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

Article · shifa liu

- Archive id: doi:10.5281/zenodo.18840393
- Published: 2025-12-31
- Venue: Zenodo (CERN European Organization for Nuclear Research)
- DOI: https://doi.org/10.5281/zenodo.18840393
- Citations: 0
- Source URL: https://doi.org/10.5281/zenodo.18840393
- Provenance: openalex

## 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 reproduced under CC-BY (publisher license, via OpenAlex)._

Topics: [Landmark Results in Pure Mathematics](https://space-kitty.com/topics/landmark-math-results)

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