We consider the problem of finding the isolated common roots of a set of
polynomial functions defining a zero-dimensional ideal I in a ring R of
polynomials over C. We propose a general algebraic framework to find the
solutions and to compute the structure of the quotient ring R/I from the null
space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous
and multi-homogeneous cases are treated. In the presented framework, the
concept of a border basis is generalized by relaxing the conditions on the set
of basis elements. This allows for algorithms to adapt the choice of basis in
order to enhance the numerical stability. We present such an algorithm and show
numerical results
