Multivariate Lagrange Interpolation and an Application of Cayley-Bacharach Theorem For it

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(1 ≤ s ≤ n) 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 = s(f1, · · · , fs)(where f1(X) = 0, · · · , fs(X) = 0 denote s algebraic hypersurfaces ) 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 f1(X) = 0, · · · , fs(X) = 0 of sufficient intersection, the set of polynomials f1, · · · , fs must constitute theH-base of ideal Is =< f1, · · · , fs >. 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.