The Julia sets of Chebyshev’s method with small degrees

The paper’s Theorem 1.1 states deg(Cp) = 3(m + n + r) − 2 − B + s and, when there are no special critical points, deg(Cp) = 3(m + n + r) − 2. Its proof proceeds by counting fixed points: deg(Cp) = m + n + r + deg(Lp), then computes deg(Lp) via factorization and cancellation at the special critical points, yielding deg(Lp) = 2(m + n + r) − 2 − B + s, hence the formula for deg(Cp) . The candidate model derives the same formula by a different route: writing Cp = N/D with D = (p')^3 and N = z(p')^3 − (p')^2 p − (1/2)p^2 p'', then computing deg N = 3d − 2, deg D = 3d − 3 (consistent with the paper’s equation (3)) and explicitly summing cancellations in gcd(N, D) at multiple roots and special critical points to obtain the same degree. Both arguments are logically sound and compatible with the paper’s definitions of special critical points and the fixed-point structure of Cp .