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Darboux Integrability via Singularities of Invariant Curves at Infinity

Hans-Christian von Bothmer

Abstract

We extend the computation of the invariant $η(ω,C,a)$ defined in arXiv:2409.01751 to special points on the line at infinity and show that, as in the affine case, its value is determined purely by the geometry of the integral curve C. By incorporating points at infinity, the invariant $η$ yields effective geometric criteria that certify Darboux integrability in cases not covered by affine data alone. As an application we construct six new codimension-11 components of the degree-3 center variety

Darboux Integrability via Singularities of Invariant Curves at Infinity

Abstract

We extend the computation of the invariant defined in arXiv:2409.01751 to special points on the line at infinity and show that, as in the affine case, its value is determined purely by the geometry of the integral curve C. By incorporating points at infinity, the invariant yields effective geometric criteria that certify Darboux integrability in cases not covered by affine data alone. As an application we construct six new codimension-11 components of the degree-3 center variety

Paper Structure

This paper contains 33 sections, 15 theorems, 122 equations, 4 tables.

Table of Contents

  1. Preliminaries
  2. The inverse problem
  3. an analytic invariant
  4. computing the invariant at infinity
  5. Zeros outside of integral curves
  6. Review of results in finite characteristic
  7. constructions
  8. $A_2\!+\!A_4$-quartic & contact conic (Steiner type $11_2$)
  9. Configuration.
  10. Darboux integrability.
  11. Explicit construction.
  12. Reduction modulo $p$.
  13. $3A_2$-quartic & $3,3$-contact conic (Steiner type $11_{25}$)
  14. Configuration.
  15. Darboux integrability.
  16. ...and 18 more sections

Key Result

Theorem 1.3

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

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Darboux 1878
  • proof
  • Lemma 1.4
  • proof
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • ...and 48 more