UNICRITICAL POLYNOMIAL MAPS WITH RATIONAL MULTIPLIERS

The paper proves that a unicritical polynomial with all rational multipliers is affinely conjugate to a power map or a Chebyshev map by passing to the standard unicritical family fc(z)=z^d+c, using dynatomic/multiplier polynomials, the splitting criterion (Proposition 10 and Corollary 42), and small-period constraints: quadratic needs periods up to 4 (Proposition 17), cubic uses periods 1–2 together with a real-multiplier reduction (Proposition 23), and degree ≥4 is forced by period 1 real multipliers (Proposition 25) to be a power map; this yields Theorem 6 (power or Chebyshev), and in the unicritical setting Chebyshev occurs only for d=2 (Remark 7). These elements are all present in the uploaded paper (Theorem 6 and setup via fc: ; multiplier/dynatomic framework: , ; quadratic case: ; cubic case: ; degree ≥4: ). The model’s solution follows this same skeleton and arrives at the same classification, but contains normalization errors (e.g., using F_α(z)=(z/α)^d+α with “critical point α,” wrong uniqueness up to a d-th root instead of a (d−1)-th root, and misstatements about the special parameters for the quadratic Chebyshev case) and a minor overstatement about discriminants. These do not affect the final classification, which matches the paper.