﻿ 多元齐次多项式插值问题研究

# 多元齐次多项式插值问题研究Research on Interpolation of Multivariate Homogeneous Polynomials

Abstract: Based on the basic theory and method of algebraic geometry, this paper studies the problem of multivariate homogeneous polynomial interpolation. In this paper, we obtain the superposition construction method of constructing multivariate homogeneous interpolation polynomial space and interpolation regular node group along multivariate homogeneous algebraic hyper surfaces. The topological structure of the regular node group of multivariate homogeneous interpolation is basically clarified, and an experimental example is given to verify the theoretical method.

1. 引言

2. 基本概念和主要定理

${H}_{n}^{\left(s\right)}=\left\{\underset{{\alpha }_{1}+\cdots +{\alpha }_{s}=n}{\sum }{x}_{1}^{{\alpha }_{1}}{x}_{2}^{{\alpha }_{2}}\cdots {x}_{s}^{{\alpha }_{s}}|{a}_{{\alpha }_{1}{\alpha }_{2}\cdots {\alpha }_{s}}\in R\right\}$

${d}_{n}^{\left(s\right)}=\left(\begin{array}{l}n+s-1\\ \text{}s-1\end{array}\right)$$A={\left\{{Q}_{i}\right\}}_{i=1}^{{d}_{n}^{\left(s\right)}}$${R}^{s}$ 中的 ${d}_{n}^{\left(s\right)}$ 个相异点，任意给定一个实数组 ${\left\{{f}_{i}\right\}}_{i=1}^{{d}_{n}\left(s\right)}$，找到一个多项式 $p\left(x\right)\in {H}_{n}^{\left(s\right)}$ 那么这个多项式需要适合以下的插值条件：

$p\left({Q}_{i}\right)={f}_{i},\text{}i=1,\cdots ,{d}_{n}^{\left(s\right)}$ (1)

${d}_{n}^{\left(s\right)}\left(k\right)=\left(\begin{array}{l}n+s-1\\ \text{}s-1\end{array}\right)-\left(\begin{array}{l}n+s-k-1\\ \text{}s-1\end{array}\right)$ (2)

$h\left({Q}_{i}\right)={f}_{i},\text{}i=1,2,\cdots ,{d}_{n}^{\left(s\right)}\left(k\right)$ (3)

3. 定理的证明

$V\left({I}_{1}\right)\subset V\left({I}_{2}\right)$$I\left(V\left({I}_{1}\right)\right)\supset I\left(V\left({I}_{2}\right)\right)$

$I\left(V\left({I}_{1}\right)\right)=\sqrt{{I}_{1}}={I}_{1}$$I\left(V\left({I}_{2}\right)\right)=\sqrt{{I}_{2}}\supset {I}_{2}$${I}_{1}\supset {I}_{2}$

${h}_{n+k}^{\left(s\right)}=q\left(x\right)r\left(x\right)$ (4)

$A$${H}_{n}^{\left(s\right)}$ 的正则插值结点组， $r\left(x\right)\in {H}_{n}^{\left(s\right)}$ 这显然与 $A$${H}_{n}^{\left(s\right)}$ 的正则插值结点组矛盾。

(5)

, (6)

(7)

4. 具体构造方法及实验算例

Figure 1. Effect diagram of bivariate homogeneous interpolation

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