多元拉格朗日插值及Cayley-Bacharach定理的一个应用
Multivariate Lagrange Interpolation and an Application of Cayley-Bacharach Theorem For it
AI总结：
本文深入研究了多元拉格朗日插值，提出了充分交的概念，利用Bezout定理等证明了沿充分交代数流形的多项式空间维数，并基于Mysovskikh定理给出了适定节点集存在性的刻画条件，最后通过推广的Cayley-Bacharach定理解决了低维代数流形上的插值问题。
摘要：
本文深入研究了n元拉格朗日插值，并给出了Cayley-Bacharach定理的一个应用。我们提出了n维复欧氏空间中s个代数超曲面的充分交的概念，并讨论了沿充分交代数流形的拉格朗日插值。借助一些定理（如Bezout定理、Macaulay定理等），我们证明了沿充分交代数流形S的多项式空间P(n)_m的维数，并利用向后差分算子给出了维数计算的便捷表达式。根据Mysovskikh定理，我们给出了沿充分交代数流形插值时任意次数的适定节点集存在性的证明及其刻画条件。进一步指出，对于充分交的s个代数超曲面，多项式集必须构成理想的H-基。作为本文的主要结果，我们推导了沿代数流形构造拉格朗日插值适定节点集的一般方法，即叠加插值过程。在论文末尾，我们利用推广的Cayley-Bacharach定理解决了沿0维和1维代数流形的拉格朗日插值问题。正是Cayley-Bacharach定理的应用构成了通过叠加插值过程沿高维代数流形构造适定节点集的起点。
英文摘要：
In this paper,we deeply research Lagrange interpolation of n-variables and give an application of Cayley-Bacharach theorem for it. We pose the concept of sufficient intersection about s algebraic hypersurfaces in n-dimensional complex Euclidean space and discuss the Lagrange interpolation along the algebraic manifold of sufficient intersection. By means of some theorems (such as Bezout theorem, Macaulay theorem and so on) we prove the dimension for the polynomial space P(n) m along the algebraic manifold S of sufficient intersection and give a convenient expression for dimension calculation by using the backward difference operator. According to Mysovskikh theorem, we give a proof of the existence and a characterizing condition of properly posed set of nodes of arbitrary degree for interpolation along an algebraic manifold of sufficient intersection. Further we point out that for s algebraic hypersurfaces of sufficient intersection, the set of polynomials must constitute the H-base of ideal. As a main result of this paper, we deduce a general method of constructing properly posed set of nodes for Lagrange interpolation along an algebraic manifold, namely the superposition interpolation process. At the end of the paper, we use the extended Cayley-Bacharach theorem to resolve some problems of Lagrange interpolation along the 0-dimensional and 1-dimensional algebraic manifold. Just the application of Cayley-Bacharach theorem constitutes the start point of constructing properly posed set of nodes along the high dimensional algebraic manifold by using the superposition interpolation process.
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