arXiv2026
We present the Robust Newton Method (RNM) for a complex polynomial $p$ of degree $n\geq2$, with continuous map $F(z)=z-p(z)\overline{p'(z)}/(9A(z)^2)$, where $A(z)=\max_{0\leq j\leq n}|p^{(j)}(z)|/j!$. Its fixed points are the roots and critical points; elsewhere it strictly decreases $|p|$ along the Newton direction. The Geometric Modulus Principle inspires the construction and supplies an escape step at nonroot critical points. Every root has an open basin. When nonroot critical points are simple, almost every seed converges to a root, and root-basin boundaries lie in countably many Lipschitz curves with Hausdorff dimension one when nonempty. These conclusions apply after any sufficiently small nonzero linear perturbation $p(z)+δz$. Every RNM orbit, with or without escapes, converges to a root or critical point. A hybrid enforcing one RNM update every $T$ iterations retains convergence and reaches $|p(z)p'(z)|\leq\varepsilon$ in $O(T/\varepsilon^2)$ updates, with constants depending on the polynomial and seed; $T=1$ gives RNM. Orbits approaching simple roots eventually reach Smale's $α$-certification, permitting a permanent Newton switch and $O(\log\log(1/\varepsilon))$ further steps for absolute root error $\varepsilon$. For cubics, deterministic critical-point initialization yields an explicit preparatory bound, retained by an RNM-guided Newton refinement. Illustrative computations cover cubics and degree-ten polynomials. For $p(z)=z^n-1$, the basins are exactly the open Voronoi regions, with boundary points converging to the origin. Polynomiographs illustrate basin geometry and iteration-based coloring.