Differential-Algebraic Finite Representation Theory and Its Extension to Analytic Equations:A Unified Framework

This paper systematically establishes a finite representation theory based on differential-algebraic operations and extends it from polynomial equations to differential equations, integral equations, difference equations, integro-differential- algebraicm equations, and related transcendental function classes. We prove that for a large class of linear differential equations with coefficients in algebraic function fields, integral equations with polynomial or rational kernels, linear difference equations, integrable nonlinear equations,and their coupled systems, their eigenvalues, eigenfunctions, and particular solutions can all be represented through finite differential-algebraic operations on the equation coefficients or parameters. We construct a hierarchical differential- algebraic operational framework, clearly distinguishing between formal representation and numerical computation, and demonstrate two parallel finite representation series:first, the roots of univariate algebraic equations can be finitely represented by trigonometric (hyperbolic) functions, elliptic functions, and hyperelliptic functions respectively; second, the solutions of linear differential, integral, and difference equations can be finitely represented by classical special functions such as exponential functions, Bessel functions, Legendre functions, and elliptic functions. These two series reveal a profound unification between algebraic equations and analytic equations at the level of representation theory. Furthermore, we propose and rigorously argue the Differential-Algebraic Spectral Theorem,providing a unified finite representation framework for all the above problems. We establish a five level classification hierarchy of transcendental functions based on ”algebraic properties of generators” , revealing the symmetry between π-type and e- type transcendental functions as standard solutions to different eigenvalue problems, and prove that they can all be finitely represented by the coefficients of their defining equations through differential-algebraic operations.