DARBOUX THEORY OF INTEGRABILITY FOR REAL POLYNOMIAL VECTOR FIELDS ON THE n−DIMENSIONAL ELLIPSOID

Jaume Llibre, Adrian C. Murza

correctmedium confidence

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

Audit review

The paper’s Theorem 6 states exactly the same upper bound for the number of invariant meridians and points to the extactic-polynomial proof used on the sphere, adapted to En (they cite their earlier S^n work) . The candidate solution reproduces that extactic argument in detail, adding a linear conjugacy En ↔ S^n to reduce to the already-settled spherical case. The hypotheses (finite number of invariant meridians; transversality) match the paper’s framework of invariance on a regular algebraic hypersurface and use of the extactic polynomial . No logical gaps or contradictions were found.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript extends established Darboux integrability techniques from S\^n to the ellipsoid E\_n and records sharp upper bounds for invariant meridians and parallels. The main meridian bound follows by the same extactic-polynomial method used on the sphere; however, the proof is deferred with a citation to prior work. Given the centrality of this bound in the ellipsoid setting, a brief, self-contained proof (or an explicit linear reduction argument) would improve clarity. Examples in E\_2 illustrating sharpness and real-vs-complex phenomena are welcome.