Normal forms of hypersurface singularities in positive characteristic

The main purpose of this article is to lay the foundations for a classification of isolated hypersurface singularities in positive characteristic. Although our article is in the spirit of Arnol'd who classified real and complex hypersurfaces in the 1970's with respect to right equivalence, several new phenomena occur in positive characteristic. Already the notion of isolated singularity is different for right resp. contact equivalence over fields of characteristic other than zero. The heart of this paper consists of the study of different notions of non-degeneracy and the associated piecewise filtrations induced by the Newton diagram of a power series $f$. We introduce the conditions AC and AAC which modify and generalise the conditions A and AA of Arnol'd resp. Wall and which allow the classification with respect to contact equivalence in any characteristic. Using this we deduce normal forms and rather sharp determinacy bounds for $f$ with respect to right and contact equivalence. We apply this to classify hypersurface singularities of low modality in positive characteristic.