Abelian integrals

TL;DR

This paper establishes a linear upper bound relative to the degree of the polynomial form on the number of real zeros of Abelian integrals over a specific real oval in the plane.

Contribution

It provides the first linear bound on the zeros of Abelian integrals for a particular family of real ovals, advancing understanding of their oscillatory behavior.

Findings

Linear bound proportional to the degree of the polynomial form

Applicable to Abelian integrals over the specific oval x^2 y (1 - x - y) = t

Enhances bounds on zeros of Abelian integrals in real algebraic geometry

Abstract

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