Journal of the
Korean Mathematical Society JKMS

Silverman [14] proved a height inequality for a jointly regular family of rational maps and the author [10] improved it for a jointly regular pair. In this paper, we provide the same improvement for a jointly regular family: let $h:\mathbb P^n_{\mathbb{Q}} \rightarrow \mathbb{R}$ be the logarithmic absolute height on the projective space, let $r(f)$ be the $D$-ratio of a rational map $f$ which is defined in [10] and let $\{f_1, \ldots, f_k ~|~ f_l : \mathbb A^n \rightarrow \mathbb A^n \}$ be a finite set of polynomial maps which is defined over a number field $K$. If the intersection of the indeterminacy loci of $f_1, \ldots, f_k$ is empty, then there is a constant $C$ such that \[ \sum_{l=1}^k \dfrac{1}{\deg f_l} h\bigl(f_l(P) \bigr) > \left( 1+ \dfrac{1}{r} \right) f(P) - C \quad \text{for all}~P\in \mathbb A^n \] where $r = \max_{l=1, \ldots, k} \left( r(f_l) \right)$.

Keywords: height, rational map, preperiodic points, jointly regular family

MSC numbers: Primary 37P30; Secondary 11G50, 32H50, 37P05