A GENERALIZED FUNDAMENTAL THEOREM OF ALGEBRA AND VIETA'S FORMULAS OVER OPERATOR ALGEBRAS: A UNIFIED FRAMEWORK FOR HIGHER-DEGREE FORMS

Abstract

This paper establishes a comprehensive and rigorous framework extending polynomial theory from Clifford algebras to operator algebras, specifically finite $C^*$-algebras with a faithful normal tracial state, and more generally to n-ary operator algebras associated with fully symmetric n-linear forms. By introducing ellipticity conditions and the concept of regular solutions, we prove that the regular solution set of a monic polynomial with real coefficients decomposes into discrete scalar roots and a finite number of smooth manifolds, each of which is a homogeneous space carrying a unique invariant probability state. Using the path ordering operator $\pathord$ to handle noncommutativity, we derive a generalized Vieta formula. We provide complete, self-contained proofs of all theorems, including: a rigorous characterization of regular solutions via minimal polynomials (Theorem \ref{thm:minimal-polynomial}); a complete proof of the Faddeev-LeVerrier algorithm with termination criterion (Theorem \ref{thm:faddeev-leverrier}); a proof that the solution manifolds are homogeneous spaces $U(d)/[U(d/2)\times U(d/2)]$ (Theorem \ref{thm:homogeneous}); a proof of the well-definedness and spectral decomposition of the path ordering operator (Theorems \ref{thm:path-ordering-well-defined}, \ref{thm:path-ordering-spectral}); a proof of its uniqueness (Theorem \ref{thm:path-ordering-unique}); and a complete derivation of the generalized Vieta formula (Theorem \ref{thm:vieta-final}). We extend the theory to non-compact cases using $L^2$ Euler characteristics (Theorem \ref{thm:noncompact-vieta}), establish connections with Shirokov's noncommutative Vieta theorem (Theorem \ref{thm:shirokov-unified}), and generalize to n-ary operator algebras (Theorems \ref{thm:nary-structure}, \ref{thm:nary-fundamental}, \ref{thm:nary-vieta-final}). We provide a complete classification of n-ary operator algebras via discriminants, Hasse invariants, and n-ary signatures (Theorem \ref{thm:classification-nary-clifford}). We construct the quantization of solution manifolds and prove the quantum generalized Vieta formula (Theorem \ref{thm:quantum-vieta}). We establish the arithmetic-geometric interpretation as conjectures connecting to Frobenius traces and Hasse-Weil L-functions (Conjectures \ref{conj:arithmetic-geometric-weak}, \ref{conj:arithmetic-geometric-strong}). We prove a conditional version of the connection with the Langlands program (Theorem \ref{thm:langlands}). Finally, we construct the Absolute Unifier $\U$, a universal object unifying all n-ary operator algebras and their polynomial theories, and prove its existence and uniqueness (Theorem \ref{thm:absolute-unifier}), with further structures presented as conjectures (Conjectures \ref{conj:unifier-ring}--\ref{conj:unifier-mtheory}). The results unify polynomial theory over quaternions, octonions, Clifford algebras, and operator algebras, providing new tools for noncommutative algebra, differential geometry, topology, number theory, and mathematical physics.

Abstract of "A GENERALIZED FUNDAMENTAL THEOREM OF ALGEBRA AND VIETA'S FORMULAS OVER OPERATOR ALGEBRAS: A UNIFIED FRAMEWORK FOR HIGHER-DEGREE FORMS" by shifa liu, https://doi.org/10.5281/zenodo.18840393, used under its Creative Commons Attribution license (CC-BY); markup removed, otherwise unmodified.