Functional Equations in Formal Power Series

Fedor Pakovich

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Theorem 1.2 proves (1)⇔(2)⇔(3) cleanly using Böttcher coordinates, transition functions, and a short coefficient-argument to get (2)⇒(3) . The model’s proof correctly establishes (1)⇒(2) and (2)⇒(1), but its key Lemma (P3) used to deduce (1)⇒(3) is false in general: one can have U∘V = z^{nm} with U,V not monomials (so P3 fails), breaking that step. The paper’s route from (2)⇒(3) via Lemma 3.4 (transition functions are iterates) and a leading-coefficient comparison is sound and avoids this flaw .

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript develops a clean and effective method to treat functional equations in the semigroup of superattracting formal series using Böttcher coordinates and transition functions. The central equivalence (Theorem 1.2) is proved with standard, transparent steps after normalization and a concise iterate/leading-coefficient comparison. The presentation is logically tight, assumptions are clearly specified (algebraically closed field of characteristic zero), and the work interfaces well with decomposition theory and symmetry characterizations.