Unified Analytic Solution of Polynomial Equations in Differential Algebraic Closure

This paper establishes a rigorous differential algebraic framework for solving polynomial equations of arbitrary degree. We prove that all roots of a degree-n polynomial can be analytically expressed within a differential algebraic closure K, which extends the coefficient field with derivative operators and their evaluations. The solution formula takes the unified form: xk = x(n−1) + n−1 ∑ m=1 Φm(y)1/nωm(k−1) n , 0 ≤k≤n−1, where x(n−1) = −a1/(na0) is the critical point from the (n −1)-th derivative, y = (y(0),...,y(n−2)) are critical values with y(j) = f(j)(x(n−1)), Φm ∈ Q(a)y are explicit polynomials with combinatorial correction terms, ωn = e2πi/n, and pm = n. We provide complete constructive proofs, derive combina torial expressions for the correction coefficients γ(n) m ,andpresent a detailed O(n2) algorithm. Extensive numerical validation across over 104 test cases demonstrates machine-precision accuracy (residuals <10−32) for degrees up to 25, including notoriously ill-conditioned problems like 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 appropriately extended differential algebraic closure K.