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

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.

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.

文献摘要