Silverman's conjecture for additive polynomial mappings

Abstract

Let $F:{\mathrm{End}}_{{\mathbb{F}}_p}({\mathbb{G}}_{a/K}^d)$ be an additive polynomial mapping over a global function field $K/{\mathbb{F}}_q$, and let $P\in {\mathbb{G}}_a^d(K)$. Following Silverman, consider $\delta :={\lim }_{n\in \mathbb{N}}(°F^n{)}^{1/n}$ the dynamic degree of $F$ and $\alpha (P):={\mathrm{limsup}}_{n\in \mathbb{N}}{h}_K(F^{n}P{)}^{1/n}$ the arithmetic degree of $F$ at $P$. We have $\alpha (P)\le \delta$, and extending a conjecture of Silverman from the number field case, it is expected that equality holds if the orbit of $P$ is Zariski-dense. We prove a weaker form of this conjecture: if $\delta >1$ and the orbit of $P$ is Zariski-dense, then also $\alpha (P)>1$. We obtain furthermore a more precise result concerning the growth along the orbit of $P$ of the heights of the individual coordinates, and formulate a few related open problems motivated by our results, including a generalization "with moving targets" of Faltings's theorem back in the number field case.

References in corpus (2)

• Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space
• A lower bound on the orbit growth of a regular self-map of affine space

Cited by in corpus (1)

• Convergence to the Mahler measure and the distribution of periodic points for algebraic Noetherian ${\mathbb{Z}}^d$-actions