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International audience; Let $f_1,\ldots,f_s \in \mathbb{K}[x_1,\ldots,x_m]$ 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 $\A := \CC[x_1, \ldots , x_m]/ \I$, 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 $\{M_{x_i}|i=1,\ldots,m\}$ of the radical $\sqrt{\I}$. We first propose a method using Macaulay type resultant matrices of $f_1,\ldots,f_s$ 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 $f_1,\ldots,f_m$ 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.

We present a method based on Dickson's lemma to compute the "approximate radical" of a zero dimensional ideal I in C[x1, . . . , xm] which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is "global" in the sense that it does not require any local approximation of the zero clusters: it reduces the problem to the computation of the numerical nullspace of the so called "matrix of traces", a matrix computable from the generating polynomials of I. To compute the numerical nullspace of the matrix of traces we propose to use Gauss elimination with pivoting, and we prove that if I has k distinct zero clusters each of radius at most e in the ∞-norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically equal to e2. We also prove that the computed approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically equal to e2. In the univariate case our method gives an alternative to known approximate square-free factorization algorithms which is simpler and its accuracy is better understood.