Applications Of Polynomial Ideals To Multivariate Interpolation

The interpolation problem is a classical mathematical problem , and at the same time a basic mathematical problem in Computational Mathematics. The initial interpolation theory and methods are now largely perfected. In the 1980s, the focus of the interpolation studies begun to turn to the multiple interpolation, especially in the multi-polynomial interpolation theories and methods has made a number of new result. It is also very concerned about the geometric structure of the set of the group. In 1998, Liang Xuezhang and Lv Chunmei discussed the Lagrange interpolation along the plane algebraic curves without multiple factors and by using the conclusions of Caramer theorem, launched a more in-depth and extensive discussion on the constructive methods of properly posed sets of interpolation nodes and gave some practical constructive methods of properly posed sets of nodes. By using the basic theories of ideals and algebraic sets in the Algebraic Geometry, the multivariate Lagrange interpolation problems in the three dimensional Euclidean Space are dealt with in this paper first. We propose a new definition i.e. Surfaces can be expressed and provide a new method for constructing the properly posed sets of nodes along algebraic surfaces i.e. Algebraic Superposition Process. Then, the multivariate Lagrange interpolation problems in the s-dimensional Euclidean Space are dealt with in this paper. We propose a new definition i.e. Hypersurfaces can be expressed. we provide a Hypersurface-Superposition Process to construct the PPSN for interpolation along an algebraic hypersurface and in polynomial space, It generalizes the result that is a Hyperplane-Superposition Process to construct the PPSN for interpolation along an algebraic hypersurface , which was X. Z. Liang, C. M. Lu and R. Z. Feng obtained in literature [17]. We also give a recursive construction method in order to constructing the properly posed sets of nodes for Lagrange interpolation in space. And as a result, we offer a clear understanding of the geometric structure of PPSN for multivariateLagrange interpolation in P_n~s, which is of great significance in surface split-joint, scattered data interpolation and fitting, etc.This article is divided into four parts, the first part is the introduction of the paper, and the first section is devoted to the basic theories of the multiple interpolation, the second section describes the basic theories of the Algebraic Geometry, especially the Lagrange interpolation problem. The third section provides a surface-Superposition Process for constructing the properly posed sets of nodes along algebraic surfaces in the three dimensional Euclidean Space and a Hypersurface-Superposition Process to construct the PPSN for multivariate Lagrange interpolation in space. We also give a recursive construction method in order to constructing the properly posed sets of nodes for Lagrange interpolation in space.