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)].


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


Zbl 1376.57033
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