Generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem in Differential Equations

This paper systematically explores generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem in the context of differential equations. First, for linear ordinary differential equations with constant coefficients, the characteristic polynomial together with the Fundamental Theorem of Algebra guarantees exactly n linearly independent solutions for an n-th order equation, which we call the Fundamental Theorem of Differential Equations; while the Vieta relations between roots and coefficients directly provide algebraic connections between solutions and coefficients. Second, for linear differential equations with variable coefficients, the Wronskian satisfies Liouville's formula, whose logarithmic derivative equals the negative of the leading coefficient, which can be viewed as a natural generalization of the sum-of-roots relation in Vieta's theorem. Furthermore, using Grassmann algebra, we establish exact relations between higher-order coefficients and determinants of higher-order exterior products of solutions, obtaining higher-order Liouville formulas. We rigorously prove that in the constant coefficient case these formulas are completely equivalent to Vieta's theorem, thus extending Vieta's theorem fully to variable coefficient linear equations. On this basis, we delve deeper into applications of Grassmann algebra in differential equations, proving the differential invariance of Pl\"ucker relations satisfied by minor vectors. Within the framework of differential algebra, we establish a rigorous algebraic formulation of a differential Vieta theorem, expressing coefficients as logarithmic derivatives of differential symmetric functions of solutions. We extend Liouville's formula to partial differential equations, obtaining a generalized Liouville formula under commutativity conditions, and demonstrate the profound connection between higher-order Liouville formulas and Lax pairs and -functions in integrable systems. Finally, we provide rigorous theorems and proofs for four emerging areas: stochastic Liouville formulas (based on It\ᵒ calculus) for stochastic differential equations, quasideterminantal Liouville formulas for noncommutative differential equations (including a complete 22 derivation and a general theorem), explicit algebraic formulations of the Bethe ansatz as a quantum Vieta theorem, and the relation between conservation laws and infinite determinants in infinite-dimensional dynamical systems via the -function expansion. Based on these, we propose deeper future research directions. These results reveal a deep unity between algebra and analysis, offering important perspectives for the theoretical study of differential equations.