Some arithmetic aspects of polynomial maps

Abstract

The Jacobian conjecture is a well-known open problem in affine algebraic geometry that asks if any polynomial endomorphism of the affine space ${\mathbb{A}}_{\mathbb{C}}^n$ ($n\ge 2$) 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.

19 pages; rewrite of the first version, references added, section 5 added; comments welcome