Differential Algebraic Closure Solution of Polynomial Equations and Its Equivalence to Transcendental Function Representations

This paper establishes a rigorous differential algebraic framework for solving polynomial equations of arbitrary degree and demonstrates its equivalence to classical transcendental function solutions. we prove that this differential algebraic solution is mathematically equivalent to classical transcendental function solutions based on elliptic, hyperelliptic, and higher-genus algebraic curves. For any degree n ≥ 5, we construct explicit isomorphisms between the differential algebraic closure Lf and function fields of associated algebraic curves, establishing that the Fourier coefficients cm =(ϕm/(na0))1/n correspond to values of Riemann theta functions on Jacobian varieties. The equivalence is demonstrated through explicit formulas: for quintic equations, the solution corresponds to elliptic theta functions; for sextic equations, to genus-2 hyperelliptic functions; and for higher degrees, to higher-genus Riemann theta functions. We provide complete algorithms with complexity analysis, numerical validation on comprehensive test suites, and detailed implementation specifications. This work unifies the differential algebraic approach with classical special function theory, providing both a computationally tractable algorithm and deep theoretical connections to algebraic geometry, while opening new research directions at the intersection of differential algebra, Galois theory, and numerical computation.