ON THE DYNAMICAL BOGOMOLOV CONJECTURE FOR FAMILIES OF SPLIT RATIONAL MAPS

The paper proves precisely the claimed uniform bound: for a split polarized endomorphism Φ on (P^1)^2 over K=Q(B) and a curve C not weakly Φ‑special, there exist ε>0 and M>0 (depending only on Φ and C) such that for all but finitely many t∈B0(Q), #{z∈Ct(Q): ĥΦt(z)<ε} ≤ M; in particular, #Ct(Q)∩Prep(Φt) ≤ M (Theorem 1.2) . The paper’s proof proceeds via a relative/unlikely-intersections route (Theorem 1.6, equidistribution, and a measure-theoretic characterization of weakly special curves), together with a height–on–total–space inequality (Theorem 1.9) stating ĥΦ(P) ≥ c1 hB(π(P)) − c2 on XΦ,⋆(Q) . The candidate solution gives a shorter argument: invoke the uniform height inequality off a proper closed exceptional set Y in the total space and then count points in fibers Yt to get a uniform bound M, excluding finitely many parameters of small base height and the finitely many vertical components. This argument matches the conclusion of Theorem 1.2. Minor gaps in the candidate’s write‑up are easily patched by the paper: (i) the height inequality needs the isotrivial case, which the paper provides for curves (Theorem 4.9) ; (ii) the finiteness of horizontal special multisections inside C is guaranteed in the proof of Theorem 1.9 (only finitely many maximal special subvarieties dominate the base) . Thus, both are correct, but the proofs are materially different.