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New solutions of the Poincaré Center Problem in degree 3

Hans-Christian von Bothmer

TL;DR

This work advances the Poincaré center problem for planar polynomial vector fields by integrating Darboux integrability with the inverse problem for plane curves. It introduces an invariant based on singularity data, enabling a quasi-homogeneous–based criterion to certify Darboux integrability and construct first principal components of the degree $3$ center variety. Through equi-singular curve configurations and careful dimension counts, the authors produce several explicit $9$-dimensional families that form new codimension $9$ components, some of which extend known Steiner ideals and Żoła̧dek’s families. The combination of theoretical framework and computer-assisted constructions significantly deepens the understanding and classification of centers in degree $3$, offering both new tools and concrete examples for broader center-variety analysis.

Abstract

Let $ω$ 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.

New solutions of the Poincaré Center Problem in degree 3

TL;DR

This work advances the Poincaré center problem for planar polynomial vector fields by integrating Darboux integrability with the inverse problem for plane curves. It introduces an invariant based on singularity data, enabling a quasi-homogeneous–based criterion to certify Darboux integrability and construct first principal components of the degree center variety. Through equi-singular curve configurations and careful dimension counts, the authors produce several explicit -dimensional families that form new codimension components, some of which extend known Steiner ideals and Żoła̧dek’s families. The combination of theoretical framework and computer-assisted constructions significantly deepens the understanding and classification of centers in degree , offering both new tools and concrete examples for broader center-variety analysis.

Abstract

Let 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.
Paper Structure (6 sections, 15 theorems, 111 equations, 5 figures, 3 tables)

This paper contains 6 sections, 15 theorems, 111 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singularities of Plane Algebraic Curves
  4. Singularities and the Inverse Problem
  5. Singularities and Darboux Integrability
  6. Constructions

Key Result

Theorem 2.3

Let $\omega$ be a differential form, $C_1,\dots,C_r$ algebraic integral curves of $\omega$ and $K_1,\dots,K_r$ their cofactors.

Figures (5)

  • Figure 1: Curve configuration for Construction \ref{['c96']}
  • Figure 2: Curve configuration for Construction \ref{['c98']}
  • Figure 3: Curve configuration for Construction \ref{['c99']}
  • Figure 4: Curve configuration for Construction \ref{['c910']}
  • Figure 5: Curve configuration used in Construction \ref{['c914']}

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Darboux 1878
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • ...and 42 more