Canonical height functions on the affine plane associated with polynomial automorphisms
Abstract
Description
Let $f: \mathbb{A}^2 \to \mathbb{A}^2$ be a polynomial automorphism of dynamical degree $δ\geq 2$ over a number field $K$. (This is equivalent to say that $f$ is a polynomial automorphism that is not triangularizable.) Then we construct canonical height functions defined on $\mathbb{A}^2(\bar{K})$ associated with $f$. These functions satisfy the Northcott finiteness property, and an $\bar{K}$-valued point on $\mathbb{A}^2(\bar{K})$ is $f$-periodic if and only if its height is zero. As an application of canonical height functions, we give an estimate on the number of points with bounded height in an infinite $f$-orbit.
The proof of (0.2) is simplified. Affine plane polynomial automorphisms except for triangularizable ones are treated
The proof of (0.2) is simplified. Affine plane polynomial automorphisms except for triangularizable ones are treated