Some Families of Polynomials Satisfying Uniform Boundedness for Rational Preperiodic Points

The paper proves Theorem 1.1: for d ≥ 2 and ψ ∈ Q(t) whose pole divisor has irreducible support with at least two geometric points, the family F_{d,ψ} = {z^d + ψ(c) : c ∈ Q} has the uniform boundedness (UB) property. The proof proceeds by (i) writing ψ(y/x) = G(x,y)/(A·H(x,y)^n) with H irreducible of degree ≥ 2; (ii) using Chebotarev to find infinitely many primes ℓ such that H mod ℓ has no linear factor, which implies uniform good reduction for z^d+ψ(c) at those ℓ for all c ∈ Q; (iii) applying Morton–Silverman’s two-prime local–global period bound to get a universal bound N=(p^2−1)(q^2−1) on exact rational periods; and (iv) invoking Doyle–Poonen to deduce a uniform bound on the number of rational preperiodic points. These steps are explicitly stated in the note (Lemma 2.1 and the argument thereafter, including the choice N=(p^2−1)(q^2−1) via Corollary B of Morton–Silverman) and match the candidate solution’s outline and details. The candidate’s proof mirrors the paper’s method, with the only difference that it leaves N unspecified rather than using the explicit (p^2−1)(q^2−1). Otherwise, the logic and hypotheses align and are correct, and no missing assumptions are identified. See the statement of Theorem 1.1 and Lemma 2.1, as well as the use of Corollary B to bound periods by N, in the uploaded paper .