Generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem in Integral Equations: A Comprehensive Treatise with Rigorous Extensions to Stochastic, Noncommutative, Quantum, and Infinite-Dimensional Settings

This monograph presents a systematic and rigorous generalization of the Fundamental Theorem of Algebra and Vieta's Theorem to the theory of integral equations. We establish a comprehensive framework that unifies classical algebraic concepts with modern analytical methods. For linear integral equations with degenerate kernels, we prove the Fundamental Theorem of Integral Equations, demonstrating that the number of eigenvalues equals the rank of the kernel, and derive integral representations for elementary symmetric functions of eigenvalues that constitute the integral analogue of Vieta's Theorem. For general linear integral equations, we develop higher-order Fredholm determinants and establish higher-order Liouville formulas, proving their exact equivalence to Vieta's Theorem in the degenerate limit. Using Grassmann algebras, we prove the differential invariance of Pl\"ucker relations for minor vectors, revealing deep geometric constraints on solution spaces. Within the differential algebra framework, we establish a differential Vieta theorem expressing kernel coefficients as logarithmic derivatives of differential symmetric functions of solutions, with a rigorous Galois-theoretic interpretation. We extend Liouville's formula to parameter-dependent kernel families and demonstrate profound connections with -functions in integrable systems. For four emerging areas, we provide complete rigorous treatments: (1) a stochastic Liouville formula for stochastic integral equations with complete It\ᵒ calculus derivations; (2) a determinantal Liouville formula for noncommutative integral equations with explicit 22 computations and general theorem proofs; (3) an explicit algebraic formulation of the Bethe ansatz as a quantum Vieta theorem with rigorous equivalence proofs; (4) the relation between conservation laws and infinite determinants via -function expansions in infinite-dimensional dynamical systems with convergence proofs. All conjectures, open problems, and heuristic derivations from the original framework are transformed into rigorously proven theorems with complete mathematical derivations. Based on these results, we propose 40 precisely formulated future research directions at the intersection of analysis, algebra, geometry, probability, and mathematical physics.