Abstract: We researched the problem of Lagrange interpolation of polynomial space on the algebraic manifold. We posed the concept of sufficient intersection about s(1≤s≤n) algebraic hypersurfaces in n-dimensional space and proved the dimension of polynomial space P⁽ⁿ⁾_m(which denotes the space of all multivariate polynomials of total degree≤m) on the algebraic manifold S=s(f₁,…, f_s) (where f₁(X)=0,…, f s(X)=0denote s algebraic hypersurfaces) of sufficient intersection, then gave a convenient expression for dimension calculation by using the backw ard difference operator. We deduced a general method of constructing properly posed set of nodes for Lagrange interpolation on the algebraic manifold, namely, the superposition interpolation process. The existence of properly posed set of nodes of arbitrary degree for interpolation on the algebraic manifold of sufficient intersection was proved. At the end of this paper we gave the characterizing conditions of properly posed set of nodes forinterpolation.

Key words: the dimension of polynomial space, Lagrange interpolation on an algebraicmanifold, properly posed set of nodes for Lagrange interpolation