This paper provides a complete solution to Hilbert’s 15th problem—“to establish a rigorous foundation for Schubert’s enumerative calculus.” We construct, for any algebraic variety X over a field of characteristic zero, a differential algebraic intersection closure KInt(X), which provides a unified and constructive framework for intersection theory. Within this closure, all intersection numbers, Schubert structure constants, and enumerative invariants are represented as explicit solutions to systems of differential equations encoding geometric conditions.We give exhaustive definitions, prove the existence and uniqueness of KInt(X) via a carefully stratified recursive construction (avoiding circularity), and establish its fundamental properties. Intersection numbers are intrinsically defined via deformation theory and characterized as constant solutions to differential equations, thereby establishing their rationality and integrality. The framework rigorously recovers all basic results of Schubert calculus, including explicit formulas for Littlewood Richardson coefficients, Giambelli’s formula, and Pieri’s rule, all derived from first principles within the differential algebraic setting.We prove full compatibility with classical theories: an explicit isomorphism is constructed between the constant subfield of KInt(X) and the rational Chow ring A∗(X) ⊗Z Q, and an embedding of the quantum cohomology ring is realized. For computing intersection numbers and Schubert constants,this paper provides effective algorithms, accompanied by certified error bounds via interval arithmetic and complexity analysis.The work further extends the framework to singular varieties (via resolution), positive characteristic (using Hasse derivatives), and connections with motivic integration and arc spaces. This solution comprehensively addresses Hilbert’s requirements: establishing a rigorous foundation for Schubert calculus, proving the rationality of enumerative numbers, developing a general intersection theory foralgebraic varieties, establishing deformation invariance—all within a constructive, computationally tractable framework.
shifa liu (Wed,) studied this question.