Generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem in Inverse Variational Problems

This paper systematically establishes deep and rigorous generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem in the context of inverse variational problems. We demonstrate that the linearized operator of a differential equation, together with its adjoint, inherits a rich algebraic structure that mirrors classical polynomial theory and provides necessary and sufficient conditions for the existence of a variational formulation. Through the systematic development of Grassmann algebra, we prove that for any k (1 k n-1), the determinant of the k-th exterior product of the fundamental matrix satisfies a differential equation whose logarithmic derivative equals a certain combination of the coefficients, extending Liouville's formula to arbitrary order. We rigorously prove that these higher-order Liouville formulas are completely equivalent to the classical Helmholtz conditions, providing a new algebraic characterization of variationality. Within the framework of differential algebra, we establish a rigorous formulation of a differential Vieta theorem, expressing the coefficients of the differential operator in terms of logarithmic derivatives of differential symmetric functions of solutions and adjoint solutions. We prove the differential invariance of Pl\"ucker relations satisfied by the vector of minors of the fundamental matrix, revealing nonlinear constraints among coefficients and minors. We extend these results to stochastic differential equations using Malliavin calculus, establishing stochastic Helmholtz conditions and a stochastic Liouville formula with complete It\ᵒ calculus derivations. For noncommutative settings, we develop quasideterminantal calculus and prove noncommutative Liouville formulas with explicit algebraic conditions. For infinite-dimensional systems, we establish rigorous connections with -functions and zeta-regularized determinants, proving that higher-order Liouville formulas generate an infinite hierarchy of conservation laws. We establish a differential Galois interpretation of these invariants, linking the existence of a variational structure to the structure of the differential Galois group. All previously stated conjectures and open problems are transformed into rigorous theorems with complete proofs, revealing a deep unity between algebra, geometry, topology, and analysis in the inverse calculus of variations. Furthermore, we provide a systematic analysis of the set-theoretic, categorical, and homotopy-type-theoretic foundations underlying each theorem, specifying the required axioms from ZF with dependent choice to Grothendieck universes and the univalence axiom.