What question did this study set out to answer?

The aim is to extend the Fundamental Theorem of Algebra and Vieta's Theorem to higher-order exterior variational equations.

March 10, 2026Open Access

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

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Generalizes algebraic theorems in variational equations, suggesting broader applications in mathematics and physics.

The aim is to extend the Fundamental Theorem of Algebra and Vieta's Theorem to higher-order exterior variational equations.
Grouped into two main types: constant and variable coefficient linear exterior variational equations.
Used characteristic forms and polynomials to prove the dimension of the solution space in constant coefficient cases.
Defined Wronskian determinants for variable coefficient cases, connecting them with logarithmic derivatives.
Applied Grassmann algebra to derive relations between coefficients and determinants of solutions.
Proved the Fundamental Theorem of Exterior Variational Equations for constant coefficients.
Established a generalized Liouville formula for variable coefficients.
Found algebraic connections between eigenvalues and coefficients, expanding Vieta's relations.
Demonstrated recursive properties of higher variations in variational problems.

shifa liu (2025) studied this question.

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