关于二元多项式插值结点平面构形问题的研究  

Research on Nodal Planar Configurations of Bivariate Polynomial Interpolation

作　　者：崔利宏[1] 刘汉香[1] 冯大雨[1] 

机构地区：[1]辽宁师范大学数学学院,辽宁大连116029

出　　处：《吉林师范大学学报（自然科学版）》2007年第4期3-5,34,共4页Journal of Jilin Normal University:Natural Science Edition

基　　金：辽宁省高等学校优秀人才支持计划(RC-04-11)

摘　　要：1977年,为了研究平面上插值结点的分布情况,使之能够惟一确定一个二元Lagrange插值多项式,Chung和Yao在[5]中首次引入几何特征(GC)这一概念,并使得所构造出的Lagrange函数是一次实系数多项式乘积的形式.1982年Gasca和Maeztu在[6]中给出了平面上任何一个满足GC条件且含(n+2)(n+1)/2个点的集合必有其中n+1点共线的猜想.后来,Carnicer与Gasca在[3]中对该猜想在n≤4的情况下给出了证明,并在[4]中从亏量的角度对满足GC条件的结点集进行了探讨.此文章则对该猜想在n=5的情形进行了研究并给出了相应的结果,该结果推广了Carnicer和Gasca在[3]与[4]中所得到的主要结论.In 1977, in order to study the distributions of interpolation nodes in the plane, in which a bivariate Lagrange interpolation polynomial can be. uniquely determined, Chung and Yao first introduced in [5] the concept Geometric Characterization （GC）, and made Lagrange functions products of 1st - degree polynomials of real coefficients. In 1982, Gasca and Maeztu put forward in [ 6] a conjecture stating that any set of （ n ＋ 2） （ n ＋ 1 ）/2 points in the plane satisfying the GC condition must contain in the set n ＋ 1 coUinear points. Later, Carnicer and Gasca gave the proof to the conjecture for the cases n ≤4 in [3] and discussed in [4] the set of nodes satisfying the GC condition from the point of defect. This paper considers then the conjecture for the case n = 5 and gives the corresponding reauhs, which generalize the conclusions of Carnieer and Gasca obtained in [3] and [4].

关 键 词：二元多项式插值 几何特征 亏量 

分 类 号：O241.3[理学—计算数学]