Withdrawn: A Unified Analytical Solution Framework for Polynomial Equations in Differential Algebraic Closure

This paper presents a comprehensive framework for solving polynomial equations through the construction of a differential algebraic closure, providing explicit analytical solutions for polynomials of arbitrary degree. We establish that all roots of a degree-n polynomial can be expressed in the form xk = x(n−1) + Pn−1 m=1 Φm(y)1/nωm(k−1) n , where x(n−1) is a critical point derived from the (n − 1)-th derivative, y is a vector of critical values, Φm are explicitly constructed polynomials with combinatorial coefficients, and ωn is a primitive n-th root of unity. We provide complete constructive proofs for the existence and form of these solutions, derive combinatorial expressions for the correction terms, and present a detailed algorithmic implementation with O(n2) complexity. Extensive numerical validation across more than 104 test cases demonstrates residuals below 10−32 for degrees up to 25, including notoriously ill-conditioned problems such as Wilkinson’s polynomial. This work reconciles with the Abel-Ruffini theorem by demonstrating that while solutions in radicals are impossible for general quintic and higher-degree equations, explicit analytic solutions exist in the differential algebraic closure. The framework presented here offers both theoretical insights and practical computational tools with applications across mathematics, physics, and engineering.