This paper develops a rigorous exterior differential algebraic framework for explicit parameterizations of exterior algebraic varieties. Building on classical differential algebra and exterior algebra, we construct explicit local parameterizations for exterior algebraic varieties using differential extensions of function fields equipped with exterior product structures. The key innovation is the systematic integration of exterior algebraic constraints into the differential algebraic parameterization process, with combinatorial correction terms derived from the geometry of tangent cones and higher-order neighborhoods in the exterior setting. We provide complete proofs for the main theorems, detailed examples including Grassmannians and exterior singularities with complete verifications, and establish precise connections with classical theories such as Schubert calculus and resolution of singularities. The framework offers new computational tools while maintaining full mathematical rigor. Our approach is restricted to irreducible exterior algebraic varieties over fields of characteristic zero, and we carefully address the technical conditions required for the existence of parameterizations, particularly the rationality of tangent cones and convergence properties in exterior differential algebraic extensions. We also provide algorithms for verifying these conditions and computing the parameterizations.
shifa liu (Wed,) studied this question.