Numerical primary decomposition

Abstract

Consider an ideal I ⊂ R = C[x1, . . . , xn] defining a complex affine variety X ⊂ C n . 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.

A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems

“…For example, [19] uses rank conditions on Macaulay matrices to locate embedded components along with other singular sets. In [6], the left null spaces of Macaulay matrices are related to syzygies.…”

A counter example to an ideal membership test

“…It is important to note that the only difference between Theorem 1 and Proposed Theorem 2 (Theorem 4.6 of [4]) is that Theorem 1 uses the whole dual space while Proposed Theorem 2 uses only the dual space of order deg g. That is, since the whole dual space can be infinite-dimensional, this difference turns the proposed ideal membership test of [4] into a finite-dimensional computation, namely Algorithm 4.7 of [4]. Unfortunately, as the example presented in Section 4 demonstrates, Theorem 4.6 and Algorithm 4.7 of [4] are invalid since truncating the dual space at order deg g is not sufficient for determining ideal membership in general.…”

“…Unfortunately, as the example presented in Section 4 demonstrates, Theorem 4.6 and Algorithm 4.7 of [4] are invalid since truncating the dual space at order deg g is not sufficient for determining ideal membership in general.…”

“…The proposed ideal membership test of [4] is based on the classical dual space ideal membership test, described in Section 2, at a generic point on each primary component of the ideal. This proposed test, described in Section 3, states that one may truncate the ideal membership test based on the degree of the polynomial being tested.…”

“…The following theorem, stated in the language of [4], is the classical dual space membership test using a primary decomposition.…”