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

TL;DR

This paper provides a method to determine if a mapping from a 2D complete intersection to the plane has only folds and cusps as singularities, and offers an effective way to count cusps with signs using quadratic form signatures.

Contribution

It introduces criteria to verify 1-genericity and a novel technique to count positive and negative cusps via quadratic form signatures for polynomial mappings.

Findings

Criteria for 1-generic mappings with only folds and cusps

An effective method to count cusps using quadratic forms

Application to polynomial mappings from 2D manifolds

Abstract

Let M be a two--dimensional complete intersection. We show how to check whether a mapping f: M-->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.