Monodromic Nilpotent Singular Points with Odd Andreev Number and the Center Problem

Claudio Pessoa, Lucas Queiroz

correcthigh confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves that the origin is a center for system (6) iff μ = 0, a50 = 0, and a11 a40 = 0 (Theorem 11), using generalized polar coordinates to get v3(T) ∝ (a11 a40 + a50) and a perturbation-to-nondegenerate-center argument to force a11 = 0 when a40 ≠ 0; sufficiency is by reversibility . The model arrives at the same classification by computing generalized (nilpotent) Lyapunov constants: V1 forces μ = 0, a strictly positive a50^3 term in V7 forces a50 = 0, and then V3 forces a11 a40 = 0; sufficiency again uses reversibility. Both deliver the same necessary and sufficient conditions, though the intermediate constants differ in form (the paper’s v3 depends only on ∫Cs^{10}, while the model’s V3 contains J10 and J16).

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper clarifies the center problem for a family with odd Andreev number and completely settles the n=3 case, identifying the precise center conditions and delineating the limits of the inverse integrating factor method. The approach is technically sound and well framed within established methods (generalized polar coordinates, focal values, and perturbations to nondegenerate centers). Some computations are presented succinctly and could be expanded for self-containment, but the results appear correct and valuable.