A Comprehensive Generalization of the Fundamental Theorem of Algebra and Vieta's Theorem in the Framework of Difference Equations

This monograph systematically explores the profound generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem within the framework of difference equations. For linear constant-coefficient difference equations, the characteristic polynomial, combined with the Fundamental Theorem of Algebra, guarantees exactly n linearly independent solutions for an nth-order equation, a result we term the Fundamental Theorem of Difference Equations. Vieta's Theorem directly provides algebraic relations between the characteristic roots and the coefficients. For linear variable-coefficient equations, the Casorati determinant satisfies a discrete Liouville formula, which can be viewed as a natural generalization of the product relation in Vieta's Theorem. By introducing Grassmann algebra, we establish precise relations between higher-order coefficients and higher-order exterior product determinants of solutions, obtaining a hierarchy of higher-order discrete Liouville formulas. We rigorously prove that in the constant-coefficient case, these formulas are exactly equivalent to Vieta's Theorem, thereby providing a complete generalization of Vieta's Theorem to variable-coefficient linear difference equations. Building on this framework, we prove the discrete invariance of Pl\"ucker relations using a coalgebraic primitive element argument, and establish a Chern-Weil type theorem for discrete curvature. Within the framework of differential algebra, we develop a differential version of Vieta's Theorem, expressing coefficients as logarithmic differences of symmetric functions of the solutions, and provide a complete Galois-theoretic interpretation. We generalize the discrete Liouville formula to partial difference equations, obtaining a generalized discrete Liouville formula under commutativity conditions, and demonstrate the profound connection between higher-order discrete Liouville formulas and -functions and Lax pairs in integrable systems. For the discrete KdV equation, we construct higher-order -functions and prove their duality relations, establishing a vertex operator algebra interpretation. Finally, we provide rigorous theorems and proofs for emerging fields: a stochastic discrete Liouville formula with explicit second-order correction terms derived from It\ᵒ calculus, a quasideterminant Liouville formula for non-commutative difference equations interpreted via Cohn's free ideal ring theory, the Bethe ansatz as an explicit algebraic formulation of a quantum Vieta Theorem derived from Baxter's TQ relation, and the relationship between conservation laws in infinite-dimensional dynamical systems and the expansion of -functions through infinite determinants. We also develop q-deformed Liouville formulas with crystal base limits, tropical conservation laws with Maslov index interpretations, and large deviation principles for random Bethe roots. These results, comprising 48 theorems with complete proofs, reveal a profound unity between algebra, analysis, geometry, and mathematical physics, offering a vital perspective for the theoretical study of difference equations.