  Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ be a system of polynomials
generating a zero-dimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary
algebraically closed field. We study the computation of "matrices of traces"
for the factor algebra $\A := \CC[x_1, ..., x_m]/ \I$, i.e. matrices with
entries which are trace functions of the roots of $\I$. Such matrices of traces
in turn allow us to compute a system of multiplication matrices
$\{M_{x_i}|i=1,...,m\}$ of the radical $\sqrt{\I}$. We first propose a method
using Macaulay type resultant matrices of $f_1,...,f_s$ and a polynomial $J$ to
compute moment matrices, and in particular matrices of traces for $\A$. Here
$J$ is a polynomial generalizing the Jacobian. We prove bounds on the degrees
needed for the Macaulay matrix in the case when $\I$ has finitely many
projective roots in $\mathbb{P}^m_\CC$. We also extend previous results which
work only for the case where $\A$ is Gorenstein to the non-Gorenstein case. The
second proposed method uses Bezoutian matrices to compute matrices of traces of
$\A$. Here we need the assumption that $s=m$ and $f_1,...,f_m$ define an affine
complete intersection. This second method also works if we have higher
dimensional components at infinity. A new explicit description of the
generators of $\sqrt{\I}$ are given in terms of Bezoutians.
