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

TL;DR

This paper studies the local behavior of smooth maps from 3-manifolds to 3-space, focusing on swallowtail singularities, and provides a method to compute the count of positive and negative swallowtails for polynomial maps using quadratic form signatures.

Contribution

It introduces a way to compute the number of positive and negative swallowtail points for polynomial maps from R^n to R^n using signatures of quadratic forms.

Findings

Provides a formula for counting swallowtail signs in polynomial maps.

Connects geometric swallowtail signs to algebraic quadratic form signatures.

Enhances understanding of singularity types in smooth mappings.

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.