Differential Algebraic Closure Solution of Polynomial Equations:A Constructive Framework Transcending the Abel-Ruffini Barrier

This paper establishes a rigorous differential algebraic framework for solving polynomial equations of arbitrary degree. We define the universal differential algebraic closure Kn for degree n, a differentially closed field extension constructed by adjoining abstract generators representing a critical point, formal derivatives, roots of unity, and nth roots of rational functions. For a concrete polynomial f of degree n, we prove the existence of a specialization homomorphism ιf : Kn → Lf into a differentially closed extension of its coefficient field. We provide a complete constructive derivation of the combinatorial coefficients and a rigorous analysis establishing the existence and uniqueness of the branch indices.The solution resides in the differentially closed extension Lf, which is generally not contained in the radical closure Frad, thus extending the admissible solution space while remaining fully consistent with the Abel-Ruffini theorem’s restriction on solutions by radicals alone. Furthermore, we present the Differential Algebraic Root-Finding (DARF) algorithm with O(n2) typical and O(n3) worst-case complexity, accompanied by comprehensive numerical validation on over 104 test cases. The validation demonstrates residuals below 10−32 for degrees up to 25 using 256-bit precision arithmetic, including notoriously ill conditioned problems like Wilkinson’s polynomial. This work provides a new paradigm for explicit polynomial solving that is algebraically transparent, numerically stable, and computationally tractable, while opening new research directions at the intersection of differential algebra, Galois theory, and numerical computation.