Some Applications of Dynamical Belyi Polynomials

The paper’s Main Theorem explicitly characterizes when a PCF polynomial of degree d has persistent bad reduction at p: exactly when d = p^n·ℓ with ℓ > p and p∤ℓ; otherwise all PCF polynomials of degree d have potential good reduction at p. This matches the candidate’s conclusion word-for-word. The forward direction in the paper uses dynamical Belyi polynomials Bd,k, Benedetto’s normalization lemma, and a Kummer-theoretic valuation check to force a negative p-adic valuation after scaling; see the Main Theorem statement and Proposition 2.4 with its proof via Kummer’s theorem and normalization (including Lemma 2.2) . The reverse implication in the paper proceeds by a Newton polygon analysis of f(z)−z and f′(z), bounding fixed and critical points at a maximal radius and using a translation argument to derive a contradiction, thus establishing potential good reduction when ℓ < p . The candidate’s solution follows the same blueprint (Belyi polynomials for existence; Newton polygon and a pigeonhole/translation argument for non-existence), hence is substantially the same proof. Two technical issues in the candidate’s write-up: (i) it incorrectly asserts vp( (d choose k) ) > 0 in the chosen case, whereas the paper shows vp( (d choose k) ) = 0 and uses vp( (d−1 choose k) ) ≥ 1 to force negativity; this is corrected in the paper’s computation ; and (ii) the “shift-by-one” rule for the derivative is invoked without full justification, while the paper replaces it by a precise slope argument ensuring equality conditions and count comparisons . These do not affect the final equivalence, so both are correct, with the paper providing the rigorous version.