On the DML(1) property for regular endomorphisms of affine spaces: multiplicities of points at infinity

Abstract

Let $f$ be a regular endomorphism of ${\mathbb{A}}_{\mathbb{C}}^N$ of algebraic degree $d$ and let $f_{\infty }$ be the induced endomorphism of the infinity hyperplane $H_{\infty }$. Suppose that for every periodic point $x_0\in H_{\infty }(\mathbb{C})$ of period $n_0$, the geometric mean of the multiplicities $e_{f_{\infty }}(x_0),\dots ,e_{f_{\infty }}(f_{\infty }^{n_0-1}(x_0))$ is strictly less than $d$. Then we show that the dynamical Mordell--Lang conjecture for $f$ holds for curves in ${\mathbb{A}}_{\mathbb{C}}^N$.

arXiv comment18 pages