Mathematical Sciences: Common Zeros of Polynomials in Several Variables and Cubature Formulae

数学科学：多变量多项式的公共零点和体积公式

• 批准号: 9500532
• 负责人: Yuan Xu
• 金额: $ 4万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-05-15 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500532&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Common; Zeros; Polynomials

9500532 Xu The purpose of this research project is to study the common zeros of polynomials in several variables in connection with the numerical approximation to integrals in several variables. Compared to the case of one variable, zeros of polynomials in several variables and high dimensional quadrature formulae are much more difficult and the results have been scattered. Recently the a new approach was used to study the topic and showed that positive high dimensional quadrature formulae can be characterized through the existence of real common zeros of certain set of quasi-orthogonal polynomials; necessary and sufficient conditions for the existence of such zeros were also obtained, which are given in terms of certain nonlinear matrix equations. The project will involve the continuation of this study and will use this approach to conduct a systematic study of this topic. It is very likely that the approach will enable one to tackle several fundamental questions, such as the connection to moment problems in several variables which may lead to an analytic characterization of common zeros, structure of minimal and ``near'' minimal cubature formulae, and construction of new efficient numerical integration formulae. The goal is to establish a unified theory for high dimensional numerical integration formulae based on the common zeros of quasi-orthogonal polynomials. The outcome of the project will help in understanding the structure of high dimensional numerical integration formulae and the structure of the common zeros of polynomials. The information will be very useful in finding new formulae for practical evaluation of high dimensional integrals, which is one of the essential questions in numerical analysis and is often taken as a test problem in high speed computing; it can also be very useful in constructing formulae with special properties, for example, the equal-weight formulae on spheres, which have applications in coding theory. The project is also motivated by the potential a pplications of the outcome in other areas of numerical mathematics, such as orthogonal polynomials in several variables and interpolation by polynomials which are basic tools for data fitting and surface reconstruction.

小行星9500532 本研究计画的目的是探讨多元积分数值逼近中多元多项式的公共零点。与单变量情形相比，多元多项式的零点和高维求积公式的求解要困难得多，且结果比较分散。最近，一种新的方法被用来研究这一问题，证明了正的高维求积公式可以通过某组拟正交多项式的真实的公共零点的存在性来刻画，并给出了这种零点存在的充要条件，这些条件是以某些非线性矩阵方程的形式给出的. 该项目将继续进行这项研究，并将利用这一方法对这一专题进行系统研究。这是很有可能的，该方法将使人们能够解决几个基本问题，如连接到矩问题的几个变量，这可能会导致一个解析表征的公共零点，结构的最小和“近”的最小容积公式，和建设新的高效数值积分公式。目的是建立基于拟正交多项式公共零点的高维数值积分公式的统一理论。 本计画之成果将有助于了解高维数值积分公式之结构及多项式之公零点结构。这些信息将是非常有用的，在寻找新的公式，为实际评估的高维积分，这是一个基本的问题，在数值分析中，往往被视为一个测试问题，在高速计算;它也可以是非常有用的构造公式具有特殊的性质，例如，等权公式的领域，其中有应用的编码理论。该项目的动机也是潜在的applications的结果在其他领域的数值数学，如正交多项式在几个变量和插值多项式，这是数据拟合和曲面重建的基本工具。