Characterization of hypersurface singularities in positive characteristic

Abstract

The classification of right unimodal and bimodal hypersurface singularities over a field of positive characteristic was given by H. D. Nguyen. The classification is described in the style of Arnold and not in an algorithmic way. This classification was characterized by M. A. Binyamin et al. [Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 61(109) (2018), no. 3, 333–343] for the case when the corank of hypersurface singularities is $\leq 2$. The aim of this article is to characterize the right unimodal and bimodal hypersurface singularities of corank $3$ in an algorithmic way by means of easily computable invariants such as the multiplicity, the Milnor number of the given equation, and its blowing-up. On the basis of this characterization we implement an algorithm to compute the type of the right unimodal and bimodal hypersurface singularities without computing the normal form in the computer algebra system $Singular$.