A Uniform Field-of-Definition/Field-of-Moduli Bound for Dynamical Systems on ${\mathbf{P}}^N$

Let $f:{\mathbb{P}}^N\to {\mathbb{P}}^N$ be an endomorphism of degree $d\ge 2$ defined over $\bar{\mathbb{Q}}$ or ${\bar{\mathbb{Q}}}_p$, and let $K$ be the field of moduli of $f$. We prove that there is a field of definition $L$ for $f$ whose degree $[L:K]$ is bounded solely in terms of $N$ and $d$.