ON THE BASINS OF ATTRACTION OF A ONE-DIMENSIONAL FAMILY OF ROOT FINDING ALGORITHMS: FROM NEWTON TO TRAUB.

The uploaded paper proves exactly the statement in the two special cases (quadratic polynomials and unicritical polynomials z^n−β) for all δ∈[0,1], and by the same core arguments used in the candidate solution: (i) local multipliers at roots (Lemma 2.1) ensure attraction (simple roots are superattracting; multiple roots are attracting for a large disk of δ that includes [0,1]) ; (ii) behavior at ∞ and degree considerations (Lemma 2.2) show ∞ is fixed and repelling for δ in [0,1] ; (iii) in the quadratic case, an explicit Möbius conjugacy yields G_δ with formula (14), G_δ is a Blaschke product for real δ, and 1 lies on the common boundary of the immediate basins of 0 and ∞; simple connectivity then holds throughout the hyperbolic component containing δ∈(0,1] (with δ=0 handled by the classical Newton case) ; (iv) in the unicritical case, rotational symmetry and a single finite pole force simple connectivity (Proposition 4.3), and unboundedness follows from the real-axis inequality 1<T(x)<x for x>1, proved via the S_{δ,n}(x) polynomial and Descartes’ rule (Proposition 4.4) . The candidate’s write-up mirrors these steps closely, including the same conjugacy to G_δ, the rotation-symmetry argument, and the real-axis proof of unboundedness. Minor differences are expository (e.g., a direct local expansion at roots versus citing Lemma 2.1), but there is no substantive disagreement with the paper.