Numerical primary decomposition
Anton Leykin · 2008
Consider an ideal I ⊂ R = C[x1,...,xn] defining a complex affine variety X ⊂ Cn. We describe the components associated to I by means of numerical primary decomposition (NPD). The method is based on the construction of deflation ideal I(d) that defines the deflated variety X(d) in a complex space of higher dimension. For every embedded component there exists d and an isolated component Y(d) of I(d) projecting onto Y. In turn, Y(d) can be discovered by existing methods for prime decomposition, in particular, the numerical irreducible decomposition, applied to X(d). The concept of NPD gives a full description of the scheme Spec(R/I) by representing each component with a 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 ideal membership problem for I.