Numerical primary decomposition

Abstract

Consider an ideal $I \subset R = \bC[x_1,...,x_n]$ defining a complex affine variety $X \subset \bC^n$. We describe the components associated to $I$ by means of {\em numerical primary decomposition} (NPD). The method is based on the construction of {\em deflation ideal} $I^{(d)}$ that defines the {\em deflated variety} $\dXd$ in a complex space of higher dimension. For every embedded component there exists $d$ and an isolated component $\dYd$ of $\dId$ projecting onto $Y$. In turn, $\dYd$ can be discovered by existing methods for prime decomposition, in particular, the {\em numerical irreducible decomposition}, applied to $\dXd$. The concept of NPD gives a full description of the scheme $\Spec(R/I)$ by representing each component with a {\em witness set}. We propose an algorithm to produce a collection of witness sets that contains a NPD and that can be used to solve the {\em ideal membership problem} for $I$.