The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

TL;DR

This paper proves that for certain polynomial maps of the affine plane, if no nonconstant rational function remains invariant, then there exists a point with a Zariski dense orbit, confirming a conjecture in the field.

Contribution

It establishes the existence of Zariski dense orbits for polynomial endomorphisms under specific conditions, confirming a conjecture for affine plane maps.

Findings

Existence of points with Zariski dense orbits under certain polynomial maps.

Validation of a conjecture by Amerik, Bogomolov, Rovinsky, and Zhang.

Conditions for dense orbits related to invariant rational functions.

Abstract

In this paper we prove the following theorem. Let $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$ be a dominate polynomial endomorphisms defined over an algebraically closed field $k$ of characteristic $0$. If there are no nonconstant rational function $g:\mathbb{A}^2-rightarrow \mathbb{P}^1$ satisfying $g\circ f=g$, then there exists a point $p\in \mathbb{A}^2(k)$ whose orbit under $f$ is Zariski dense in $\mathbb{A}^2$. This result gives us a positive answer to a conjecture of Amerik, Bogomolov and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.