Differential Algebraic Closure Framework for D-Module Theory: Constructive Solutions to Systems of Linear Partial Differential Equations

This paper establishes a comprehensive differential algebraic framework for constructing explicit solutions to systems of linear partial differential equations within the theory of D-modules. We extend the previously developed differential geometric closure KDiffGeo and linear exterior differential closure KLEDE to define the D-module differential closure KDMod, a differentially closed field extension constructed through recursive adjunction processes that incorporate D-module structures, characteristic varieties, holonomic solutions, and fundamental solutions. Within this closure, we prove that solutions to broad classes of D-module equations-including holonomic systems, regular singular D-modules, and microdifferential equations—admit unified representations that respect the underlying algebraic and geometric structures. The framework rigorously addresses the algebraic complexity of D-module systems while preserving their analytic and geometric properties. We provide detailed constructive proofs, derive explicit solution formulas with rigorous error bounds, and establish convergence criteria in appropriate function spaces. Comprehensive algorithms with precise complexity analysis are presented, including stability guarantees and adaptive precision control with certified error bounds. A rigorous validation framework employing interval arithmetic and computational D-module theory demonstrates the practical effectiveness of our approach. The work demonstrates that explicit analytic solutions to D-module systems exist within appropriately constructed differential algebraic closures, providing new algebraic perspectives on D-module solvability while maintaining consistency with classical theory. Extensions to Bernstein-Sato polynomials, regular singularities,and the Riemann-Hilbert correspondence establish connections across algebraic analysis, representation theory, and mathematical physics.