ON THE CYCLICITY OF HYPERBOLIC POLYCYCLES

The paper proves that the cyclicity of a hyperbolic polycycle Γ_n is at least Δ(Γ_n) in both the smooth and polynomial settings (Theorem 1), introducing Δ(Γ_n) via partial products of the hyperbolicity ratios and maximizing over permutations, and supporting the analysis with parameter-regular Dulac maps and a non-subsequent-saddle displacement construction (Sections 2–3) . It also records a nested first-return equation (56) consistent with Dulac-composition asymptotics and leverages polynomial approximation of bumps for the polynomial case . The candidate solution reaches the same lower bound using a different but compatible strategy: it builds a multi-parameter displacement expansion for the first return map with leading monomials s^{R_{i,σ}} and isolates Δ(Γ_n,σ) simple zeros via scale separation and sign control, invoking the same Dulac-regularity input and a bypass to realize any ordering of exponents. The paper’s proof proceeds by iteratively ‘casting out’ saddles to produce sub-polycycles of alternating stability, using Poincaré–Bendixson to create the cycles step-by-step (a different construction) . Minor details the model sketches but does not fully justify—such as precise handling of the r_i=1 case and uniform positivity/sign control of coefficients—are addressed in the paper through tailored displacement maps and invariant-region arguments (e.g., Proposition 7) . Overall, the theorem statements align and the approaches are mathematically consistent, but the proofs are materially different.