Generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem in Antidifference Equations: A Comprehensive Treatise with Rigorous Extensions to Discrete, 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 antidifference equations (discrete analogues of differential equations). We establish a comprehensive framework that unifies classical algebraic concepts with modern discrete analysis methods. For linear antidifference equations with constant coefficients, we prove the Fundamental Theorem of Antidifference Equations, demonstrating that the number of linearly independent solutions equals the order of the equation, and derive discrete analogues of Vieta's Theorem relating the roots of the characteristic equation to the coefficients. Using discrete Grassmann algebras, we prove the invariance of discrete Pl\"ucker relations for Casorati matrices, revealing deep geometric constraints on solution spaces. Within the difference algebra framework, we establish a discrete Vieta theorem expressing coefficients as elementary symmetric functions of the characteristic roots. We extend discrete Liouville's formula to parameter-dependent families and demonstrate connections with discrete integrable systems. For four emerging areas, we provide complete rigorous treatments: (1) a stochastic discrete Liouville formula for stochastic antidifference equations with complete discrete stochastic calculus derivations; (2) a determinantal Liouville formula for noncommutative antidifference equations with explicit computations; (3) an explicit algebraic formulation of the Bethe ansatz for discrete quantum integrable systems as a quantum Vieta theorem; (4) the relation between conservation laws and infinite determinants in infinite-dimensional discrete dynamical systems. 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 discrete analysis, algebra, geometry, probability, and mathematical physics.