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

This paper systematically explores the generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem to exterior differential equations. First, for constant coefficient linear exterior differential equations, by introducing characteristic forms and the characteristic polynomial of the exterior differential 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 Differential Equations. The Vieta relations establish algebraic connections between eigenvalues (generalized roots) and coefficients. Second, for variable coefficient linear exterior differential equations, we define the Wronskian determinant of exterior differential 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 differential equations. Building on this, we deeply explore applications of Grassmann algebra in exterior differential 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 differential systems (Pfaffian systems), obtaining a generalized Liouville formula in a general context. Finally, for four emerging fields---stochastic exterior differential equations, noncommutative exterior differential equations, quantum integrable systems, and infinite-dimensional exterior differential systems---we provide rigorous theorems and proofs, and based on these, propose deeper future research directions. These results reveal a profound unity between algebra and analysis, providing important perspectives for the theoretical study of exterior differential equations.