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In this paper, an intersection theory for generic differential polynomials ispresented. The intersection of an irreducible differential variety of dimension$d$ and order $h$ with a generic differential hypersurface of order $s$ isshown to be an irreducible variety of dimension $d-1$ and order $h+s$. As aconsequence, the dimension conjecture for generic differential polynomials isproved. Based on the intersection theory, the Chow form for an irreducibledifferential variety is defined and most of the properties of the Chow form inthe algebraic case are established for its differential counterpart.Furthermore, the generalized differential Chow form is defined and itsproperties are proved. As an application of the generalized differential Chowform, the differential resultant of $n+1$ generic differential polynomials in$n$ variables is defined and properties similar to that of the Macaulayresultant for multivariate polynomials are proved.

Intersection Theory in Differential Algebraic Geometry: Generic Intersections and the Differential Chow Form