Construction of Algebraic Cubature Rules Using Polynomial Ideal Theory

Abstract

Cubature formulae of fixed degree using the minimum number of nodes, the common zeros of a set of polynomials, are constructed by a number of techniques based on the theory of polynomial ideals. Examples demonstrate that known lower bounds to the number of nodes can be attained though usually these bounds are too severe. One example is shown to give rise to an interlacing family of rules, a two dimensional analogue of the Clenshaw–Curtis quadrature. An effective numerical procedure is also given for finding all the common zeros of a set of polynomials.