This paper systematically generalizes the theory of higher-order variations, duality, descent hierarchies, geometric realizations, and arithmetic correspondences to the realm of polynomial equations. We define higher-order variation operators in the algebraic context using formal residues, prove the algebraic versions of the Great Descent Theorem and the Great Ascent Theorem, and introduce spectral curves as the fundamental geometric objects. Descent towers are constructed using Hilbert schemes of points on spectral curves, while ascent towers are given by the corresponding intermediate Jacobians. We develop a Hierarchical Period Number Theorem and a duality pairing of period lattices. A Hierarchical Unified Rank Correspondence is established, linking geometric, algebraic, arithmetic, and analytic ranks. We formulate a Hierarchical Birch-Swinnerton-Dyer Conjecture for algebraic curves and prove it in the function field case. The theory is applied to classify algebraic curves by their descent length, including elliptic curves, hyperelliptic curves, and Fermat curves. Furthermore, we develop a combinatorial interpretation of higher variations using Bell polynomials and Fa\`a di Bruno formulas. The entire framework is extended to higher-dimensional spectral varieties. Finally, an axiomatic formulation is presented, capturing the universal duality principle underlying all these structures. All theorems are provided with complete, rigorous proofs using algebraic methods.
S. B. Liu (Wed,) studied this question.