Finiteness of Stationary Configurations of the Planar Four-vortex Problem. II

The paper proves that for nonzero vorticities there are finitely many stationary configurations of four planar vortices, completing the collapse case via a Puiseux-series/singularity analysis of a polynomially embedded system (2.10)–(2.11), and then deriving explicit upper bounds using Bézout-type degree counts; see the statement of Theorem 1.3 and the setup around (2.6), (2.10)–(2.11), and the collapse finiteness Theorem 6.1, with quantitative bounds in Section 7 . By contrast, the model’s argument hinges on (i) asserting a Dziobek-type factorization Γ_iΓ_j/|z_i−z_j|^2 = σ + λ A_iA_j for vortex central configurations, and (ii) claiming compactness of the normalized distance-shape space after fixing one distance (s12=1), neither of which is established. The paper does not derive or use such a Dziobek relation for the vortex equations, and the compactness claim is false: fixing one edge length does not bound the remaining distances a priori. The model also claims zero-dimensionality from a heuristic independence check without a rigorous elimination/dimension argument. Hence the paper’s result is correct, while the model’s proof is flawed/incomplete.