CHARACTERIZATION OF CENTERS BY ITS COMPLEX SEPARATRICES

The paper’s Theorem 2 states that for a real-analytic planar vector field with a monodromic singularity, and any real-analytic invariant curve F with cofactor K, the principal value integral ∫_{γ̂_{ρ0}} K̂ exists and the origin is a center iff that integral vanishes for all small ρ0. The proof proceeds via a weighted polar blow-up, reduction to the ϕ-time desingularized field X̂=∂ϕ+F(ϕ,ρ)∂ρ, and the identity X̂(F̂)=K̂F̂, yielding K̂ as the ϕ-derivative of log|F̂| along orbits. After excising neighborhoods of finitely many angles where Θ=0, the PV integral equals log|F̂(2π,ρ(2π;ρ0))|−log|F̂(0,ρ0)|, i.e., log|F̂(0,Π(ρ0))|−log|F̂(0,ρ0)|, using 2π-periodicity (formula (23) in the paper). This is exactly the route taken in the paper’s Section 5.1, including the PV construction and the reduction to the Poincaré map Π (proof steps around (19)–(23) and the PV argument) . The candidate solution follows the same structure: the same blow-up and desingularization, the same derivative-of-log identity, the same PV excision argument, and the same linkage to the return map. Where the candidate improves the paper is in justifying the implication “PV integral ≡ 0 ⇒ Π=Id”: it explicitly proves injectivity of ρ ↦ F̂(0,ρ) for small ρ from its Puiseux-type leading term cρ^m+O(ρ^{m+1}), hence strictly monotone on (0,ε), which forces Π(ρ0)=ρ0 from f(Π(ρ0))=f(ρ0). The paper states only that F̂(0,ρ) “depends on ρ” and concludes Π(ρ0)=ρ0 “clearly,” a step that benefits from the model’s monotonicity/injectivity clarification. All other key ingredients—choice of (p,q)∈W(N(X)), the explicit expression K̂=D(ϕ)K(ρ^p cosϕ,ρ^q sinϕ)/(ρ^rΘ), and the existence of a uniform band S^1×(0,m] on which F̂≠0—are common to both arguments and are established in the paper (Theorem 2 statement and setup; blow-up and orbit equation; explicit K̂; existence of S^1×I; PV identity) .