On Properly Posed Sets Of Nodes For Multivariate Graded Interpolation

Content: Multivariate interpolation is a classical and complicatedmathematical problem. Recent years people have come to focus onmultivariate interpolation especially on multivariate graded interpola-tion. The reason is that multivariate interpolation has a widespreadapplication in many fields, such as neural network technology, geo-logical simulation and weather forecast. So the problem is researchedfurther in this paper. And the poisedness of set of nodes is describedby algebraic geometry theory.The paper is composed of four charpters. In the first charpter, weintroduce the basic theory of multivariate polynomial interpolationand recent production. In the second charpter, using the results ofvariety in algebraic geometry, we give Gro¨bner Bases'method and in-terpolation in algebraic manifold. In the third charpter, On the basicof bivariate Lagrange interpolation along an algebraic curve withoutmultiple factors, we give a kind of method of constructing properlyposed set of nodes for multivariate Lagrange interpolation in R3. Inthe four chapter, we mainly deal with the poisedness of set of nodes inmultivariate graded polynomial interpolation and give new construc-tive methods of properly posed set of nodes in bivariate graded in-terpolation , i.e., Line-Superposition Process and Conic-SuperpositionProcess. Also we prove it reasonable by example.