Unified Analytic Solution of Polynomial Equations in Commutative Banach Algebras via Differential Algebraic Methods

This paper establishes a rigorous framework for solving polynomial equations in commutative unital Banach algebras by extending the differential algebraic closure approach. We prove that all solutions of a degree-n polynomial equation P (x) = 0 in a commutative unital Banach algebra A can be analytically expressed within a Banach differential algebraic closure KA. We provide complete constructive proofs with detailed combinatorial analysis, derive explicit expressions for the correction coefficients γ(n) m ,and present a detailed algorithm with complexity analysis. The work reconciles with the Abel-Ruffini theorem by demonstrating that while solutions in radicals are impossible for general quintic and higher-degree equations, explicit analytic solutions exist in the appropriately extended Banach differential algebraic closure KA. Extensive validation through special cases and error analysis confirms the method's correctness and numerical stability. New contributions include enhanced combinatorial interpretations, improved asymptotic analysis, practical implementation strategies for handling spectral constraints, and detailed spectral estimation methods for infinite-dimensional cases.