A Differential-Algebraic Solution to Hilbert's 21st Problem: A Unified Framework for Regular and Irregular Singularities

This paper presents a comprehensive differential-algebraic framework for Hilbert’s 21st problem on the Riemann–Hilbert correspondence, providing a unified treatment that encompasses both regular and irregular singularities. We constructively resolve the historical challenges posed by Bolibruch’s counterexamples by introducing a novel concept of Riemann–Hilbert differential-algebraic closures.Our approach synthesizes techniques from differential algebra, complex analysis, and asymptotic analysis to establish a complete solution that includes:1. Precise regularity criteria distinguishing when a monodromy representation can be realized by a Fuchsian system;2. Systematic classification and explanation of Bolibruch-type obstructions with minimal irregularity bounds;3. Recursive algorithms for constructing differential equations from monodromy and Stokes data with certified error bounds using interval arithmetic;4. Extensions to higher-dimensional complex manifolds;5. Connections with modern developments in isomonodromic deformations, Painlev´e equations,harmonic metrics, Higgs bundles, and the geometric Langlands program.All proofs are presented with complete mathematical rigor, including detailed derivations of formal asymptotic expansions, rigorous analysis of Stokes phenomena, and verification of compatibility conditions through differential-algebraic methods. This paper generalizes the differential-algebraic framework for the one-dimensional Riemann Hilbert problem to the case of higher-dimensional compact complex manifolds. Starting from first principles, we recursively construct a Riemann-Hilbert differential-algebraic closure which naturally encodes all necessary information for a flat connection with prescribed monodromy representation. This framework unifies and extends classical existence theories, not only recovering results like the higher-dimensional Plemelj theorem and regular singularity theory but also providing constructive solutions. We systematically address higher-dimensional specificities such as combinatorial correctionterms (e.g.,polylogarithmic crossing terms)and branchs election issues,giving the ircomplete parametric formulas and derivations.As an application,we implement within the differential-algebraic closure a constructive iterative algorithm for harmonic metrics, thereby establishing an effective link with non-abelian Hodge theory. Concrete examples and algorithmic descriptions are provided,demonstrating the computational feasibility of the framework. The main results include the construction of the differential closure KRHX(ρ) (Proposition3.1), the complete combinatorial structure of formal solutions (Theorem5.1),the parameterization of branch data (Theorem5.6),and the differential-algebraic Riemann-Hilbert correspondence (Theorem5.7).