Unified Solving of Multivariate Polynomial Systems via Hierarchical Differential Algebraic Closure: A Foundation for Symbolic-Numeric Quantum Computation
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Key Points
All solutions to multivariate polynomial systems can be expressed analytically through a representation theorem.
Our method achieves machine-precision accuracy with complexity O(dn) for sparse systems, outperforming Gröbner basis methods.
Extensive validation across various systems confirms the robustness and numerical stability of the hierarchical closure approach.
The proposed framework extends to hybrid symbolic-numeric computation and quantum acceleration with generalized symmetries.
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Implication
This framework demonstrates machine-precision solutions in polynomial systems, highlighting efficiency and numerical stability.