On linear deformations of Brieskorn singularities of two variables into generic maps

TL;DR

This paper investigates how linear deformations of Brieskorn singularities in two variables produce maps with only indefinite fold and cusp singularities, providing estimates on cusp counts and implications for Morse singularities.

Contribution

It demonstrates that such deformations generically yield only indefinite fold and cusp singularities and establishes that the number of cusps is exactly three for Morse singularities with linear terms.

Findings

Deformed Brieskorn polynomials have only indefinite fold and cusp singularities.

Number of cusps in these deformations is estimated and shown to be three for Morse singularities.

Results contribute to understanding the singularity structure of polynomial deformations.

Abstract

In this paper, we study deformations of Brieskorn polynomials of two variables obtained by adding linear terms consisting of the conjugates of complex variables and prove that the deformed polynomial maps have only indefinite fold and cusp singularities in general. We then estimate the number of cusps appearing in such a deformation. As a corollary, we show that a deformation of a complex Morse singularity with real linear terms has only indefinite folds and cusps in general and the number of cusps is 3.