Rigorous Analytic Solutions of Univariate Real-Power Equations in Transcendental Differential Algebraic Closure

This paper establishes a rigorous framework for solving univariate real-power equations through transcendental differential algebraic closures. We consider equations of the form N i=0aixri = 0 with real exponents ri ∈ R and coefficients ai ∈ C. The main contributions are: 1. Construction of a well-defined transcendental differential algebraic closure KT that extends classical differential algebraic closures by adjoining exponential, logarithmic, and necessary special functions, with explicit branch selection rules for multi-valued functions. 2. Derivation of explicit closed-form solutions using generalized Lambert W functions and their hypergeometric series expansions, complete with rigorous convergence analysis and truncation error bounds. 3. Development of an adaptive-precision numerical algorithm with comprehensive error control, achieving residuals below 10−100 using 256-bit precision, verified through extensive statistical validation.4. Theoretical reconciliation with the Abel-Ruffini theorem, demonstrating that while general transcendental equations lack algebraic solutions, explicit analytic solutions exist in appropriately extended closures, with complete differential Galois group analysis. 5. Comprehensive numerical validation on benchmark problems including illconditioned cases, with comparison to state-of-the-art solvers (Mathematica, Maple, PARI/GP), demonstrating superior accuracy and robustness. The work bridges differential algebra, special function theory, and numerical analysis, providing both theoretical insights and practical computational tools. All algorithms are implemented in open-source software with complete reproducibility materials.