Bounded limit cycles of polynomial foliations of $\mathbb CP^2$

TL;DR

This paper presents a new proof demonstrating that generic polynomial vector fields in complex two-dimensional space have infinitely many bounded, homologically independent limit cycles, improving understanding of their dynamical behavior.

Contribution

The authors introduce a novel proof technique that avoids integral estimates, refines the exceptional set for quadratic fields, and guarantees bounded limit cycles.

Findings

Generic polynomial vector fields in $\\mathbb{C}^2$ have countably many homologically independent limit cycles.

The new proof does not rely on integral estimates, simplifying the analysis.

Limit cycles can be confined within bounded domains, enhancing control over their behavior.

Abstract

In this article we prove in a new way that a generic polynomial vector field in $\mathbb C^2$ possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.