  In this paper,we deeply research Lagrange interpolation of n-variables and
give an application of Cayley-Bacharach theorem for it. We pose the concept of
sufficient intersection about s algebraic hypersurfaces in n-dimensional
complex Euclidean space and discuss the Lagrange interpolation along the
algebraic manifold of sufficient intersection. By means of some theorems (such
as Bezout theorem, Macaulay theorem and so on) we prove the dimension for the
polynomial space P(n) m along the algebraic manifold S of sufficient
intersection and give a convenient expression for dimension calculation by
using the backward difference operator. According to Mysovskikh theorem, we
give a proof of the existence and a characterizing condition of properly posed
set of nodes of arbitrary degree for interpolation along an algebraic manifold
of sufficient intersection. Further we point out that for s algebraic
hypersurfaces of sufficient intersection, the set of polynomials must
constitute the H-base of ideal. As a main result of this paper, we deduce a
general method of constructing properly posed set of nodes for Lagrange
interpolation along an algebraic manifold, namely the superposition
interpolation process. At the end of the paper, we use the extended
Cayley-Bacharach theorem to resolve some problems of Lagrange interpolation
along the 0-dimensional and 1-dimensional algebraic manifold. Just the
application of Cayley-Bacharach theorem constitutes the start point of
constructing properly posed set of nodes along the high dimensional algebraic
manifold by using the superposition interpolation process.
