Local theory of singularities of three functions and the product maps

Abstract

Suppose that a smooth map $(f,g,h):\mathbb{R}^n \rightarrow \mathbb{R}^3$, where $n \geq3$, has a stable singularity at the origin. We characterize the stability of the function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ and the map $(f,g):\mathbb{R}^n \rightarrow \mathbb{R}^2$ at the origin in terms of the discriminant set of $(f,g,h)$.