Circular Curve On Lagrange Interpolation Problem

In Computational Mathematics，Interpolation problem is one of the basic problems, andmultivariate interpolation is an important research direction, Because of its widely used inmultifunction calculation, the surface of the exterior design, as well as in our real life (forexample securities investment analysis, the survey charts), recent years has become the objectof many mathematical scholars. Multivariate polynomial interpolation is not a simpleextension of a polynomial interpolation, It must first address the properly posed of theproblem, which many theoretical issues need to be solved in the practical application ofresearch. Professor Liang Xuezhang the first time in [1] took the multivariate Lagrangeinterpolation properly posed of the problem into a geometric problem, this is abreakthrough point of the interpolation problem, which by means of the methods of algebraicgeometry to study the theory and method of construction of multivariate polynomialinterpolation properly posed set of nodes. This article is on the basis of the above researchresults, continue to explore the Lagrange interpolation in R2, introduce the concept of weakGr bner basis, and combine with the conclusions of the well-known Cayley-Bacharachtheorem in algebraic geometry, get the new method of construct along the circumferencecurve interpolation properly posed set of nodes, which is using circumference curveintersecting with any algebraic curve, as well as some inferences and examples.In the preamble, mainly introduces interpolation problems of the development processand research background, and previous interpolation research achievements and conclusions.This paper is divided into three chapters. In the first chapter introduces the basic concepts andtheoretical methods of multivariate interpolation, and the recent development of multivariateinterpolation, which results are summarized, the introduction of the concept of weak Gr bnerbasis is to extend the results. The second chapter is based on the definition of binary Lagrangeinterpolation appropriate set of nodes by Liang Xuezhang, studies circumference curves andstraight lines intersecting to construct node group which meet the conditions, focus onexploring the constructor along the circumference curve interpolation appropriate set of nodes,and prove and deduce the conclusion. In order to more in-depth and extensive discussion ofthe results, applies the conclusions of the theorem of Cayley-Bacharach, and gives someconstructor to facilitate the practical application. The third chapter contains some corollariesand proof of the theorems.