Darboux Integrability via Singularities of Invariant Curves at Infinity
Hans-Christian von Bothmer
Journal of Singularities
volume 30 (2026), 81-115
Received: 14 October 2025. In revised form: 23 April 2026
Abstract:
We extend the computation of our invariant \eta(\omega,C,a) 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 \eta 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.
Author(s) information:
Hans-Christian von Bothmer
Fachbereich Mathematik der Universität Hamburg
Bundesstrass e 55
20146 Hamburg, Germany
email: hans.christian.v.bothmer@uni-hamburg.de