A Generalized Fundamental Theorem of Algebra and Vieta's Formulas Over Clifford Algebras: A Unified Framework for Higher-Degree Forms

This paper establishes a comprehensive and rigorous framework extending polynomial theory from normed division algebras to real Clifford algebras p,q and, more generally, to n-ary Clifford 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 real roots and a finite number of smooth manifolds, each of which is a homogeneous space carrying a unique invariant measure. Using the path ordering operator P to handle noncommutativity, we derive a generalized Vieta formula ˜ek = (−1)kaN−k + δk, where the correction terms δk are given by products of Euler characteristics of the solution manifolds and combinatorial coefficients arising from the underlying form. We provide complete, self-contained proofs of all theorems, including: (1) a rigorous characterization of regular solutions via minimal polynomials and the FaddeevLeVerrier algorithm with complete proof of termination; (2) a corrected proof that the solution manifolds are homogeneous spaces Pin(p, q +1)/ Spin(p, q), with explicit coordinate charts and proof of transitivity using the CartanDieudonn´e theorem and Witt extension; (3) a proof of the well-definedness and spectral decomposition of the path ordering operator via the tensor algebra construction, including its interpretation as an E∞-operad structure; (4) a complete derivation of the combinatorial coefficients as Pfaffians, with explicit generating functions and proof of the relation to hyperdeterminants; (5) a proof of the topological origin of correction terms using the Atiyah-Singer index theorem and explicit heat kernel asymptotics, including the Getzler symbol calculation and localization formula; (6) an extension to non-compact cases using L2 Euler characteristics and relative measures; (7) a unification with Shirokov’s noncommutative Vieta theorem; (8) a generalization to ternary Clifford algebras associated with cubic forms; (9) a universal framework for n-ary Clifford algebras associated with arbitrary fully symmetric n-linear forms, including the universal Wick theorem with precise definition of sign factors, n-ary Pfaffians, and the universal generalized Vieta formula; (10) a complete classification of n-ary Clifford algebras via discriminants, Hasse invariants, and n-ary signatures; (11) the construction of the Absolute Unifier U, a universal object unifying all n-ary Clifford algebras and their polynomial theories. The results unify polynomial theory over quaternions, octonions, Clifford algebras, and higher-degree generalizations, providing new tools for noncommutative algebra, differential geometry, topology, number theory, and mathematical physics.