GENERALIZED FUNDAMENTAL THEOREM OF ALGEBRA AND VIETA FORMULAS ON EXTERIOR ALGEBRAS: A UNIFIED FRAMEWORK FOR HIGHER ORDER FORMS
Description
This paper establishes a complete and rigorous framework generalizing polynomial theory from division algebras to exterior (Grassmann) algebras. By introducing the concept of regular solutions, we prove that the regular solution set of a real-coefficient polynomial on an exterior algebra consists solely of discrete real roots. Using path-ordering operators to handle non-commutativity, we derive classical Vieta formulas and prove that all topological correction terms vanish due to the nilpotent nature of exterior algebras. Furthermore, we provide a tensor-algebraic construction of the path-ordering operator, prove its well-definedness, basis independence, spectral decomposition, and intrinsic $E_\infty$ operad structure. Through a Wick theorem on exterior algebras, we compute expectations of symmetrized products and discuss connections with the cohomology of Grassmann manifolds. This work unifies polynomial theory on exterior algebras and provides foundational tools for broader non-commutative algebraic geometry.
Files
exterior_basic_theorem.pdf
Additional details
Additional titles
- Alternative title (English)
- GENERALIZED FUNDAMENTAL THEOREM OF ALGEBRA AND VIETA FORMULAS ON EXTERIOR ALGEBRAS
Dates
- Submitted
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2025-12-31
References
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