In this paper we explore the interrelationship between the theory of polynomial ideals and certain branches of analysis including multivariate approximation theory and linear partial differential equations. Specifically, we use the perturbation technique from analysis to study the codimension of a multivariate polynomial ideal. Let K be a field. In this paper, K will often be the field C of com...

Abstract. Let K be a field and S = K[x1, . . . , xn] be a polynomial ring. A single spot ideal I ⊂ S is a graded ideal whose local cohomology H m (S/I), i < dimS/I and m = (x1, . . . , xn), only has non-trivial value N , a finite length module, at i = depthS/I. We consider characterization of single spot ideals in terms of (long) Bourbaki sequences. The codimension 2 case has been fairly well i...

In this paper we develope a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known Morsification results for nonisolated singulatities and generalize them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects t...

This paper investigates normal surface theory in topologically finite 3–manifolds with ideal triangulations. Treated are closed and non– compact normal surfaces, the projective solution space and the projective admissible solution space, as well as the leaf spaces of the transversely measured singular codimension–one foliations defined by admissible solutions. AMS Classification 57M25, 57N10

We present examples which show that in dimension higher than one or codimension higher than two, there exist toric ideals IA such that no binomial ideal contained in IA and of the same dimension is a complete intersection. This result has important implications in sparse elimination theory and in the study of the Horn system of partial differential equations.

In this paper we extend the properties of ordinary points of curves [10] to ordinary closed points of one-dimensional affine reduced schemes and then to ordinary subvarieties of codimension one.