A Unified Theory for Solving Polynomial Equations via Differential Algebraic Closure and Transcendental Function Representation

The problem of solving polynomial equations by radicals has been clearly delineated since Abel and Galois: general polynomials of degree five and above have no radical solutions. Traditional research introduces transcendental functions (elliptic functions, modular functions, etc.) to express the solutions of higher-degree equations, but these methods lack uniformity. This paper proposes a unified framework based on differential algebraic closures, incorporating the differential information of the polynomial itself into the construction of solutions, reopening the possibility of explicit solutions by extending the operation set (adding derivation and evaluation operations). This paper first establishes the differential algebraic solution theory for univariate polynomial equations, proving its equivalence to transcendental function solutions; then explores the precise mathematical relationship between genus and degree; subsequently extends the framework to multivariate polynomial systems, analyzing the essential limitations of solvability; and finally extends to transcendental equations, delineating the boundaries of finite representability based on H¨older’s theorem and others.