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

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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.