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

9500532徐，这个研究项目的目的是研究多元多项式的公共零点与多元积分的数值逼近有关。与一元情形相比，多元多项式的零点和高维求积公式要困难得多，而且结果比较分散。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些拟正交多项式集的实公共零点的存在来刻画，也得到了此类零点存在的充要条件，并以某些非线性矩阵方程的形式给出了这些条件。该项目将涉及这项研究的继续，并将使用这一方法对这一专题进行系统研究。这种方法很可能使人能够解决几个基本问题，例如与可能导致公共零点的解析表征的多变量矩问题的联系、最小和“接近”最小体积公式的结构，以及新的有效的数值积分公式的构造。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。该项目的成果将有助于理解高维数值积分公式的结构和多项式的公共零点的结构。高维积分是数值分析中的一个基本问题，也是高速计算中经常被作为测试问题的一个问题，这些信息将非常有用地寻找新的公式用于实际计算；它也可以非常有用地构造具有特殊性质的公式，例如在编码理论中有应用的球面上的等权公式。该项目的动机还在于该成果在其他数值数学领域的潜在应用，如多元正交多项式和多项式插值，这些都是数据拟合和曲面重建的基本工具。

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.