= Solution
For $P=[x_0:\cdots:x_N]\in\mathbb P^N(\mathbb Q)$, choose coprime integer coordinates and define the <projective height>
$$
H(P)=\max_i|x_i|.
$$
Write the morphism $F:\mathbb P^1\to\mathbb P^1$ as $F=[F_0:F_1]$, where $F_0,F_1\in\mathbb Z[X,Y]$ are homogeneous of degree $d$ with no common zero. Bounding each polynomial by the sum of the absolute values of its coefficients gives
$$
H(F(P))\leq c_2H(P)^d.
$$
Because $F_0$ and $F_1$ have no common projective zero, the <Projective Nullstellensatz> gives an integer $m\geq d$ and homogeneous polynomials $A_{ij}$ such that suitable nonzero integer multiples of $X^m$ and $Y^m$ lie in the ideal $(F_0,F_1)$. Evaluating at primitive coordinates and using the same coefficient bound gives
$$
H(P)^m\leq C H(P)^{m-d}H(F(P)),
$$
after absorbing the bounded common divisor of $F_0(x,y)$ and $F_1(x,y)$ into $C$. Therefore
$$
\boxed{c_1H(P)^d\leq H(F(P))\leq c_2H(P)^d.}
$$
This is <height growth under a morphism of the projective line>.
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