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 $d\ge2$ defined over $\overline{\mathbb{Q}}$ or $\overline{\mathbb{Q}}_p$, 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 $d$.