Superposition interpolation process in Cn

In this paper, we deeply research Lagrange interpolation by polynomial in several variables and give an application of Cayley-Bacharach theorem for it. Particularly on the sufficiently intersected algebraic manifold (or SIAM, for short), we introduce a general method of constructing properly posed set of nodes (or PPSN, for short) for Lagrange interpolation, namely the superposition interpolation process. Then we give an equivalent condition about a PPSN along a SIAM. Further we introduce a relation between the sufficiently intersected algebraic hypersurfaces and H-basis. At the end of this paper, we use the extended Cayley-Bacharach theorem to resolve some problems of Lagrange interpolation along the zero-dimensional and one-dimensional algebraic manifolds.