Standard bases with respect to the Newton filtration

Abstract

The aim of this article is to introduce standard bases of ideals in polynomial rings with respect to a class of orderings which are not necessarily semigroup orderings. Our approach generalises the concept of standard bases with respect to semigroup orderings described by Graebe and Greuel/Pfister. To compute these standard bases we give a slightly modified version of the Buchberger algorithm. The orderings we consider are refinements of certain filtrations. In the local case these filtrations are Newton filtrations. For a zero dimensional ideal, an algorithm converting standard bases with respect to local orderings is given. As an application, we show how to compute the spectrum of an isolated complex hypersurface singularity $f:(C^n,0)\to(C,0)$ with nondegenerate principal part.