A Unified Numerically Stabilized Framework for Sextic, Octic, and Decic Equations via Higher-Degree Tschirnhaus Transformations with Newton Refinement

This paper presents a unified numerically stabilized framework for solving polynomial equations of degrees 6, 8, and 10 based on higher-degree Tschirnhaus transformations. Unlike classical algebraic reduction, which leads to high-degree resolvent equations, the elimination of odd-degree terms is formulated as a nonlinear system in the transformation parameters and solved numerically using Newton iteration. The transformed equation contains only even powers and is reduced to a lower-degree polynomial in w = y^2, solved using backward-stable solvers (cubic, quartic, or quintic). The method is proven to be locally convergent and backward stable under standard floating-point arithmetic. Numerical experiments demonstrate relative errors below 10⁻¹⁴ across a wide range of test cases.