On the number of cusps of perturbations of complex polynomials. (English) Zbl 1445.57023

Inaba, Kazumasa

Kodai Math. J. 42, No. 3, 593-610 (2019).

The main result of the paper is the following : Let \(f(z)\) be a complex polynomial and let \(k\geq 2\) be the multiplicity of \(f\) at the origin. If a linear perturbation \(f_t\) of \(f\) is an excellent map for \(0<|t|\ll 1\), then the number of cusps of \(f_t|_U\) is equal to \(k+1\), where \(U\) is a sufficiently small neighborhood of the origin (Theorem 1). An interesting estimation of the number of cusps of \(f_t\) in terms of the degree of \(f\) is given in Corollary 1. Another paper by the author directly connected to this topic is [K. Inaba et al., Tohoku Math. J. (2) 69, No. 1, 85–111 (2017; Zbl 1376.57033)].

Reviewer: Dorin Andrica (Riyadh)

MSC:

57R45	Singularities of differentiable mappings in differential topology
58K05	Critical points of functions and mappings on manifolds
58K60	Deformation of singularities

Keywords:

excellent map; cusp; complex polynomial

Citations:

Zbl 1376.57033