Bifurcations of polynomial functions with diffeomorphic fibers

Abstract

The phenomena that cause a value of a polynomial function to be a bifurcation one are yet to be described when the fibers have dimension higher than $1$. In this note, the main result is the construction of a polynomial submersion function of $\mathbb{R}^3$ with connected fibers having a bifurcation value such that close enough to it the fibers are mutually diffeomorphic. We also present an example of a polynomial submersion function of $\mathbb{R}^2$ having a bifurcation value such that close enough to it the fibers are mutually diffeomorphic.