Approximate Radical for Clusters: A Global Approach Using Gaussian Elimination or SVD

Abstract

Abstract. We present a method based on Dickson's lemma to compute the "approximate radical" of a zero dimensional idealĨ 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 ofĨ. To compute the numerical nullspace of the matrix of traces we propose to use Gauss elimination with pivoting or singular value decomposition. We prove that ifĨ has k distinct zero clusters each of radius at most ε in the ∞-norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically equal to ε 2 . We also show that the (k + 1)-th singular value of the matrix of traces is proportional to ε 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 equal to ε 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.

Moment matrices, trace matrices and the radical of ideals

“…The advantage of the latter is that it can be generalized for higher dimensional ideals (see for example [25]). We note here that an advantage of the method using matrices of traces is that it behaves stably under perturbation of the roots of the input system, as was proved in [23]. Other methods to compute the radical of zero dimensional ideals include [24,16,28,29,30,39].…”

“…This paper is a continuation of our previous investigation in [22,23] to compute the approximate radical of a zero dimensional ideal which has zero clusters. The computation of the radical of a zero dimensional ideal is a very important problem in computer algebra since a lot of the algorithms for solving polynomial systems with finitely many solutions need to start with a radical ideal.…”

“…The theoretical basis of the symbolic-numeric algorithm presented in [22,23] was Dickson's lemma [14], which, in the exact case, reduces the problem of computing the radical of a zero dimensional ideal to the computation of the nullspace of the so called matrices of traces (see Definition 14): in [22,23] we studied numerical properties of the matrix of traces when the roots are not multiple roots, but form small clusters. Among other things we showed that the direct computation of the matrix of traces (without the computation of the multiplication matrices) is preferable since the matrix of traces is continuous with respect to root perturbations around multiplicities while multiplication matrices are generally not.…”

“…is an (approximate) multiplication matrix of x k for the (approximate) radical of I. See [23] for the definition of (approximate) multiplication matrices. Note that a generating set for the radical √ I can be obtained directly from the definition of multiplication matrices, in particular, it corresponds to the rows of the matrices M x1 , .…”

On the computation of matrices of traces and radicals of ideals

“…We now recall the definition of the multiplication matrices and the matrix of traces as presented in [26]. The matrix of traces is the N × N symmetric matrix:…”