New solutions of the Poincar\'e Center Problem in degree 3

TL;DR

This paper introduces new criteria for Darboux integrability in degree 3 plane systems with quasi-homogeneous singularities, leading to the discovery of previously unknown components of the center variety.

Contribution

It provides novel criteria for Darboux integrability and constructs new components of the center variety in degree 3.

Findings

New criteria for Darboux integrability in degree 3 systems.

Construction of previously unknown components of the center variety.

Application to systems with quasi-homogeneous singularities.

Abstract

Let $\omega$ be a plane autonomous system and C its configuration of algebraic integral curves. If the singularities of C are quasi-homogeneous we we present new criteria that guarantee Darboux integrability. We use this to construct previously unknown components of the center variety in degree 3.