ABSTRACT
The relation between Gaussian quadrature and zeros of orthogonal polynomials are well-known. This chapter gives a quick tour of what is known on the zeros of orthogonal polynomials in several variables and its relation to cubature formulae. It provides a complete characterization of the common zeros of certain quasi-orthogonal polynomials and of the related cubature formulae. The chapter presents two preliminary results which motivated study; the second one gives the necessary conditions for the existence of minimal cubature formulae and defines the set of polynomials whose common zeros we will study. The word cubature is used for the higher dimensional quadrature which seems to be well accepted among numerical analysts, we shall adopt this terminology throughout this monograph. Starting with the three-term relation, many properties of orthogonal polynomials in several variables have been derived.
