Approximate radical for clusters: a global approach using Gaussian elimination or SVD

Abstract

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 Gaussian elimination with pivoting or singular value decomposition. We prove that if I has k distinct zero clusters each of radius at most epsilon in the max-norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically bounded by epsilon^2. We also show that the (k+1)-th singular value of the matrix of traces is O(epsilon^2). The resulting approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically bounded by epsilon^2. 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.