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

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)].

### 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

Zbl 1376.57033
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