2204.08812
The number of relative equilibria in the PCR4PB
Jordi-Lluís Figueras, Warwick Tucker, Piotr Zgliczynski
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves for all ordered masses m in M that the critical equation ∇z V(z;m)=0 has exactly 8, 9, or 10 solutions in the annulus C by a new analytic/computer-assisted framework (partitioning mass space, exclusion regions, Krawczyk operator, and Lyapunov–Schmidt reductions), culminating in Theorem 6. The model reaches the same conclusion via a topological-degree argument on C with small disks excised near p1,p2 and an external enumeration result to bound the number of minima; apart from a correctable technical slip (Δ(1/r)=0 in 2D), the reasoning is coherent. The two approaches are substantively different: the paper’s proof is validated-analytic/computational; the model’s is index/degree-theoretic combined with literature on bifurcations/minima counts.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper provides a clean, validated-analytic proof that recovers the 8/9/10 count without algebraic elimination, using set-valued numerics and Lyapunov–Schmidt reduction. The methodology is likely portable to more complex problems. Documentation of the computational steps is transparent. Minor clarifications on the bifurcation set geometry and a few technical remarks would further improve clarity.