Abstract: Let $f : \mathbb{P}^m \to \mathbb{P}^m$ be a morphism of degree $d \ge 2$ defined over a number field $K$ of degree $n$. Let $h$ denote the standard height on $\mathbb{P}^m(K)$, and let $\hat{h}_f$ denote the canonical height attached to $f$. Given a real number $x$, we wish to determine (1) the number of points $P$ in $f(\mathbb{P}^m(K))$ such that $h(f(P)) < x$, and (2) the number of points $P$ in $\mathbb{P}^m(K)$ such that $\hat{h}_f(P) < x$. It turns out that the answer to both questions depends on the quantity $N(x) := \{P \in \mathbb{P}^m(K) : h(f(P)) < x\}$. In this talk, we will illustrate the connection between (1), (2), and $N$, and then sketch a proof that $N(x)$ is asymptotic to a constant $c_K(f)$ times $e^{n(m+1)x/d}$, giving a decomposition of $c_K(f)$ as a product of local factors over all places of bad reduction of $f$.
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