On Perturbations of Ideal Complements

Let F[x] be the space of polynomials in d variables, let GN be the Grassmannian of N -dimensional subspaces of F[x] and let JN stand for the family of all ideals in F[x] of codimension N . For a given G ∈ GN we let JG := {J ∈ JN : J ∩G = {0}} Is it true, that (with appropriate topology on JN ) the set JG is dense in JN? In general the answer is ”No”. What is even more surprising, that there are ”good ideals” J ∈ JN such that every ”neighborhood” U(J) ⊂ JN has a non-empty intersection with JG for any G ∈ GN and there are ”bad” ideals J ∈ JN (for d ≥ 3) such that some ”neighborhoods” U(J) ⊂ JN have an empty intersection with JG for some G ∈ GN . This contrast illuminates the non-homogeneous nature of JN . 1. Preliminaries Let F be a normed linear space, let G (F ) be the Grassmannian of N -dimensional subspaces of F and let GN (F ) denotes the Grassmannian of all subspaces of F of codimension N . For a given G ∈ GN (F ) let GG(F ) := {J ∈ GN (F ) : J ∩G = {0}} i.e., GG(F ) is a family of all subspaces J in F that are ”missing” G or, equivalently, the family of all subspaces of F that complement G. It is well-known and easy to see that (with appropriate topology on GN (F )) for any G ∈ G (F ), the set GG(F ) is an open and dense subset of GN (F ). The main focus of this article is an investigation of the following ”ideal” version of this statement. Let F = F[x] = [x1, ..., xd] be the space of polynomials in d variables over the field F of real or complex numbers. Let JN stand for the family of all ideals in F[x] of codimension N . Let G := G (F[x]). For a given G ∈ G we let JG := {J ∈ JN : J ∩G = {0}} Question : Is it still true, that JG is dense in JN? In general the answer is ”No”. What is even more surprising, that there are ”good ideals” J ∈ JN such that every ”neighborhood” U(J) ⊂ JN has a non-empty intersection with JG for any G ∈ G and there are ”bad” ideals J ∈ JN (for d ≥ 3) such that some ”neighborhoods” U(J) ⊂ JN have an empty intersection with JG for some G ∈ G . This contrast illuminates the non-homogeneous nature of J (as oppose to GN ). 1991 Mathematics Subject Classification. Primary 46H10,46J05,46J20; Secondary 14C05.