NEWTON POLYGONS AND BÖTTCHER COORDINATES NEAR INFINITY FOR POLYNOMIAL SKEW PRODUCTS

The paper proves that for a polynomial skew product f(z,w)=(p(z),q(z,w)) with deg p=δ≥2 and a Newton-polygon–selected dominant term z^γ w^d of q, there is an invariant wedge U near infinity and a biholomorphic Böttcher conjugacy φ on U to the monomial model f0=(a_δ z^δ, b_γd z^γ w^d), under the hypotheses d≥2 or (d=1 and δ≠T_k) (Proposition 1.1 and Theorem 1.2). The wedge U and its cases (1–4) are explicitly determined by the Newton polygon and the intercepts T_k, and the construction proceeds via careful dominance estimates and a limit of compositions f0^{-n}∘f^n, with blow-ups reducing Cases 2–4 to a Case-1 form (see the case split and statements in the introduction and main results, and the proofs via blow-up in Sections 2–4). These claims and methods are laid out in the paper’s main statements and technical sections . The candidate solution establishes the same conclusion on the same wedge regions using a different, cohomological-equation approach for the fiber coordinate: (i) precise Newton-polygon dominance and forward invariance on U; (ii) a standard 1D Böttcher coordinate φ1 for p; (iii) a solution φ2 via s∘f = G^{-1} s^d, with a weighted series for d≥2 and exponential-decay control for d=1, δ≠T_k; and (iv) near-identity estimates. This aligns with the paper’s hypotheses, region U, and exclusion of the resonant case δ=T_k when d=1 (cf. Remark 1.3) . Two minor gaps in the model’s write-up are: (a) the injectivity/biholomorphy is argued via “log-coordinates + Cauchy estimates,” which is not fully rigorous as written; the standard fix is to construct the inverse as the limit of f^{-n}∘f0^{n}, the method used in the paper (Section 8) ; and (b) for d=1, the claimed exponential decay along orbits needs an explicit inequality deriving from the Newton-polygon drift—this is plausible and consistent with the non-resonant condition, but should be spelled out. With these minor corrections, the model’s proof is valid. Overall: both establish the same result under the same conditions, by different methods.