The purpose of this monograph is to study the common zeroes of families of polynomials in several variables which are related to higher dimensional quadrature. It aims to introduce a new approach and use it to conduct a systematic study.

Preface -- 1. Introduction -- 1.1 Review of the theory in one variable -- 1.2 Background to the theory in several variables -- 1.3 Outline of the content -- 2. Preliminaries and Lemmas -- 2.1 Orthogonal polynomials in several variables -- 2.2 Centrally symmetric linear functional -- 2.1 Lemmas -- 3. Motivations -- 3.1 Zeros for a special functional -- 3.2 Necessary conditions for the existence of minimal cubature formula -- 3.3 Definitions -- 4. Common Zeros of Polynomials in Several Variables: First Case -- 4.1 Characterization of zeros -- 4.2 A Christoffel-Darboux formula -- 4.3 Lagrange interpolation -- 4.4 Cubature formula of degree 2n — 1 -- 5. Moller’s Lower Bound for Cubature Formula -- 5.1 The first lower bound -- 5.2 Moller’s first lower bound -- 5.3 Cubature formulae attaining the lower bound -- 5.4 Moller’s second lower bound -- 6. Examples -- 6.1 Preliminaries -- 6.2 Examples: Chebyshev weight function -- 6.3 Examples: product weight function -- 7. Common Zeros of Polynomials in Several Variables: General Case . 85 -- 7.1 Characterization of zeros 86 -- 7.2 Modified Christoffel-Darboux formula 93 -- 7.3 Cubature formula of degree 2n — 1 96 -- 8. Cubature Formulae of Even Degree99 -- 8.1 Preliminaries 99 -- 8.2 Characterization 101 -- 8.3 Example 105 -- 9. Final Discussions 108 -- 9.1 Cubature formula of degree 2n — s 108 -- 9.2 Construction of cubature formula, afterthoughts 112 -- References.