Computing Approximate Radicals using Bezoutians

Consider I ⊂ C[x1, . . . , xm], a zero dimensional complete intersection ideal, with I = (f1, . . . , fm). Assume that I has clusters of roots, each cluster of radius at most ε in the ∞-norm. We compute the “approximate radical” of I, i.e. an ideal which contains one root for each cluster, corresponding to the center of gravity of the points in the cluster, up to an error term asymptotically bounded by ε. The main bottleneck of our method is the computation of the “matrix of traces”, a matrix with entries which are trace functions of the roots of I with respect to some basis of C[x1, . . . , xm]/I. We investigate how to compute a matrix of traces from the Bezoutian of f1, . . . , fm and their Jacobian J . We use work by Mourrain and Pan to reduce the Bezout matrix to a matrix of traces with respect to a certain canonical basis of C[x1, . . . , xm]/I.