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.