Limit cycles appearing from the perturbation of a cubic isochronous center

The paper proves the bound by rewriting the Abelian integral via polar coordinates, reducing it to integrals L_k(h) with a known structure, and then controlling zeros after one differentiation; this yields at most 4⌊(n+1)/2⌋+1 zeros on (0,1) and hence the same bound for limit cycles. The key steps and identities (first integral H, integrating factor μ, r(θ), use of Green’s formula, reduction to L_k, and the final derivative structure) are all present and consistent with standard results and cited literature, even if some zero-counting details are condensed . By contrast, the candidate solution hinges on a “reduction lemma” claiming that every integral ∫ cos^{2p}θ sin^{2q}θ/(A cos^2θ + B sin^2θ)^2 dθ can be written as P(A,B)S0+Q(A,B)S1 with polynomial P,Q. This is false as stated: for instance, even J(A,B)=∫ dθ/(A cos^2θ+B sin^2θ)=2π/√(AB) cannot be expressed as a polynomial combination of S0 and S1 since any polynomial combination (A+B)P+(B−A)Q necessarily vanishes at (A,B)=(0,0), so it cannot represent constants; more generally, the asserted polynomial reduction would require identities that do not hold. Because the bound on denominators and the degree estimate in the candidate’s proof depend on this incorrect lemma, the model’s solution is not valid.