Counting signed swallowtails of polynomial selfmaps of {\mathbb {R}^3}

For a generic \({f \in C^\infty({\mathbb {R}}^3,{\mathbb {R}}^3)}\), there is a discrete set of swallowtail critical points. At any swallowtail point p there exists a well-oriented coordinate system centred at p, and a coordinate system centred at f(p), such that locally f has the form \({f_\pm(x, y, z) = (\pm xy+x^2 z+x^4, y, z)}\), so one may associate with p a sign \({I(f, p) \in \{\pm 1\}}\). We shall show how to compute the number of swallowtail points having the positive/negative sign, in the case where \({f : {\mathbb {R}}^3 \rightarrow {\mathbb {R}}^3}\) is a polynomial mapping, in terms of signatures of quadratic forms.