Explicit Canonical Heights for Divisors Relative to Endomorphisms of \(\mathbb {P}^N\)

Given an endomorphism f of \(\mathbb {P}^N\) ${\mathbb{P}}^N$ and a divisor D on \(\mathbb {P}^N\) ${\mathbb{P}}^N$ , both defined over \(\overline{\mathbb {Q}}\) $\overline{\mathbb{Q}}$ , we exhibit explicit bounds on the difference between the naive height of D and the canonical height relative to f.