Newton’s method applied to rational functions: Fixed points and Julia sets

The paper’s Theorem A states exactly what the candidate claims: any Newton map with exactly two attracting fixed points, one exceptional, is Möbius-conjugate to N_R with R(z)=z^d/p(z), p monic, p(0)≠0, deg p=d; in the generic case there is a unique conjugacy class represented by z^{d+1}+(d−1)z^d; and for d=3,4,5 the counts are 3,5,8, respectively . The paper’s proof normalizes the exceptional fixed point to ∞ and the other to 0, notes the conjugate is still a Newton map and (because ∞ is exceptional) a polynomial, and then derives R(z)=z^d/p(z) with the explicit Newton map formula, matching the candidate’s outline . The multiplier facts used by the candidate (finite zeros give multipliers 1−1/m; poles give 1+1/n; and ∞ is superattracting when deg numerator = deg denominator) are exactly Lemma 2.2 in the paper . Minor issue: the candidate wrote “∞ is attracting exactly when deg numerator equals deg denominator,” which is too strong in general—attracting occurs when d≤e, with superattracting precisely when d=e—but the intended use here (exceptional ⇒ superattracting) still requires equality, so the conclusion remains correct . The paper also explicitly notes that a Newton map with an exceptional point is conjugate to a polynomial, justifying the normalization step , and provides the polynomiality constraint g(z)≡const for N_{z^d/p} that underlies the low-degree enumeration, aligning with the candidate’s enumeration claims . Overall, both arguments coincide in structure; the candidate’s “ODE reduction” for the generic case is asserted without details, but the final statements match the paper.