Presents a systematic study of the common zeros of polynomials in several variables which are related to higher dimensional quadrature. The author uses a new approach which is based on the recent development of orthogonal polynomials in several variables and differs significantly from the previous ones based on algebraic ideal theory. Featuring a great deal of new work, new theorems and, in many cases, new proofs, this self-contained work will be of great interest to researchers in numerical analysis, the theory of orthogonal polynomials and related subjects.

Introduction
Preliminaries and lemmas
Motivations
Common zeros of polynomials in several variables: first case
Moller's lower bound for cubature
formula
Examples
Common zeros of polynomials in several variables: general case
Cubature formulae of even degree
Final discussions

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