Criteria for singularities for mappings from two-manifold to the plane. The number and signs of cusps

Abstract

Let $M \subset \mathbf{R}^{n+2}$ be a two-dimensional complete intersection. We show how to check whether a mapping $f : M \rightarrow \mathbf{R}^2$ is 1-generic with only folds and cusps as singularities. In this case we give an effective method to count the number of positive and negative cusps of a polynomial $f$, using the signatures of some quadratic forms.