A Differential-Algebraic Solution to Hilbert’s 13th Problem: A Constructive Framework Based on Universal Closure and Specialization Homomorphism

This paper presents a novel, constructive differential-algebraic solution to the algebraic variant of Hilbert’s 13th problem—namely, whether the roots of an arbitrary degree-n polynomial equation can be represented as a composition of finitely many continuous functions of two variables. We construct a universal differential algebraic closure, providing a unified solution formula for all degree-n equations. The core theory lies in proving that for any concrete polynomial f, there exists a specialization homomorphism ιf : Kn → Lf, mapping this abstract formula to a concrete root of f in a certain differentially closed extension field Lf.This framework is strictly distinct from traditional radical solving, as the solution field Lf generally does not lie within the radical closure, thus being fully compatible with the Abel-Ruffini theorem. Using n = 7 as an example, we provide a detailed exposition of the complete derivation from the abstract formula to the concrete root expression, and demonstrate that this expression can be decomposed into a finite composition of binary elementary operations. This research not only offers a concrete, computable answer to Hilbert’s 13th problem but also extends polynomial solving theory from the realm of radicals to the differential-algebraic realm, establishing a profound connection with classical transcendental function theory.Through an in-depth study of the structure of the differential Galois group of Lf and its relationship with the classical Galois group, we develop a complete classification theory for ”differentially algebraically solvable” equations, revealing the profound essence of the universal solvability of all polynomial equations within the differential-algebraic framework.