中文摘要：

研究高维空间中代数流形上多项式空间的Lagrange插值问题,给出了n维空间中s（1≤s≤n）个代数超曲面充分相交的概念，证明了n元m次多项式空间Pm^（n）在充分相交的代数流形S=s（f1，…，fs，）（f1（X）=0，…，f1（X）=0表示s个代数超曲面）上的维数，并利用倒差分算子给出一个方便计算的表达式；构造了沿代数流形上插值适定结点组的叠加插值法；证明了在充分相交的代数流形上任意次插值适定结点组的存在性；给出代数流形上插值适定结点组的性质和判定条件．

英文摘要：

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 Pm^（n） which denotes the space of all multivariate polynomials of total degree ≤m） on the algebraic manifold S = s （f1,…, fs ） （ where f1（X） = 0,…, fx （X） = 0 denote s algebraic hypersurfaces） of sufficient intersection, then gave a convenient expression for dimension calculation by using the backward 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 for interpolation.