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

2020/05/05 by Tuyen Trung Truong, Truong, Tuyen Trung
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2005.02260

openalex publication_date 2020/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Some approaches toward the Jacobian conjecture
• An Optimization Approach to Jacobian Conjecture

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