Ja n 20 08 NUMERICAL PRIMARY DECOMPOSITION

Consider an ideal I ⊂ R = C[x] = C[x 1 ,. .. , x n ] defining a complex affine variety X = V (I) = {x ∈ C n | ∀f ∈ I, f (x) = 0}. We describe the associated components VAss(I) = {V (P) | P ∈ Ass(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) = V (I (d)) in a higher-dimensional complex space. For every embedded component in VAss(I), there exists d such that VAss(I (d)) contains an isolated component Y (d) projecting onto Y. In turn, Y (d) can be discovered by the existing methods for numerical irreducible decomposition of 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.