On the Equality of Algebraic and Analytic Ranks for Higher-Dimensional Varieties: ADifferential Algebraic Approach

This paper establishes a comprehensive generalization of the equality between algebraic and analytic ranks from algebraic curves to higher-dimensional algebraic varieties defined over number fields. We construct specialized differential algebraic closures that simultaneously capture the algebraic structure of intermediate Jacobians and the analytic properties of Hasse-Weil L-functions in arbitrary dimension. Within these closures, we develop explicit higher-dimensional parameterizations of algebraic varieties and derive spectral representations of geometric partial differential equations governing the arithmetic and analytic structures. The main theorem demonstrates that the algebraic rank of an appropriate Albanese or intermediate Jacobian variety equals the analytic rank of the associated Hasse-Weil L-function, providing a unified geometric-analytic perspective on this fundamental arithmetic relationship. Our approach synthesizes techniques from differential algebra, algebraic geometry, analytic number theory, and spectral theory, offering new insights into the geometric foundations of higher-dimensional analogues of the Birch and Swinnerton-Dyer conjecture. All constructions are carried out with complete mathematical rigor, and explicit computational frameworks with certified error bounds are provided.