Bifurcations of polynomial functions with diffeomorphic fibers

TL;DR

This paper constructs examples of polynomial functions with higher-dimensional fibers that exhibit bifurcation phenomena, expanding understanding of how polynomial fibers change topology near bifurcation points.

Contribution

It provides explicit constructions of polynomial submersion functions with connected fibers in dimensions 2 and 3 that demonstrate bifurcation behavior.

Findings

Constructed a polynomial submersion in $\\mathbb{R}^3$ with a bifurcation value and diffeomorphic fibers nearby.

Presented an example in $\\mathbb{R}^2$ with similar bifurcation properties.

Showed that bifurcation phenomena can occur even when fibers are mutually diffeomorphic near the bifurcation point.

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.