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2402.13712

ON MULTIPLICATIVE DEPENDENCE BETWEEN ELEMENTS OF POLYNOMIAL ORBITS

Marley Young

correcthigh confidence
Category
math.DS
Journal tier
Strong Field
Processed
Sep 28, 2025, 12:56 AM

Audit review

The paper’s Theorem 1.3 exactly formalizes the equivalence between (A) infinitely many rank‑1 multiplicative dependencies in Of(x) × Og(y) and (B) the existence of integers i,j ≤ 2, coprime k,ℓ, semiconjugacies via power maps P_ℓ, P_k, and a root-of-unity twist ξ so that f̂ and ĝξ share a common iterate; see the statement of Theorem 1.3 and its proof outline in §4.2 . The forward direction (B ⇒ A) in the paper is handled by the explicit identities f^{iN}(X^ℓ)=f̂^N(X)^ℓ and g^{jN}(X^k)=ĝ^N(X)^k, followed by a careful choice of base points (x non-preperiodic for f and y an ℓ-th root of x) and an adjustment for the ξ‑twist to produce infinitely many rank‑1 relations (eq. (4.1) and subsequent lines) . In contrast, the model’s (B ⇒ A) argument incorrectly treats P_k(ĝ^{rn}(u)) and P_k((ĝ)_ξ^{rn}(u)) as interchangeable using “rescaling by ξ commutes with taking powers,” overlooking that the semiconjugacy P_k∘ĝ = g^j∘P_k does not hold for ĝ_ξ and that the ξ‑twist must be absorbed by a careful choice of base points, exactly as done in the paper. The (A ⇒ B) direction in the model mirrors the paper’s core steps: (i) rule out opposite‑sign exponents via specialization and S‑unit finiteness (Proposition 4.3(a)) ; (ii) bound and then fix the exponents k,ℓ and a root of unity ζ using superelliptic bounds (Theorem 2.2 and Proposition 4.3(b)) ; (iii) force a perfect‑power times monomial shape in an iterate unless only finitely many solutions occur (Proposition 4.3(c)) ; and (iv) pass from multiplicative relations to orbit intersections and then to common iterates via GTZ (Theorem 1.1) . The paper’s specialization framework (Proposition 4.1 and Proposition 4.2) ensures preservation of non‑preperiodicity and shape under specialization, which the model cites in spirit but not with the same precision . Overall, the paper is correct and complete; the model’s proof has a critical flaw in the (B ⇒ A) step regarding the ξ‑twist and semiconjugacy.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The work gives a sharp and natural extension of the GTZ theory of orbit intersections to multiplicative rank‑1 constraints. The classification via power semiconjugacies and a root‑of‑unity twist is compelling and useful. The paper is technically solid, with well-chosen tools (specialization, S‑unit finiteness, superelliptic bounds, and GTZ). Minor clarifications around the ξ‑twist in the (B ⇒ A) direction and explicit emphasis on the i,j ≤ 2 caveat would further aid readability.