On the number of cusps of perturbations of complex polynomials

Abstract

Let f be a 1-variable complex polynomial such that f has an isolated singularity at the origin. In the present paper, we show that there exists a perturbation f_t of f which has only fold singularities and cusps as singularities of a real polynomial map from R² to R². We then calculate the number of cusps of f_t in a sufficiently small neighborhood of the origin and estimate the number of cusps of f_t in R².