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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 $\mathbb P^m (K)$?

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    $\begingroup$ Why do you mean by determine? You can plug in $\alpha=(1:0:\ldots:0)$ compute $h=H(\phi(\alpha))$, find some $h'$ such that if $H(\beta)>h'$, then $H(\phi(\beta))>h$, list all $\beta, H(\beta)\le h'$ and find the one for which $\phi(\beta)$ has minimal height. $\endgroup$ Jul 7, 2014 at 18:33
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    $\begingroup$ That looks as hard as deciding the existence of a rational point on an arbitrary variety $V/K$. Having found one $\phi(\beta)$ of low height, you can effectively find all points of lower height, but the preimage of each one is some variety $V/K$, and finding a rational point on $V$ is a system of Diophantine equations that can be as intractable as you wish. (It must be possible to rig things so that there's only one plausible candidate of height lower than $\phi(\beta)$ and this candidate yields your favorite $V$.) $\endgroup$ Jul 7, 2014 at 18:43
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    $\begingroup$ @TonyShaska: Since $\phi$ is a morphism of degree $d$, you have $H(\phi(\beta)) > c H(\beta)^d$ for some constant $c > 0$ independent of $\beta$ and explicitly computable from the coefficients of the polynomials $p_i$. (This amounts to an application of effective nullstellensatz). You are then guaranteed that $H(\phi(\beta)) > h$ as soon as $H(\beta) > c^{-1/d}h^{1/d}$, giving your $h'$. $\endgroup$ Jul 7, 2014 at 20:36
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    $\begingroup$ @TonyShaska You probably want to specify that $p_0,\ldots,p_m$ have no common roots in $\mathbb{P}^m(\overline{K})$ to get the (effective) estimate $H(\phi(\beta)) \ge c(\phi)H(\beta)^d$ mentioned in Vesselin's answer. $\endgroup$ Jul 7, 2014 at 20:42
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    $\begingroup$ @TonyShaska: Since you wrote $\forall \alpha \in \mathbb{P}^n$, I assumed your $\phi$ was a morphism rather than a rational map; that is, defined on all of $\mathbb{P}^n$ rather than on a Zarisiki-dense open set. In that case note that $m \geq n$, if $\phi$ is non-constant; a non-constant morphism from a projective space is finite, and N. D. Elkies' remark does not apply since then $V$ is necessarily finite. Indeed, in that case you have the estimate I mention. If on the other hand $\phi$ is just a rational function $\mathbb{P}^n \dashrightarrow \mathbb{P}^1$, then Elkies' remark is in force. $\endgroup$ Jul 8, 2014 at 5:06

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