Let $K$ be an algebraically number field and $$\phi : \mathbb P^n (K) \to \mathbb P^m (K)$$ a polynomial map, such that $\forall \alpha \in \mathbb P^n$, where $\alpha = [\alpha_0, \dots , \alpha_n]$, $$\phi (\alpha) = \left[ p_0 (\alpha_0, \dots , \alpha_n), \dots , p_m(\alpha_0, \dots , \alpha_n) \right],$$ and $p_0, \dots , p_m$ are homogenous polynomials of degree $d$.
Let $P \in \mathbb P^n (K)$ with coordinates $P=[ x_0, \dots , x_n]$, $x_i \in K$. The height of $P$ is defined as $$H_K (P) = \prod_{\nu \in M_K} \mbox{max } \left\{ |x_0|_\nu^{n_\nu} , \dots , |x_n|_\nu^{n_\nu} \right\}, $$ where $M_K$ is the set of absolute values of $K$ and $n_\nu = [K_\nu : \mathbb Q_\nu ]$ for completions $K_\nu$, $\mathbb Q_\nu$ of $K$ and $\mathbb Q_\nu$ respectively.
Question: Is there any way to determine a point $\beta \in \mathbb P^n (K)$ such that $\phi (\beta)$ has minimal height in $\mathbb P^m (K)$?