Unified Solving of Multivariate Polynomial Systems via Hierarchical Differential Algebraic Closure: A Foundation for Symbolic-Numeric Quantum Computation

This paper presents a unified theoretical and computational framework for solving multivariate polynomial systems through a novel construction of hierarchical differential algebraic closure (DAC). Building upon the univariate differential algebraic method established in [3], we generalize this approach to arbitrary dimensions by constructing a recursively layered closure space incorporating partial derivative operators. For a system of m polynomials f(x) = 0 in n variables with maximal total degree d, we prove that all solutions can be expressed analytically through an explicit representation theorem involving representation-theoretic components. Our method achieves machine-precision accuracy (residuals < 10−28) with complexity O(dn) for sparse systems, significantly outperforming traditional Gr¨obner basis and homotopy continuation methods. Extensive validation across binary, ternary, and high-dimensional systems confirms the robustness, efficiency, and numerical stability of our approach. Furthermore, the proposed framework naturally extends to incorporate generalized symmetries and tensor network representations, opening up novel pathways for hybrid symbolic-numeric computation and quantum acceleration. We also establish the theoretical consistency of our approach with the Abel-Ruffini theorem, demonstrating that while solutions in radicals are impossible for general quintic and higher-degree equations, explicit analytic solutions exist in the appropriately extended differential algebraic closure K.