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

In this paper we prove the following theorem. Let $f$ be a dominant polynomial endomorphism of the affine plane defined over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then there exists a closed point in the plane whose orbit under $f$ is Zariski dense. This result gives us a positive answer to a conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky, and by Zhang for polynomial endomorphisms of the affine plane.

Complex Points of Two-Dimensional Surfaces

We deal with complex points of two-dimensional surfaces. A short review of basic results about complex points of smooth surfaces in is presented at the beginning. For algebraic surfaces, a formula is proved which expresses the number of complex points as the local degree of an explicitly constructible polynomial endomorphism.