Variation of the Canonical Height in a Family of Polarized Dynamical Systems

The paper’s Theorem 1 proves the asymptotic with power savings: ĥ_{f_t}(P_t) = ĥ_f(P) h_B(t) + O(h_B(t)^{2/3}) in general, and O(h_B(t)^{1/2}) when B is rational, exactly as stated (see the statement and discussion of Theorem 1 and its proof outline, including the optimization over k via d^{3k} ≈ h_B(t) to get 2/3, and the P^1 improvement via d^{2k} ≈ h(t) leading to 1/2) . The method relies on explicit elimination bounds and the curve-height comparison h_{B,D} = h_{B,E} + O(h_{B,E}^{1/2}) for heights of the same degree on a curve, to pass from D(F,P)(t) to ĥ_f(P)h_B(t) with O(h^{1/2}) error on the base . The candidate model gives a different route: it sketches a local-to-global decomposition using adelic local heights and then bounds the resulting truncated proximity/gcd term by subspace-theorem estimates (2/3 on a general curve, 1/2 on P^1). While the model’s outline leaves technical details implicit (notably the precise truncated-proximity theorems invoked and the rigorous construction of the local decomposition and auxiliary functions), its conclusion matches the paper’s theorem and is consistent with known subspace-theoretic bounds. Hence, the two arrive at the same main statement by different methods; the paper’s proof is fully detailed and complete in the uploaded text, and the model’s proof is plausible but would need additional citations and hypotheses to be fully rigorous.