用里特-吴特征集解代数方程直至重数的一个方法(英文)  

A Method to Solve Algebraic Equations Up to Multiplicities via Ritt-Wu′s Characteristic Sets

作　　者：李邦河[1] 

机构地区：[1]中国科学院数学与系统科学研究院,北京100080

出　　处：《应用泛函分析学报》2003年第2期97-109,共13页Acta Analysis Functionalis Applicata

基　　金：Supportedpartiallybythe973programofP.R.China,andInnovitionFundofAMSS

摘　　要：　里特-吴特征集提供了用计算机解代数方程的有效方法,但迄今为止,还不能由这一方法给出孤立解的重数.文章给出了孤立解的重数的两个定义,它们是等价的,并且在范德瓦尔登的定义有意义时与后者一致.一个定义是在非标准分析的框架中,另一个则是标准分析的.在证明与范德瓦尔登的定义一致时,非标准分析的定义是本质的.通过再一次在计算机上应用里特-吴方法于由原方程得到的含无穷小参数的代数方程,可以得到原方程的孤立解的重数.文中给出一个例子的计算机计算结果:首先得出有八个解,然后给出它们的重数:其中有两个的重数为六重,另六个为单根.Ritt\|Wu′s method of characteristic sets is an effective method to solve algebraic equations by computer, while one cannot get the multiplicity of an isolated solution by this method so far. We introduce two definitions for the multiplicity of an isolated solution, they are equivalent and coincide with that of Van der Waerden in the case the later works. One of our definitions is in the framework of nonstandard analysis, and another is in standard analysis. To prove the equivalence with that of Van der Waeden, the nonstandard one is essential. By using Ritt\|Wu′s method on computer again to the algebraic equations with some infinitesimals as parameters obtained by modifying the original equations, we can then determine the multiplicity of an isolated solution of the original equations.\;An example by computer is given: first get 8 solutions, then compute their multiplicities\|6 of them are simple roots, while each of the other two has multiplicity 6.

关 键 词：里特-吴特征集 代数方程 孤立解 重数 标准分析 范德瓦尔登 定义 计算机 

分 类 号：O241[理学—计算数学] O151.1[理学—数学]