A Unified Theory of Differential-Algebraic Geometry with Application to Hilbert's 20th Problem

This paper constructs a unified differential-algebraic geometric framework from first principles of differential algebra, completely solving Hilbert’s 20th problem and extending beyond classical theory. We systematically construct the differential-algebraic geometric closure KDA(X), proving its core properties: differential closedness, geometric completeness,and model completeness. Based on this framework, we develop a combinatorial correction theory, deriving explicit formulas for higher-dimensional cases and mechanisms for logarithmic term generation. By establishing profound connections with motivic integration and perverse sheaf theory, we reveal the geometric essence of the combinatorial coefficients γm,α.Our work not only recovers classical results such as Picard-Vessiot theory and differential Galois theory but also yields novel transcendence results, including a new classification criterion for higher-dimensional regular singularities and a decidability criterion for algebraic integrability of nonlinear equations. We also provide complete algorithmic implementations and computational examples.