Some Researches On The Problem Of Multivariate Function Interpolation Along An Algebraic Curve

This paper introduces and studies the problem of Multivariate polynomial simply(that is, choosing compatible set of Nodes guarantee the problem of existence and uniqueness of Multivariate polynomial interpolation), and states we deeply discuss and study the problem of Multivariate polynomial interpolation with the aid of the viewpoint of algebraic geometry. The first chapter introduces some theories about polynomial interpolation, a simple polynomial interpolation problem and to review the existing research results are briefly reviewed. The second chapter,This study through the use of the concept and property of H-bases, gets a new method that use the transverse intersection of two algebraic curves to construct the Properly Posed Set of Nodes on a plane curve:Theorem2.4Assume there exist ak times algebraic curve with no repeat component q(x,y)=0and al times algebraic curve q(x,y)=O meet in lk points nicely, and these points are different, note that C={Qi}i=1,lk but{p,q} is H-bases of ideal I=<p, q>, if B∈In(2)(q)(n≥k-2) and B∩C=(?),then we haveTheorem2.5Assume there exist a k times algebraic curve q(x,y)=0and a conic algebraic curve q(x, y)=0meet in2k different points nicely, note that C={Qi}i=12k, but B∈In(2)(n≥k-2) and B∩C=(?),then we have B∪C＝In+l(2)(q)We get the new methods that Theorem2.4and Theorem2.5generalize the old results of this research direction to a more general situation, at the same times, we deeply master the geometry and basic features of Multivariate polynomial interpolation Properly Posed Of Nodes, and get a more general method and more practical inference, this inference used for structure plane algebraic curve interpolation Properly Posed Of Nodes. The inference following: Inference2.1Assume there exist a k times algebraic curve q(x,y)=0and a conic algebraic curve q(x,y)=0meet in2k different points nicely, then{p,q} must be H-bases of ideal I=<p, q>.The above results have the important practical application value, and provides the theory basis for the practical application of the multivariate polynomial interpolation in manufactures industrial products design and finite element.