  In this paper, we describe new methods to compute the radical (resp. real
radical) of an ideal, assuming it complex (resp. real) variety is finite. The
aim is to combine approaches for solving a system of polynomial equations with
dual methods which involve moment matrices and semi-definite programming. While
the border basis algorithms of [17] are efficient and numerically stable for
computing complex roots, algorithms based on moment matrices [12] allow the
incorporation of additional polynomials, e.g., to re- strict the computation to
real roots or to eliminate multiple solutions. The proposed algorithm can be
used to compute a border basis of the input ideal and, as opposed to other
approaches, it can also compute the quotient structure of the (real) radical
ideal directly, i.e., without prior algebraic techniques such as Gr\"obner
bases. It thus combines the strength of existing algorithms and provides a
unified treatment for the computation of border bases for the ideal, the
radical ideal and the real radical ideal.
