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March 3, 2026Open Access

Generalizations of the Fundamental Theorem of Algebra and Vieta's Theorem in Integral Equations: A Comprehensive Treatise with Rigorous Extensions to Stochastic, Noncommutative, Quantum, and Infinite-Dimensional Settings

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This monograph demonstrates rigorous extensions of classical algebraic theorems to integral equations, indicating profound mathematical implications.

The aim is to generalize classical algebraic theorems, specifically the Fundamental Theorem of Algebra and Vieta's Theorem, within the framework of integral equations and modern analytical methods.
Develop a comprehensive framework unifying algebraic concepts with analytical methods.
Prove the Fundamental Theorem of Integral Equations for linear equations with degenerate kernels.
Establish higher-order Fredholm determinants and their relation to Vieta's Theorem.
Use Grassmann algebras to prove differential invariance of Pl"ucker relations.
Extend Liouville's formula to parameter-dependent kernel families.
Demonstrated that the number of eigenvalues equals the rank of the kernel in integral equations.
Provided integral representations for elementary symmetric functions of eigenvalues.
Established rigorous connections between Vieta's Theorem and various mathematical frameworks.
Presented a stochastic Liouville formula and other rigorous treatments for four emerging areas.
Transform conjectures and open problems into rigorously proven theorems with mathematical derivations.

shifa liu (2025) studied this question.

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This monograph demonstrates rigorous extensions of classical algebraic theorems to integral equations, indicating profound mathematical implications.