【作者】 徐艳；

【导师】 崔利宏；

【作者基本信息】 辽宁师范大学 ， 应用数学， 2007， 硕士

【摘要】 本文对多元多项式分次插值适定结点组的构造理论进行了深入的研究与探讨。在沿无重复分量代数曲线进行Lagrange插值的基础上，我们给出了沿无重复分量分次代数曲线进行分次Lagrange插值的方法，并利用这一结果进一步给出在R~2上构造分次Lagrange插值适定结点组的基本方法。另外，利用弱Grobner基这一新的数学概念，以及构造平面代数曲线上插值适定结点组的理论，我们进一步给出了构造平面分次代数曲线上分次插值适定结点组的方法。同时，利用代数几何学中关于理想和代数簇的理论，研究了代数超曲面上分次插值适定结点组的几何结构，通过上述理论的研究，并利用无重复分量代数超曲面上的分次插值适定结点组的构造方法，我们又得到了构造高维空间中分次插值适定结点组的递归构造方法，从而基本上弄清了多元分次Lagrange插值适定结点组的几何结构和基本特征。另一方面，在研究插值结点组的移动对插值适定性的影响这一问题时，我们对以往一个重要定理加以推广并给出了证明，从而得到了更一般性的结论。更多还原

【Abstract】 The constitution theory of a properly posed set of nodes forthe multivariate polynomial graded interpolation is studied deeply inthe paper. On the basis of Lagrange interpolation which along the al-gebraic curve without multiple factors,we give the approach of gradedLagrange interpolation which along the algebraic curve without mul-tiple factors. Futhermore,using this result we give a basic method toconstruct the graded Lagrange interpolation in R~2. In addition,usingweak Grobner basis’ method which is a new mathematic concept andthe theory for constructing the properly posed set of nodes for inter-polation on plane algebraic curve,we give the method to construct theproperly posed set of nodes for graded interpolation on plane algebraiccurve accordingly. At the same time, using the results of algebraic vari-ety and ideal in algebraic geometry, we study the geometrical structureof properly posed set of nodes for graded interpolation on algebraic hy-persurface, more over,we give a Hyperplane Superposition Process toconstruct the properly posed set of nodes for graded interpolation onalgebraic hypersurface,therefore we make clear the geometrical struc-ture of properly posed set of nodes for multivariate graded interpola-tion basically.On the other hand,when we study the effect of movinginterpolation nodes to the property for interpolation,we improve andprove an important theorem gao, thus we acquire the general conclu-sion.更多还原