Mappings from R^3 to R^3 and signs of swallowtails

Abstract

Let M be an oriented 3-manifold. For a generic f \in C^ \infty(M,R^3), there is a discrete set of swallowtail critical points. In that case, at any swallowtail point p there exists a well-oriented coordinate system centered at p, and a coordinate system centered 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\}. A geometric definition of the sign associated with a swallowtail was recently introduced by Goryunov. We shall show how to compute the number of swallowtail points having the positive/negative sign, in the case where f : R^n \rightarrow R^n is a polynomial mapping, in terms of signatures of quadratic forms.