Differential Algebraic Intersection Theory: A Unified Framework with Explicit Computations

This paper develops a comprehensive differential algebraic framework for intersection theory,unifying classical methods with modern computational and theoretical approaches. We construct the differential intersection closure KInt(X), a differentially closed field extension that encodes geometric information about subvarieties, their tangent cones, and intersection multiplicities. Within this closure, we derive explicit formulas for local intersection numbers using combinatorial correction terms derived from higher-order tangent geometry. Our formulas are valid for arbitrary isolated intersections and provide algorithmic implementations even in non-transverse and singular settings,without requiring Cohen–Macaulay hypotheses. The framework extends naturally to singular varieties via weighted parametrizations, to arithmetic geometry via p-adic and analytic closures, and to virtual intersection theory via differential derived closures. We provide complete proofs, detailed algorithms with certified error bounds, and implementations in computer algebra systems. The work establishes new connections between differential algebra, classical intersection theory, enumerative geometry, and arithmetic geometry, and points to potential links with emerging methods in theoretical physics.