Unified Analytic Solution of Polynomial Equations in Vector Algebraic Closure

This paper extends the differential algebraic framework for solving polynomial equations to the vector algebraic setting. We establish a rigorous theoretical foundation for solving systems of multivariate polynomial equations and matrix eigenvalue problems within a vector algebraic clo sure V. The solution methodology provides explicit analytical expressions for roots of multivariate polynomial systems and eigenvalues of matrices, taking the unified form: n−1xk =x(n−1) + j=1Φj(Y)1/nωj(k−1) n, 0 ≤k≤n−1,where x(n−1) is the multivariate critical point, Y = (y(0),…,y(n−2)) are vector critical values, Φj ∈ Q(A)Y are explicit vector polynomials with combinatorial correction terms, and ωn = e2πi/n. We provide complete constructive proofs, derive combinatorial expressions for the correction coefficients γ(n)j , and present detailed algorithms with complexity analysis. Extensive numerical validation demonstrates machine-precision accuracy across various test cases, including ill-conditioned multivariate systems and challenging eigenvalue problems. This work reconciles with classical impossibility results while demonstrating that explicit analytic solutions exist in appropriately extended algebraic structures.