Some observations on the properness of identity plus linear powers: part 2

TL;DR

This paper investigates the properness of certain polynomial maps related to the Jacobian conjecture, introduces criteria for non-properness, and constructs a counter-example to a recent proof of the conjecture.

Contribution

It develops new criteria for non-properness of polynomial maps and provides a counter-example to a recent proposed proof of the Jacobian conjecture.

Findings

Non-properness set $S_f$ contains 0 under generic conditions.

A counter-example to the proof of the Jacobian conjecture from arXiv:2002.10249.

Insights into the properness of polynomial maps related to the Jacobian conjecture.

Abstract

This paper develops our previous work on properness of a class of maps related to the Jacobian conjecture. The paper has two main parts: - In part 1, we explore properties of the set of non-proper values $S_f$ (as introduced by Z. Jelonek) of these maps. In particular, using a general criterion for non-properness of these maps, we show that under a "generic condition" (to be precise later) $S_f$ contains $0$ if it is non-empty. This result is related to a conjecture in our previous paper. We obtain this by use of a "dual set" to $S_f$, particularly designed for the special class of maps. - In part 2, we use the non-properness criteria obtained in our work to construct a counter-example to the proposed proof in arXiv:2002.10249 of the Jacobian conjecture. In the conclusion, we present some comments pertaining the Jacobian conjecture and properness of polynomial maps in general.