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 for any degree n polynomial, all roots can be expressed analytically as: xk=x(n−1)+∑m=1n−1Φm(y)1/pmωnm(k−1) for 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 explicitly defined polynomials, pm∈Z+, and ωn=e2πi/n. Comprehensive validations for degrees 2-6 demonstrate machine-precision accuracy with errors <10−12. This work refines the Abel-Ruffini theorem, showing that while radical solutions are impossible for quintic+ equations in elementary functions, explicit analytic solutions exist in differential algebraic closure.