Perturbation of Polynomial Ideals

Abstract

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 complex w x Ž ww xx. numbers. We denote by K Z , . . . , Z resp. K Z , . . . , Z the ring of 1 s 1 s Ž . polynomials resp. the ring of formal power series in s indeterminates w x over K. Let I be an ideal of K Z , . . . , Z . The codimension of I is the 1 s w x dimension of the quotient space K Z , . . . , Z rI over K. If this dimension 1 s is finite, then I is said to be of finite codimension. The perturbation technique for polynomial ideals was used in the study of polynomial mappings on C . Suppose p , . . . , p are homogeneous 1 s w x polynomials in C Z , . . . , Z such that the ideal I generated by them is of 1 s finite codimension. In this case, the origin is the only common zero of p , . . . , p . Let F be the mapping from C s to C s given by 1 s