Topic #10 -------------- OP-SF NET --------------- February 9, 1995 ~~~~~~~~~ From: Yuan Xu
Subject: New book on Orthogonal Polynomials in Several Variables The following Research Note appeared: Yuan Xu "Common Zeros of Polynomials in Several Variables and Higher Dimensional Quadrature" Pitman Research Notes in Mathematics Series, 312 Essex, 1994. Abstract: This research note presents a systematic study of the common zeros of sets of polynomials in several variables which are related to higher dimensional quadrature (often called cubature). Just like the classical Gaussian quadrature formula, a cubature formula of degree 2n-1 needs at least N nodes, where N denotes the dimension of the subspace of polynomials of degree at most n-1. A cubature formula whose number of nodes is equal to N exists if, and only if, the corresponding orthogonal polynomials of degree n have N real and distinct common zeros. However, since such a cubature does not exist in general, one is led to study cubature formulae with more nodes, the most interesting cases being the ones with a minimal number of nodes. In the recent development of orthogonal polynomials in several variables, the common zeros of orthogonal polynomials are characterized as the joint eigenvalues of a family of block Jacobi matrices defined through the coefficient matrices of a three-term relation in vector-matrix form. The main results in this book show that, in general, cubature formulae are based on the real and distinct common zeros of a family of quasi-orthogonal polynomials, and the existence of these common zeros can be characterized through the solvability of certain nonlinear matrix equations derived from the modified block Jacobi matrices. The approach is parallel to that of one variable, which differs significantly from earlier ones based on the algebraic ideal theory. The book is in essence a research paper; a good portion of the theorems are new and, in many cases, new proofs are given for known results. The content is basically self-contained, including one section that summarizes the recent results on the general structure of orthogonal polynomials in several variables.