QUADRATIC RATIONAL MAPS WITH INTEGER MULTIPLIERS

The paper proves that if a quadratic rational map f has all multipliers for cycles of periods up to 5 in the ring RD of an imaginary quadratic field, then f is conjugate to a power map, a Chebyshev map, or a Lattès map (Theorem 9) . The proof uses multiplier/dynatomic polynomials together with the holomorphic fixed-point identity 1/(1−λ1)+1/(1−λ2)+1/(1−λ3)=1, a reduction via moduli parameters (σ1,σ2), and a finite case analysis split into three lemmas: (i) no superattracting or multiple fixed point ⇒ power or Lattès (Lemma 28), achieved by enumerating fixed-point multipliers from RD and testing splitting of M3, M4, M5, with non-Lattès/non-power triples excluded by irreducibility/degree arguments over quadratic extensions ; (ii) with a superattracting fixed point, working in the polynomial normal form z↦z^2+c and using discriminants and splitting of M1, M3 (and checking M4) forces c∈{−2,0}, i.e., Chebyshev or power (Lemma 29) ; and (iii) multiple fixed points are excluded (Lemma 30) . The splitting criterion relating multipliers and multiplier polynomials is stated explicitly (Corollary 20) , and the dependence of M_n on (σ1,σ2) is given (Proposition 23) . The candidate solution follows the same architecture: dynatomic/multiplier polynomial technology; Milnor–Silverman moduli; finite Diophantine analysis of fixed-point multipliers; and the same trichotomy. Minor discrepancies include: referring to the parameter as 0 or ±2 in the superattracting normal form (the paper finds c∈{−2,0} in z^2+c), and mentioning M2 where the paper uses M1 and M3; these do not change the classification. The model’s outline also correctly notes that quadratic Lattès multipliers lie in RD (Milnor) and uses the paper’s list of quadratic Lattès fixed-point multipliers as a recognition tool (Corollary 27) . Overall, the proofs are substantially the same and correct.