On Topological Approaches to the Jacobian Conjecture in $\mathbb{C}^n$

TL;DR

This paper explores topological methods and structural properties of the nonproperness set of polynomial mappings in complex n-space, providing insights into the Jacobian conjecture and conditions for potential counterexamples.

Contribution

It establishes a structure theorem for the nonproperness set and links topological properties to the Jacobian conjecture in complex n-space.

Findings

If a counterexample exists, the nonproperness set is a hypersurface intersecting every complex hyperplane.

The nonproperness set's structure relates to polynomial submersions and hypersurfaces.

Topological approaches offer new perspectives on the Jacobian conjecture in higher dimensions.

Abstract

We obtain a structure theorem for the nonproperness set $S_f$ of a nonsingular polynomial mapping $f:\mathbb{C}^n \to \mathbb{C}^n$. Jelonek's results on $S_f$ and our result show that if $f$ is a counterexample to the Jacobian conjecture, then $S_f$ is a hypersurface such that $S_f\cap Z \neq \emptyset$, for any $Z\subset \mathbb{C}^n$ biregular to $\mathbb{C}^{n-1}$ and $Z = h^{-1}(0)$ for a polynomial submersion $h: \mathbb{C}^n \to \mathbb{C}$. Also, we present topological approaches to the Jacobian conjecture in $\mathbb{C}^n$.