Generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem in Exterior Variational Equations: Complete Theory with Rigorous Proofs

This paper systematically generalizes the Fundamental Theorem of Algebra and Vieta's Theorem to exterior variational equations. The core insight is that higher-order variational problems are understood as reaching first variation through multiple variations---this is the essence of variational problems, and higher variations naturally possess their exterior forms. First, for constant coefficient linear exterior variational equations, by introducing characteristic forms and the characteristic polynomial of the variational operator, we prove using the Fundamental Theorem of Algebra that the dimension of the solution space equals the order of the operator---this is called the Fundamental Theorem of Exterior Variational Equations. The Vieta relations establish algebraic connections between eigenvalues (generalized roots) and coefficients. Second, for variable coefficient linear exterior variational equations, we define the Wronskian determinant of exterior variational forms and prove that it satisfies a Liouville formula, where its logarithmic exterior derivative equals the negative of the leading coefficient multiplied by a fixed 1-form. This can be viewed as a natural generalization of the sum-of-roots relation in Vieta's Theorem. Furthermore, using Grassmann algebra, we establish precise relations between higher-order coefficients and higher exterior wedge product determinants of formal 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 completely to variable coefficient linear exterior variational equations. Building on this, we deeply explore applications of Grassmann algebra in exterior variational equations, proving the differential invariance of Pl\"ucker relations satisfied by subdeterminant vectors. Within the framework of differential algebra, we establish a rigorous algebraic formulation of the Differential Vieta Theorem, expressing coefficients as logarithmic derivatives of differential symmetric functions of formal solutions. We generalize the Liouville formula to first-order exterior variational systems (Pfaffian systems), obtaining a generalized Liouville formula in a general context. We establish the recursive nature of higher variations, showing that an $m$th-order variational problem reduces to a first-order system through the introduction of intermediate variables $\omega^i = \delta^i \omega$. This recursive reduction is the geometric essence of higher-order variational problems. Finally, for emerging fields---stochastic exterior variational equations, noncommutative exterior variational equations, quantum integrable systems, and infinite-dimensional exterior variational systems---we provide rigorous theorems and proofs, establishing a unified categorical framework. We also develop spectral sequences for recursive filtrations, connect recursive variations with deformation theory and quantization, and explore deep connections with mirror symmetry and Donaldson-Thomas theory. These results reveal a profound unity between algebra, analysis, geometry, and physics, providing important perspectives for the theoretical study of exterior variational equations.