A Numerically Stable Framework for Nonic Equations via Cubic Tschirnhaus Transformation, Newton Refinement and Reduction to a Cubic

This paper presents a numerically stable framework for solving real-coefficient nonic equations (degree 9). The method is based on a cubic Tschirnhaus transformation y = x^3 + \alpha x^2 + \beta x + \gamma that approximately eliminates the y^8, y^7, and y^6 terms. The parameters (\alpha, \beta, \gamma) are determined by solving a 3\times3 nonlinear system using Newton's method with an LU/SVD fallback to handle ill-conditioned Jacobians. After refinement, the transformed nonic reduces to the near