Some arithmetic aspects of polynomial maps

TL;DR

This paper surveys various results related to the Jacobian conjecture, emphasizing an arithmetic perspective that links polynomial automorphisms to counting points over finite fields, highlighting recent progress and open questions.

Contribution

It discusses an arithmetic approach to the Jacobian conjecture, connecting polynomial automorphisms with counting points over finite fields, and reviews recent developments in this area.

Findings

Explores the relationship between the Jacobian conjecture and counting $ ext{F}_p$-points.

Highlights cases where the arithmetic approach provides insights into the conjecture.

Discusses the connection between polynomial automorphisms and affine schemes over finite fields.

Abstract

The Jacobian conjecture is a well-known open problem in affine algebraic geometry that asks if any polynomial endomorphism of the affine space $\mathbb{A}_{\mathbb{C}}^{n}$ ($n\geq2$) with jacobian $1$ is an automorphism. We present a survey about some results around this conjecture and we discuss an arithmetic aspect of this conjecture due to Essen-Lipton. We investigate some cases of this arithmetic approach showing the close relationship between the Jacobian Conjecture and the problem of counting $\mathbb{F}_p$-points of an affine scheme.