Characterization of hypersurface singularities in positive characteristic

Amir Shehzad,Muhammad Ahsan Binyamin,Hasan Mahmood

doi:10.33044/revuma.v61n2a17

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 ≤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.

Highlights

Let K[[x1, . . . , xn]] be the local ring of formal power series in n variables, m its maximal ideal, K an algebraically closed field of characteristic p > 0, and R = AutK (K[[x1, . . . , xn]]), the set of all K-automorphisms of K[[x1, . . . , xn]]
Let f and g ∈ m. f is said to be right equivalent to g, f ∼r g, if there exists an automorphism φ ∈ R such that φ(f ) = g
Arnold introduced in the seventies [2, 3, 4] the notion of modality in singularity theory for real and complex singularities

Summary

Introduction

Arnold introduced in the seventies [2, 3, 4] the notion of modality in singularity theory for real and complex singularities He classified simple, unimodal, and bimodal hypersurface singularities with respect to right equivalence. Unimodal, and bimodal hypersurface singularities with respect to right equivalence These are classifications with respect to contact equivalence. Greuel and Nguyen [8] classified the simple hypersurface singularities in characteristic p > 0 with respect to right equivalence. These classifications are characterized in [1]. Nguyen gave the classification of right unimodal and bimodal hypersurface singularities in positive characteristic [10].

Characterization of hypersurface singularities of corank 3

Short description of the classifier