HOLOMORPHICALLY CONJUGATE POLYNOMIAL AUTOMORPHISMS OF C2 ARE POLYNOMIALLY CONJUGATE

The uploaded paper proves Theorem A: if two loxodromic (Hénon-type) polynomial automorphisms f,g of A^2 over a subfield K of C are conjugate by a biholomorphism φ, then φ is polynomial and defined over a finite extension of K; moreover there exists ψ ∈ Aut(A^2_K) with ψ ∘ f = g ∘ ψ and deg ψ ≤ 257 (deg f·deg g)^29. The proof proceeds by reducing to Hénon form, showing φ preserves the canonical Green currents/functions up to positive scalars, deriving polynomial growth and hence algebraicity of φ, and then combining a finiteness/field-of-definition argument with an effective birational-conjugacy bound and a lemma that any birational conjugacy between Hénon maps is in fact regular. All of these steps are explicitly present in the text, including the constant 257 in the degree bound and the birational→regular lemma. The candidate solution follows essentially the same architecture (Friedland–Milnor reduction; Bedford–Smillie currents; centralizer finiteness; Blanc–Cantat bound; birational conjugacy is regular), so the core reasoning aligns with the paper. Two minor discrepancies: (i) the candidate asserts G_g^±∘φ = G_f^± (i.e., scale 1), whereas the paper obtains equality up to a positive scalar c and c1=1/c, which suffices for the argument and is what is proved; (ii) the candidate misquotes the explicit constant as 2^{57} instead of the paper’s 257, though this does not affect correctness of existence of a bounded-degree conjugacy. Overall, both present the same proof strategy and conclusions, with the paper’s version being precise on normalizations and constants.