On bifurcations of cusps (Q2317150)

Let \(F_t : (\mathbb{R}^2, 0) \to (\mathbb{R}^2, 0),\) where \(t \in \mathbb{R},\) be an analytic family of mappings with \(F_0\) having a critical point at the origin. Under some natural assumptions there is a finite family of cusp points of \(F_t\) bifurcating from the origin. (We know from [\textit{H. Whitney}, Ann. Math. (2) 62, 374--410 (1955; Zbl 0068.37101)] that critical points of a generic plane-to-plane mapping are folds and cusps.) In the article under review, the author provides effective algebraic methods of computing the number of cusps of \(F_t\) for all \(0 \ne t\) sufficiently close to zero: the number of cusps is expressed in terms of the local topological degree of certain mappings. Some illustrative computational examples are also given in the paper.