Multivariate Stieltjes type theorems and location of common zeros of multivariate orthogonal polynomials

Invariant factors of bivariate orthogonal polynomials inherit most of the properties of univariate orthogonal polynomials and play an important role in the research of Stieltjes type theorems and location of common zeros of bivariate orthogonal polynomials. The aim of this paper is to extend our study of invariant factors from two variables to several variables. We obtain a multivariate Stieltjes type theorem, and the relationships among invariant factors, multivariate orthogonal polynomials and the corresponding Jacobi matrix. We also study the location of common zeros of multivariate orthogonal polynomials and provide some examples of tri-variate.