A Unified Solution Formula for Polynomial Equations in Noncommutative Operator Algebras via Differential-Combinatorial Methods

This paper establishes a comprehensive framework for solving degree-npolynomial equations P(x) = 0 in unital noncommutative operator algebras, including C*-algebras, von Neumann algebras, and general operator systems. We construct a noncommutative operator differential-algebraic closure that extends the coefficient algebra by adjoining critical values of P, all roots of unity, and is closed under the extraction of p-th roots via holomorphic functional calculus and under solving polynomial equations with ordered noncommutative coefficients.