Hölder Estimates and Uniformity in Arithmetic Dynamics

The paper’s Theorem A explicitly asserts a dense Zariski-open V ⊂ Rat_{d1} × Rat_{d2} and an integer N = N(d1,d2) with #(Preper(f) ∩ Preper(g)) ≤ N for all (f,g) ∈ V(C), exactly the statement the candidate gives via “take V = S \ Z” with Z proper closed. The proof strategy in the paper proceeds by establishing a uniform positive lower bound on the Arakelov–Zhang pairing on a dense open set, and combining this with a quantitative energy/height inequality to bound the size of a set of common preperiodic points; see the theorem statement and proof outline in the introduction and Sections 5–6. The candidate’s explanation mirrors this logic and cites the same core result. In short, the model’s solution is a faithful restatement of the paper’s main result (with a sketch of the mechanism), not a distinct proof. Key matches: the main theorem statement (Theorem A) , the two-step proof schema (nonvanishing measure and pairing positivity; Hölder/height inequalities) , the quantitative inequality (Theorem B) , and the final deduction of a uniform bound via a positive pairing on a Zariski-open set .